dr. krzysztof ostaszewski, fsa, cfa, maaa actuarial...

Dr. Krzysztof Ostaszewski, FSA, CFA, MAAA

Actuarial Program Director and Professor of Mathematics Illinois State University, Normal, IL 61790-4520, U.S.A.

Tel. 1-309-438-7226 Fax: 1-309-438-5866

http://www.math.ilstu.edu/krzysio E-mail: [email protected]

Valuation and Pricing of Bank and Insurance Products

Copyright © 2006-2007 by Krzysztof M. Ostaszewski. All rights reserved.

Problems from the past actuarial examinations are copyrighted by the Society of Actuaries and/or Casualty Actuarial Society and are used here with permission.

Copyright © 2006-2007 by Krzysztof M. Ostaszewski - 1 -

1. Financial Assets A real asset is defined as an asset that produces income, or is utilized in production of income. A factory is a real asset, machinery in a factory is a real asset, computer is a real asset. Precious metals are real assets. It is a common misconception that real assets must be tangible. They need not be. For example, a patent is a real asset. Human capital (ability to earn income) is a real asset. A financial asset (also called a security, or a capital asset) is defined as a claim on the income produced by real assets. In other words, financial assets direct the distribution of income produced in the economy. For example, you may use your human capital to get a paycheck, but then pay a portion of it to your mortgage, that mortgage is a financial asset of its owner, giving that owner the right to a portion of the income produced by your human capital. There are two main types of financial assets:

- Bonds (or loans): assets that specify in advance the amount of income that will be forwarded to the financial asset holder.

- Stocks (or shares): assets that allow the asset holder to share in both good and bad fortunes of income producer, but do not specify in advance what the amount of income forwarded to the financial asset holder will be.

Technically, government’s rights to incomes of its citizens, i.e., taxes, are also financial assets, but governments rarely present balance sheets, instead producing only income statements (and those on cash basis, as opposed to accrual basis used by businesses). While gold, silver, or other precious metals, are treated as real assets, because they are used in the process of producing income, money is treated as a financial asset, as it represents a claim to consumption of goods and services, and income really means ability to obtain goods and services. You might want to ask what money is. Economists usually do not pose this question directly, as it is probably too philosophical for most of them, and their philosophical approaches to money differ. This author believes (and believes that his belief is derived directly from Adam Smith’s An Inquiry Into The Origin of the Wealth of Nations) that: money is an objective measure of value of human labor derived from continuously negotiated subjective evaluations of such value by all interested human parties. Most excellent classical economist, David Ricardo, is known as the creator of the labor theory of value, which proclaimed that value of a good or a service is the accumulated amount of labor used in its production. The labor theory of value was later adopted by Marx and by Marxists. Subsequently, this labor theory of value was sharply criticized by the Austrian School of Economics, notably Ludwig von Mises, who pointed out that value exists only subjectively: as no matter how hard you work on something, if there is no willing buyer for your product or service, its value is zero. The definition given here attempts to point out that, as in all things, Mises was right and Marx was wrong, but David Ricardo was not all that wrong, because while the evaluation process is subjective, it is not one-sided: it is a process of negotiation between the buyer and the seller, and the value established in the negotiation is in fact the value of the labor input in a good or a service traded. However, in this work, we are not interested in money, but in financial assets.


The two main types of financial assets: bonds and stocks, are not completely mutually exclusive. If you give a loan to a business, you are promised specific payments of interest and a repayment of principal. If however, the borrower business is unable to make the promised payments and becomes bankrupt, you, the lender, are likely to end up with a share of the business, instead of the promised payments of interest and principal, thus becoming a shareholder of the business, albeit inadvertently. Thus your bond (loan) is partially a stock, and the riskier the bond is, the more it becomes like a stock in the borrowing company. There is also a third group of financial assets: derivative securities. A derivative security has its cash flows derived from cash flows of other securities. It does not relate directly to income produced in economic activity, but rather, it relates to it indirectly, through other securities. Assets issued by financial intermediaries: banks, insurance companies, investment companies, etc. are derivatives. A bank deposit is a derivative security. But the most important derivatives are those that are simple, building blocks for more complicated securities, and they include: forwards, options, futures, swaps. Those “building blocks” derivatives are commonly embedded in derivatives issued by financial intermediaries, and even securities issued by businesses (e.g., bonds are commonly issued with a right to pay off the loan early, which is a form of a call option). Derivatives are used for a variety of purposes: risk management, speculation, as an alternative to investment in primary securities (especially if investment in derivatives reduces costs), or to address/manage regulatory requirements. Financial engineering is the science of design and pricing of financial assets out of other financial assets. Financial assets can be purchased in individual transactions or in financial markets (or capital markets), i.e., markets created specifically for financial assets. Examples of financial markets are: New York Stock Exchange (located at Wall Street in New York City), or Giełda Papierów Wartościowych in Warsaw, Poland. The risk of investing in financial assets is divided into diversifiable risk and non-diversifiable risk (also known as market risk). Diversifiable risk is called so because it can be reduced or managed by combining securities in a larger portfolio (this process is called diversification). The main purpose of existence of financial markets is to create a place for sharing of non-diversifiable risk within the society. Such risk can be either avoided (thus reducing economic activity, and reducing society’s overall welfare), accepted by the government (resulting in government ownership of means of production and subsequent risks of political influences on the economic process, with likely reduction in economic output and threat to domestic tranquility), or traded in the markets (resulting in unequal distribution of risks, and likely accumulation of wealth to those bearing higher risks, i.e., unequal distribution of risk causing unequal distribution of wealth). Financial assets can also be purchased from and sold to dealers, who hold inventory of them, instead of trading in an exchange. This is called the over-the-counter market. The price at which a financial asset is offered for sale in the market (or by a dealer, so by any market-maker in general) is called the ask price, while the purchase price offered (by


a market-maker) for it is called the bid price. The difference between the two is called the bid-ask spread. Exercise 1.1 Mr. Romuald Carcosheek has proposed Ms. Berlin Marriott an unusual real estate transaction. Mr. Carcosheek believes that the prices of real estate in New York City will decline substantially over the next year, and he has asked Ms. Marriott to let him borrow a hotel in New York City she owns, sell it short, and repurchase it in the future, returning it to Ms. Marriott. Ms. Marriott requires a 10% haircut (deposit) on the transaction, and she will hold it, together with the proceeds of the short sale, in trust, earning 5% on the funds, and reinvesting interest. They agree to proceed with the transaction. Mr. Carcosheek sells the hotel for $25 million today, and has to pay a 3% commission on the transaction out of his own pocket, because the proceeds of the sale will be held in trust. In three years, Mr. Carcosheek buys the hotel back for $20 million, with the commission paid by the seller. Calculate the difference between the annual rate of return earned by Mr. Carcosheek over those three years, and the annual rate of return earned by Ms. Marriott (including her unrealized gain/loss on the market value of the hotel). Solution. Mr. Carcosheek has to put down $2.5 million for the haircut. He sells the hotel for $25 million and has to pay 0.03 !25,000,000 = 750,000 in commission. Thus his total cash outlay is $3.25 million. In three years, he buys the hotel back realizing a profit on the transaction of $5 million. Moreover, he only gets $2.5 million, the haircut, back, but the sales commission is not returned. Thus his annual rate of return is

7.5

3.25

!"#

$%&

1

3

'1 ( 32.1476%.

Ms. Marriott owns a $25 million building. After the short sale, she receives $25 million plus $2.5 million haircut, for a total of $27.5 million. In three years, this will accumulate to $27.5 !1.053 " $31.83 million. She will return $27.5 million of that, and keep $4.33 million. But her $25 million building is now a $20 million building, so she has an unrealized loss of $5 million, for a net loss of $0.67 million on a $25 million capital over three years and the annual rate of return of

25 ! 0.6725

"#$

%&'

1

3

!1 ( !0.9014%.

The difference is 32.1476% ! !0.9014( )% = 33.0490%. Valuation of Stocks We generally assume that the time horizon for stocks (also called shares, or equity) is infinite. Stocks represent fractional ownership in limited liability corporations. Cash flows generated from holding stocks are uncertain: dividends may or may not be paid out, and dividends amounts may vary because they depend on a company’s growth, profitability and investment opportunities, as well as behavior of management, taxes, and regulatory environment. If you are an investor in stocks, you must decide whether you want to receive income from owning the stock or whether you want to capitalize on


company’s growth. In the first case, you prefer companies that pay their profits out in dividends, while in the second case you prefer companies that reinvest profits in their operations. Companies that pay stable dividends usually try to keep their dividend level or increasing, within a certain percentage range of their earnings per share (EPS). For example, a company has a policy of paying 40% of its earnings in dividends, then dividends per share are 40% of EPS. In the case of such policy, dividends tend to grow at the same rate as earnings. Some corporations use up their retained earnings before paying dividends. This is called a residual dividend policy. Companies that follow a residual dividend policy only pay dividends if they have some leftover income that they do not need to invest in projects that will result in growth of their business. If a dividend paid at time t is denoted by D

t and the amount of dividend is known with

certainty, the price P0 of stock at time 0 is

P0=

D1

1+ k+

D2

1+ k( )2+

D3

1+ k( )3+…,

where k is the applicable discount rate. If dividend grows at a constant rate g, then

P0=

D1

k ! g.

The model giving this last formula is called the Dividend Discount Model, and the formula is also called the dividend discount formula. Exercise 1.2 The cash flows of the stock of Universe Inc. are discounted by the rate k = 10%, and pays a dividend that grows at a rate of g = 6%. If the dividend paid at the time t = 0 is $1, what is the price of the stock based on the Dividend Discount Model? Solution. Since the dividend is $1 now, it will be $1.06 at the end of the year. Therefore,

P0=

D1

k ! g=

$1.06

0.10 ! 0.06=$1.06

0.04= $26.50.

Exercise 1.3 Assume current market price is $23, the stock will pay a dividend of $1.242 one year from now, and it will grow at a constant rate of 8% in the future. What is the rate of return k expected on this stock? Solution. We solve the dividend discount formula for k and obtain

k =D1

P0

+ g =1.242

23+ 8% = 5.4% + 8% = 13.4%.


The ratio D1P0

is called the dividend yield of the stock. If a stock’s dividend does not

increase, or decrease, then a stock is valued as a perpetuity. This can be handled also by simply assuming that g is 0% in the Dividend Discount Model. Price-Earnings Ratio The price-earnings ratio of a company is defined as the ratio of the price per share to the earnings per share (the earnings are generally profits expected at the end of the year). The P/E ratio is generally used for relative valuation of firms, i.e. to compare financial performance among firms. High P/E ratio means higher value for the same amount of profits. Some research shows that investors can be interested in firms that have a lower P/E ratio based on the belief that these companies might be under-priced by the market. Generally, firms that focus more on growth and re-invest more of their earnings tend to have a higher P/E ratio than firms that don’t reinvest as much. Firms that have a higher Return on Equity (ROE) tend to have a higher P/E ratio. Finally, firms perceived as being “riskier” (the ones for which investors require a higher rate of return k) have a smaller P/E ratio. Fixed-income securities In the United States bonds and other types of fixed-income securities (such as mortgage-backed securities or asset-backed securities) are issued by corporations, federal, state and local governments, and federal agencies that wish to borrow money from investors. There is no centralized place or exchange on which bonds are traded. They are traded generally over-the-counter via a dealer system, electronically. It is not possible to issue bearer bonds in the United States, as all bonds must have registered owners. Coupons are typically paid twice a year. In Poland bonds are traded on the Warsaw Stock Exchange. There are two types of debt instruments: state Treasury bonds and corporate bonds. State Treasury bonds have 2-year, 3-year, 5-year and 10-year maturities, and they all have a nominal value of 1000. Coupons can be paid quarterly or once a year. Bonds with variable coupon payments are called floating-rate bonds. Bonds that pay a variable coupon have coupon rates periodically reset according to a specified market rate, e.g., Treasury Bill rate or LIBOR (London Inter-Bank Offered Rate, the most widely short-term interest rate index in U.S. dollars). In the US, Treasury Bonds and Notes are issued with a fixed coupon rate. Corporate bonds can be issued with fixed or variable rates. In Poland, state Treasury Bonds can pay a fixed coupon rate (paid once a year) or a floating coupon rate (paid quarterly). The floating rate is based on the average yield (interest rate) of 13-week Treasury Bills if the coupon rate is paid quarterly, or on the average yield of 52-week Treasury Bills if the coupon rate is paid once a year. A call provision in a bond allows a corporation to “recall” (or re-buy) all or part of the debt issue prior to maturity. In the U.S., Treasury-issued securities are not callable, but corporate or municipal issues typically are. Generally, the issuer will pay a premium over


the nominal or face value of the bond when the bond is called. A bond with a call provision is called a callable bond. A put provision gives the bondholder the right to sell a bond back to the corporation at the nominal or face value, generally after a relatively short period (about 3 to 5 years after the bond was first issued). A bond with a put provision is called a putable bond. A convertible bond gives bondholders the option to exchange the bond for a pre-specified number of shares of the company.

Exercise 1.4 Assume a $10,000 face value bond is convertible into 40 shares of stock. At which stock price does it become beneficial for the investor to convert the bonds into stock? Solution. Actually, this question is a bit complicated, because the answer depends on the current level of interest rate in relation to the coupon rate of the bond. If this bond is trading at par, the answer would be

10,000

40= 250.

Bond provisions for risk of default • Sinking fund: allows the issuer to periodically repurchase some proportion of the outstanding bonds prior to maturity. • Subordination clause: refers to the situation when a firm re-issues additional bonds or notes (“junior issues), these stipulate that previous “senior bondholders” will be paid first in the event of bankruptcy and that “junior” bondholders will be paid only after that. • Collateral: bonds can be issued with a specific asset pledged against possible default of the company. Collaterals are generally assets of the firm that investors in the bond will receive if the firm defaults in its payment: property, equipment, and so on. Risks associated with investing in bonds • Inflation risk. • Liquidity risk: refers to the risk associated to re-trading a bond in the secondary

market (no one wants it) • Maturity risk: refers to the fact that longer-term bonds prices react more to interest

rate changes that shorter-term bonds. • Default risk: risk that the company will default on the coupon payment or on the face

value of the bond Bond Ratings Since the early 1900s, bonds have been assigned quality ratings that reflect their probability of going into default. In the U.S, the three major rating agencies are: Moody’s, Standard & Poor’s (S&P) and Fitch Investors Services. These agencies also


analyze the credit quality of non-U.S. corporations. Bond rating criteria are both qualitative and quantitative. They can be based on: o The corporation’s various financial ratios (debt, liquidity or profitability ratios). o Subordination provisions. Is the bond subordinated to another debt instrument? o Maturity of the bond. o Stability: are the bond issuer’s sales and revenues stable? o Overseas: political climate in other countries? o Environmental factors. o Product liability: are the firm’s products safe? o Labor unrest. Bond ratings are important to both the firm and the investors in the firm, because bond ratings assess a firm’s risk of default. As a result, the lower the ratings, the higher the risk, the higher the return required by the investor, the more costly it will be for the firm to issue debt. 2. Forwards and Options Forwards A forward contract is a sale transaction, which is consummated in the future, but with all details of the transaction specified in the present. The time at which the contract settles is called the expiration date. The asset (often a commodity) on which the forward is based is called the underlying. A forward contract generally requires no upfront payment. It should be noted that both sides of a forward are exposed to a significant risk of non-performance of the other party (i.e., credit risk), and for that reason, forwards are rarely used in practice, and even if they are used, they are private transactions, not traded on an exchange. Forwards are negotiated and customized for specific parties. They are created on commodities, financial assets, indices, or currency exchange rates. They are arranged by brokers or dealers, who make money on the spread. Note that the price paid in the market for a given asset is called the spot price (as opposed to the price named in the forward contract, the forward price). The person buying the underlying in a forward contract is said to be long forward, and the person selling the underlying is said to be short forward. We have the following Payoff to long forward = Spot price at expiration – Forward price Payoff to short forward = Forward price – Spot price at expiration By analyzing the payoffs of a forward we can also conclude that Long forward = Underlying’s Spot Price – PV(Forward Price) Forward price (price at which the contract is agreed upon) depends on the spot price of the underlying, and converges to it as the contract nears maturity (otherwise arbitrage opportunities would be present). If forward is created so that no cash changes hands at inception, then Forward price = Accumulated value (Spot Price), F = S ! e

" t.


Consider a forward on an n-year zero-coupon bond, for the time k in the future. Let us write F 0,k( ) for the forward price. Let P n + k( ) be the price of a $1 risk-free zero-coupon bond maturing at time n + k, and let P k( ) be the price of a $1 risk-free zero coupon bond maturing at time k. This case does not seem to allow for a straightforward application of the formula above. However, we have the following: PV(Forward Price) = PV F 0,k( )( ) = F 0,k( ) !P k( ). The underlying in this case is a zero-coupon bond maturing at time n + k, issued at time n, which effectively is a unit monetary amount paid at time n + k, so that its present value is P n + k( ). Because we assume that a long forward, with Long forward = Underlying – PV(Forward Price), is costless to enter, we must have 0 = P n + k( ) ! F 0,k( ) "P k( ), so that

F 0,k( ) =P n + k( )

P k( ).

You also can show this by using a no-arbitrage argument (assume all maturity values are units): - Buy a k-year zero-coupon bond and then buy a forward contract to purchase an n-year

bond k years from now; you are thus entitled to a unit payment at time n + k. - Purchase an (n + k)-year zero coupon bond now with the same unit terminal value. - Since these strategies would produce identical payoffs, they must cost the same:

P k( ) !F 0,k( ) = P n + k( ), or F 0,k( ) =P n + k( )

P k( ).

Note that the key formula

F 0,k( ) =P n + k( )

P k( )

effectively says that the forward price is the accumulated price of the underlying, accumulated at a risk free interest force of interest r (credit risk is ignored in this model). This is equivalent to the standard relationship of interest rates:

F 0,k( ) =1

1+ fn,n+ k( )k=P n + k( )

P k( )=

1

1+ sn+ k( )n+ k

1

1+ sk( )k

,

or 1+ sn+ k( )

n+ k= 1+ sk( )

k! 1+ fn,n+ k( )

k

. General relationship between spot and forward prices Let S be the current spot price at time. Then the relationship between spot and forward prices is:

F t( ) = S ! ert " D t( ),


where F t( ) is the futures price to be paid at time t agreed upon at time 0, D t( ) is the cash flows produced by the asset from time 0 to time t, accumulated to time t, and r is the risk-free force of interest. This relationship is known as the forward-spot parity (it also applies to futures). If the cash flow is a dividend, and the dividend is payable at a continuous fashion, with the force of interest ! for the rate of dividend payment, then F t( ) = S ! e

"# t( ) ! ert . For commodities, the underlying does not produce income, but it requires the cost of carry (the total cost of “carrying”, i.e. storing, transporting etc.), and if that cost is expressed as a continuously compounded annualized rate c, we obtain this relationship F t( ) = S ! e

ct( ) ! ert . This relationship is known as the cost of carry relation between forward (also applies to futures, see below) and spot prices. A futures contract is basically like a forward, but with a lot of things added in order to make it tradable on the exchange, and eliminate the counterparty credit risk. Futures are exchange-traded and are standardized with regard to maturity, size, and the underlying asset. Futures are also marked-to-market each day (with cash flows required from all parties): this means that each side of the trade must deposit a certain required margin and the margin must be sufficient for the position held not just at the beginning, but each day the position is held. Also, the exchange may impose a price limit on a daily movement in the futures price (once that limit is reached, trading is halted for a specified period of time). While there may be some slight differences between prices of forwards and futures in practice, due to margin requirement and intermediate cash flows in future contracts, we will generally assume that pricing formulas for futures are the same as pricing formulas for forwards that we developed above. We will discuss possible divergence between forward price and futures price later.

Forwards versus futures overview: Security feature Forward Futures Type of market Dealer or broker Exchange Liquidity Low (almost zero) Very high Contract form Custom Standard Performance guarantee Creditworthiness Mark-to-market Transaction costs Bid-ask spread Fees or commission

Because futures are usually traded for hedging or speculation, without actual intent of buying or selling the underlying, many futures contracts are settled in cash, without the delivery of the underlying. Stock index futures, for example for the S&P 500 index, are always settled in cash.


Hedging spot prices with futures Assume a company knows now that it will want to issue bonds at a later date and it is afraid that rates will rise between now and that date, forcing it into higher cost of debt. The company can hedge its position by holding a short position in Treasury-Bond futures:

- If rates go up, it will be forced to issue debt at a higher rate but this loss will be offset by a gain from the short futures position.

- If rates go down, it can issue debt at a lower rate, but this gain will be offset by a loss from the short futures position.

- This kind of a hedge is called a cross hedge, because the underlying of the hedge (futures) is different than the position hedged. As a result the company may still be assuming basis risk: the risk of divergence between the hedge underlying and the security hedged; even if Treasury rates remain level, the company may be forced into higher rates if corporate bonds spreads widen.

Stock index futures No futures are written on the DJIA (Dow Jones Industrial Average), but there are futures contracts on the S&P 500, the NYSE composite, the MMI (Major Market Index, closely correlated with DJIA), and the Nikkei (Tokyo market index). MMI futures contracts sell for 250 times the index value. Suppose you want to hedge a portfolio of X dollars of a diversified portfolio of stocks of large companies (commonly called blue chips). You can’t use DJIA futures, but MMI is available. You can hedge the portfolio by shorting n futures contracts, where n is obtained from n !250 !S = X where S is the spot index level. This creates a hedge: if the index rises, the loss on the futures contracts is offset by a gain in the portfolio, and if the index falls, the loss on the portfolio is offset by a gain in the futures contracts. In general, for index futures, F t( ) = S ! e r"#( )t

, where ! is the dividend yield of the index. This shows us that when the dividend yield is greater than the risk-free rate, the futures price will be less than the current index price. Exercise 2.1 Assume it is now June 30, 2002, and a Treasury Bill maturing September 30, 2002 with $10,000 face value is selling for $9,955.20. Current spot price for gold is $315 per ounce. What is the price of the futures contract for gold with September 30, 2002 delivery? Assume that there is neither any cost of storage for gold, nor any convenience yield to owning it. Solution. The information about the Treasury Bill gives us information about the risk-free interest rate, i.e,, force of interest r applicable must satisfy:

9,955.20 ! er !3

12 = 10,000. Since gold does not pay any dividends, all we have is this relationship between spot and futures price (spot-futures parity):

F = S ! er !3

12 = $315 !10,000.00

9955.20= $316.42.


Options In general, an option is a contract in which one side acquires the right to buy (or sell, but only one of these two) the underlying at a predetermined price (might be a function of something, but the conditions are stated in advance) at a predetermined time or by predetermined time. In hedging with forwards and futures, upside potential must be sacrificed in order to receive downside protection. When hedging with options, this is not the case. Options have asymmetrical payoff patterns such that they only pay off when the index/security/commodity price moves in a specific direction. In an option contract, the long side had the right, but not the obligation, to purchase or sell (depending on the option) a security at a specified price. The other (short) side must be the counterparty and provide the market for the right of the long side. This sounds good for the long side, bad for the short side, but … the long side must pay an up-front premium to the short side. Option terminology

- American options are exercisable at any time prior to expiration. - European options can be exercised only at maturity. - Bermuda options can only be exercised during prescribed periods before maturity,

but not the entire period from now till maturity. - A call option gives the long side the right to buy the underlying at a fixed price,

called the strike price, or exercise price. - A put option gives the long side the right to sell the underlying at a fixed price,

called the strike price, or exercise price. Remember that every right (to buy or to sell) of the long side originates from the obligation of the short side to accommodate the right granted to the long side. The process of creation an option by the short side is called option writing. Let us now assume that the underlying is a stock. If S

t is the stock price at the time of

expiration of a European option, and K is the strike price, then the following table summarizes the long side payoffs of call and put options:

St< K S

t= K S

t> K

Call Option Payoff 0 0 St! K

Put Option Payoff K ! St 0 0

Equivalently (for the long side),

Call option payoff = max St! K ,0( ) = S

t! K( )

+.

Put option payoff = max K ! St,0( ) = K ! S

t( )+.

The above are true for American options, but the stock price need not be the price at expiration of the option; it can be the price at any time until the expiration. The value that an option has if it were exercised instantly is known as its intrinsic value. Before its expiration, the option may have a price higher than intrinsic value, and the difference between the two is called the time value of the option.


An option whose intrinsic value is positive is said to be in-the-money. If the underlying is trading at exactly the exercise price, then the option is said to be at-the-money. An option that is at-the-money has intrinsic value of zero, but the opposite is not necessarily true. If an option has intrinsic value of zero, but it is not at-the-money, we say that the option is out-of-the-money. Options generally have positive time value before expiration. This is, in fact, a must for an American option. The price of American options can never be less than their intrinsic value, since arbitrage opportunities would exist otherwise (someone could buy the American option and immediately exercise it, profiting a gain). What happens if an option is issued on a stock that pays a dividend? In a sense, nothing. The dividend has no effectively no influence on the situation of the parties involved in the option trade. The only influence of the dividend is that affected on the stock price itself. When a dividend is paid, stock price is reduced by the amount of the dividend. Effectively, before the time of the payment, the stock trades with a dividend, and after that, without a dividend. But the price that applies to the option contract is the price of the stock, either the one with the dividend, or without it. Exercise 2.2 Mr. Romuald Carcosheek writes a six-month put option on the MIDWIG (an index of the Warsaw Stock Exchange in Warsaw, Poland) index with the exercise price of 2750. In order to be able to write options, he must put down a margin deposit of 1000. He does not receive interest on his margin deposit. The put option he writes on MIDWIG sells for a premium of 75. In six months, MIDWIG index stands at 2700. Assuming that Mr. Carcosheek earns interest on the option premium at a nominal annual rate of 4% compounded semiannually, and assuming that at option expiration he buys the index at 2750 from the long side of the option contract and sells it immediately in the market, calculate the effective annual rate of return Mr. Carcosheek will have earned over the six month period. Solution. Mr. Carcosheek puts down an investment in the amount of 1000. He receives the option premium of 75, and that premium, with interest, after six months is worth

75 ! 1+0.04

2

"#$

%&'2!1

2

= 75 !1.02 = 76.50.

When he buys MIDWIG for 2750 and sells it for 2700, he suffers a loss of 50. Thus his net cash flow at the end of the six-month period is 26.50, and his effective annual rate of return earned is

1+26.50

1000

!"#

$%&2

'1 ( 5.37%.

Exercise 2.3 Mr. Romuald Carcosheek owns a home that costs 20,000,000 PLN. The house is so well built and so well protected that it can only suffer damage from a fire or an earthquake. Mr. Carcosheek purchases an insurance policy that will pay for the damage to his house due to fire or earthquake any time during the next year, with a 1,000,000 PLN deductible,


for a premium of 100,000 PLN. Assuming that the value of the house does not change for any other reason than a fire or an earthquake, describe an option contract that Mr. Carcosheek can use to obtain the same protection as that given by the insurance policy. Solution. Let us write S for the value of the house. The payoff (not counting the insurance premium) of the insurance policy is

0, if S ! 19,000,000,

19,000,000 " S, if S < 19,000,000,

#$%

&'(= max 19000000 " S,0( ).

This is the same as the payoff of an American put option on the house, with exercise price of K = 19,000,000 and expiration date equal to the last day the insurance policy is valid. Exercise 2.4 Countrybank is entering the Polish market and has decided to offer a new attractive Certificate of Deposit, tied to the WIG index (an index of the Warsaw Stock Exchange in Warsaw, Poland). The certificate is issued for two years, and it promises to pay the full amount deposited plus 70% of the performance of the WIG index over that period, assuming that performance is positive, and zero otherwise. Express the payoff of the certificate in terms of an appropriate option contract. Solution. Assume for simplicity that the amount of the initial deposit K in the certificate is the current value of WIG. Let S stand for the value of the index in two years. In two years, the certificate will be worth

K + 0.7 ! S " K( ), if S > K ,

K , if S # K ,

$%&

'()= K + 0.7 !max S " K ,0( ).

Therefore, Countrybank certificate simply pays 70% of the payoff of a European call option on WIG with exercise price of K, which expires when the certificate expires, as the return on the deposit. 3. Insurance Created With Options, Collars, Floors, Caps, Spreads, Straddles As we had already pointed out in an exercise in Section 2, some options positions result in the same payoffs as certain insurance policies. For example, a put option with the exercise price of $50 on a stock selling now for $60 provides insurance against the fall of the price below $50. This insurance provided by a put is sometimes called a floor, as it puts a “floor” under losses that the long position can suffer. On the other hand, there are situations when an opposite form of insurance is needed. Suppose that an insurance company promises to pay its customer the rate of return on a stock market index, and the customer holds an account valued at $1,000,000, and the current value of the index is 1000. Thus the customer holds 1000 units of the stock market index. In order to protect itself (i.e., insure itself) against a sharp increase in the stock market index (which would result in a corresponding sharp increase in the liability


to the customer), the insurance company can buy 1000 calls on a unit of the stock market with the exercise price of 1000. Then any increase in the stock market index above the current value of customer’s account would be fully covered by the increase in the value of these calls, and the company’s liability would be fully insured. This form of insurance is sometimes called a cap, as it “caps” the value of the liability of the company. Caps and Floors Caps and floors, however, are most commonly used in the context of insuring against interest rate risk. They provide one-sided protection against movements in a floating interest rate. They are commonly embedded in adjustable rate mortgages in the United States. The cost can either be paid for up-front or embedded in the interest rate paid. An interest rate cap consists of a series of caplets, and an interest rate floor consists of a series of floorlets. An interest rate caplet is analogous to a call option on the level of interest rates: at expiration, if the interest rate is above the strike rate, the caplet pays i ! k( ) times the notional, and zero otherwise ( i is the index rate, k is the strike rate). An interest rate floorlet is analogous to a put option on the level of interest rates; at expiration, if the interest rate is below the strike rate, the caplet pays k ! i( ) times the notional, and zero otherwise ( i is the index rate, k is the strike rate). An interest rate collar is a long position in a cap plus a short position in a floor; it effectively makes a payment to its holder whenever the index rate is outside the boundaries set by the strike rates (the floor rate is below the cap rate, and the range between them leaves the floating rate alone). Covered and naked option writing If you own the underlying and write an option, this is called covered writing. If you do not own the underlying, and write an option, this is called naked writing. Exercise 3.1 Mr. Romuald Carcosheek owns 100,000 shares of Megabank, a Polish bank whose shares are traded on the Warsaw Stock Exchange. Mr. Carcosheek is concerned that the current price of PLN 50 of the Megabank stock is too high, and it is likely to decline to PLN 45 range. He decides to protect himself, at least partially, by writing at-the-money calls on his shares, which currently sell for 1 PLN per share. He is able to invest the premium received from writing these calls in a risk-free one-year Polish government bond earning 6%. In one year, the shares of Megabank are selling for PLN 47. Calculate Mr. Carcosheek’s total rate of return over that one-year period, including his unrealized loss on the shares, premiums received for options written, and interest earned on the premiums invested in a risk-free one-year Polish government bond. Solution.


Mr. Carcosheek begins the year with 100,000 shares of Megabank worth PLN 50 each, for a total initial investment of PLN 5,000,000. By writing covered calls, he earns PLN 100,000, which accumulate to PLN 106,000 by the end of the year by being invested in a government bond. But at the end of the year, his 100,000 shares are worth only PLN 4,700,000, so that his total amount at the end of the year is PLN 4,806,000. His rate of return is

4,806,000

5,000,000!1 = !3.88%.

While this may look bad, note that the shares declined by 6%, so was able to cushion his loss by utilizing the strategy of writing covered calls. An important observation to make concerning European calls and puts is that if a European call and a European put have the same underlying, same maturity date, and the same exercise price, then Call ! Put = Forward. In other words, a investor who buys a European call with exercise price K and writes a European put on the same underlying, same maturity date, and the same exercise price, will experience exactly the same payoff as an investor who enters into a long forward position with the same maturity date and the same exercise price K. Short forward position is replicated by a portfolio of a long put and short call (both European). Put-Call Parity Let us make the following assumptions:

- No dividends payable on the underlying. - All investors may borrow and lend at the risk-free rate. - There are no transaction fees or taxes. - Short selling and borrowing are allowed, fractional shares may be traded. - There are no arbitrage opportunities.

Under these assumptions, consider the following two portfolios: Portfolio 1: Long European call option with maturity date t years from now, plus Ke

!rt invested in the risk-free asset. Portfolio 2: Long European put and one share of the underlying.

We assume that the call and put have the same exercise price and maturity. Then the value of these two portfolios is as follows:

ST< K S

T> K

Long call 0 ST! K

Ke!rt in risk-free asset K K

Total Portfolio 1 K ST

Long put K ! S

T 0

One share of stock ST

ST

Total Portfolio 2 K S

T


We see that these two portfolios produce identical payoffs regardless of the price of the underlying. Therefore, these two portfolios must sell for the same price, i.e.,

c + Ke!rt

= S + p where c is the call price, and p is the put price. This relationship is called the put-call parity. Exercise 3.2 The current price, as of June 30, 2002, of a $325 call on September 30, 2002, gold is $12, with the spot price being $315 per ounce, and a three-month Treasury-Bill maturing on September 30, 2002 with $10,000 maturity value selling now for $9,955.20. Find the price of a September 30, 2002, $325 (exercise price) gold put as of June 30, 2002. Assume that all conditions for put-call parity to hold are satisfied. Solution. We use the put-call parity formula c ! p = S ! PV (K ). In this case, c = 12, S = 315, and the present value of a cash flow paid on September 30,

2002, as of June 30, 2002, is established by multiplying that cash flow by 9955.2010000

. We

get

12 ! p = 315 !9955.20

10000" 325 # !8.54.

This gives us p ! 20.54. Exercise 3.3 You are given the following information: - An option market satisfies the condition for put-call parity. - The current underlying security price is 100. - A call option with a strike price of 105 and maturity one year from now has a current price of 4. - A put option with a strike price of 105 and maturity one year from now has a current price of 6. Determine the one-year risk-free interest rate. Solution. We use the put-call parity relationship

c ! p = S ! PV K( ). Substituting the data given in the problem we get

4 ! 6 = 100 !105

1+ i,

or 102 = 1051+ i

. The solution is i = 3

102! 2.94%.

Spreads, Collars, Straddles


An option spread is a position of only calls or only puts, in which some options are purchased and some are written. Here are some of these strategies: • A bull spread consists of a long call with a lower exercise price and short call with higher exercise price, both calls expiring at the same time. • A bear spread consists of a long call with a higher exercise price and a short call with a lower exercise price, both calls expiring at the same time. • A box spread consists of a long call and short put with the same exercise price, and another position of a long put and a short call with the same exercise price, but different than the previous one. This amounts to being long one forward and short another forward. The strategy is purely a means of borrowing or lending money. • A ratio spread is constructed by buying by buying m calls at one exercise price and selling n calls at a different exercise price, with same maturity and same underlying. • A collar is created by purchasing a put option and writing a call option. A reverse position is a short collar. A collar width is the difference between the call and put strike prices. A zero-cost collar is created by adjusting strike prices so that there is no cost or income to the transaction. • A straddle is a position consisting of a long call and a long put with the same exercise price. This position benefits from high volatility. Short straddle benefits from low volatility. • A strangle consist of a long out-of-the-money call and a long out-of-the-money put. This is a strategy similar to a straddle, but at a lower cost. • A butterfly spread consists of a short straddle combined with a long out-of-the-money put and a long out-of-the-money call. The position could be symmetric or asymmetric. 4. Managing Risk With Derivatives • Hedging with forwards or futures A long position in stocks or commodities can be hedged by holding a short position in a forward or in futures. • Portfolio insurance You can insurance a minimum value of a portfolio by buying a put option. • Insurance by selling a call Used to hedge the risk of a long portfolio of stocks or commodities. This does not provide complete protection, but reduces risk with premium income. But recall put-call parity: c ! p = S ! PV K( ), so that S ! c = PV K( ) ! p. Thus the long position insured with a short call can be replicated by buying a bond and a selling a put option. Why do firms manage risk? Financial risk management should be viewed from the perspective of the Modigliani-Miller Theorem: financing mode affects value only if it • increases firm’s output (without increasing costs, or at least not increasing them above the benefit created),


• reduces taxes, • reduces bankruptcy costs, or • reduces agency costs. But there may be good reasons not to hedge: • Derivatives involve high transaction costs. • Hedging requires costly expertise. • Hedging increases agency costs (e.g., rogue traders). • Hedging increases taxes or other government costs (e.g., regulatory). Cross-Hedging There are situations when hedging is achieved not by a position in a security identical or directly related to the originally position held, but a more remotely related security. This means practically that the movements of prices of the hedged security will not be mimicked by the hedge, but will only be related somehow. Suppose that an investor holds Q units of a security whose price is S and hedges it with a short position in H units of a security whose price is F, with !

S being the standard deviation of the changes in price of

a unit of the hedged instrument, !F

being the standard deviation in the changes in price of the hedging instrument, and ! being the correlation of the price changes of the two instruments. Then the investor’s total position is ! = QS " HF. The variance of this position is Var !( ) = Var QS " HF( ) = Q2#

S

2+ H

2#F

2 " 2QH$#S#

F.

We want to derive the value of H, which minimizes this variance. Note that the value of Q is given. We take the derivative of Var !( ) with respect to H and set it equal to 0

!Var "( )

!H= 2H#

F

2 $ 2Q%#S#

F= 0.

Note that

!2Var "( )

!H2

= 2#F

2> 0.

The value of H at which the derivative of Var !( ) with respect to H is equal to 0, i.e.,

H = Q!"

S

"F

,

produces the lowest variance portfolio. Also,

h =H

Q=!" S

" F

is called the optimal hedge ratio or minimum variance hedge ratio. Consider a company that is hired to repair roads in a small town every year. The materials used in the process of road repair are not traded in financial markets, but they are produced from crude oil, an asset for which futures contracts are readily available. Suppose that this company is paid P

r per kilometer of a road repair, and that N

c barrels

of crude oil are needed to repair Nr kilometers of roads. Let P

c be the crude oil price per


barrel. Then this company’s profit (we ignore other possible expenses and sources of profits for now) is ! = N

r"P

r# N

c"P

c.

Now suppose that this company enters into H futures contracts on crude oil, assuming for simplicity that each contract covers one barrel. If F is the futures price, the profit on such a hedged position is !

H= N

r"P

r# N

c"P

c+ H P

c# F( ).

Now suppose that the objective is to minimize the variance of the hedged position. The variance of the hedged position is (we make a simplifying assumption that F does not vary, as we are doing this calculation for this moment in time, based on historical estimates of volatility and correlations) !

"H

2= N

r

2#!

Pr

2+ H $ N

c( )2

#!Pc

2+ 2 H $ N

c( ) #Nr#Cov P

r,P

c( ). The variance-minimizing hedge ratio is

H*= N

c! N

r"Cov P

r,P

c( )#

c

2.

In this hedge ratio, we can interpret the first term as hedging costs, and the second term as hedging revenue. The coefficient that N

r is multiplied by is the result of regression of

the price of a kilometer of a road on the price of a barrel of oil. The resulting minimum variance is !

"H*

2= N

r

2 #!Pr

2 # 1$ %Pr,Pc

2( ). This positive variance indicates that there is risk left in this position, due to possible divergence of the revenue earned from the road and the price of oil. This kind of risk due to divergence of the value of the portfolio hedged and the value of the hedge used is called the basis risk. 5. More on Forwards and Futures Alternative ways to buy a stock • Buy the stock directly, for cash. • Fully leveraged purchase: borrow all money used to buy the stock. • Prepaid forward contract: pay for the stock today, receive it at time t in the future. • Forward contract. Prepaid forward contract In the absence of dividends, the price on the prepaid forward contract is today’s stock price S. The reason is simple: you will own the stock anyway, so you should pay its market price. We will denote the prepaid forward price by F

0,t

P. If we use a subscript in

the stock price notation to denote the time of the price, we have F0,t

P= S

0. If the prepaid

forward is on a stock that pays a dividend, then the forward contract holder does not receive that dividend, and the prepaid forward price is the price of the stock without any of the dividends paid through time t.


Forward contract Let us denote by F

0,t the forward price for transaction to occur at time t. As we had

already noted before, F0,t

= S0! e

rt, where r is the risk-free force of interest between time

0 and time t. If the stock pays a dividend, the price of the stock should be used without any of the dividends payable between time 0 and time t. The forward premium is defined

as the ratio F0,t

S0

. The annualized forward premium is 1t! ln

F0,t

S0

"

#$%

&', and, in the absence of

arbitrage opportunities, it equals the risk-free force of interest (or the difference between the risk-free force and the force of dividend). Note that the payoff of a long forward position is S

t! F

0,t. This payoff can be replicated

by buying 1 share of stock with borrowed funds of e!rt "F0,t. This shows again that

F0,t

= S0! e

rt. It also illustrates the fact that

Forward = Stock – Risk-Free Zero-Coupon Bond in the amount e!rt "F0,t,

and Stock = Forward + Risk-Free Zero-Coupon Bond in the amount e!rt "F

0,t,

and Risk-Free Zero-Coupon Bond in the amount e!rt "F

0,t= Stock – Forward.

Note that given the market prices of a stock and of a forward on that stock, we get the implied risk-free interest rate from the last identity. That implied interest rate is called the implied repo rate. A transaction in which you buy the underlying and short the offsetting forward is called a cash-and-carry. As we see from the above, this is equivalent to a purchase of a risk-free zero-coupon bond. Market makers in forwards often offset their position by buying the underlying, and creating the cash-and-carry position. A reverse cash-and-carry is created by being short underlying and long forward, and is equivalent to borrowing at the risk-free rate. We should note that the rate on cash-and-carry is the risk-free rate only if the forward is prices by the familiar equation F

0,t= S

0! e

rt. Otherwise, arbitrage can be

utilized to earn riskless profit. Recall that: Arbitrage: Creation of a portfolio requiring no capital outlay, but never losing

money, and producing positive payoff with positive probability. In practice, however, an attempt to arbitrage the difference between F

0,t and S

0! e

rt involve transaction costs. Suppose that the spot bid and ask prices are Sb < Sa and that the forward bid and ask prices are Fb

< Fa. Assume also that there is a transaction cost k

for a transaction in the stock or its forward, and that the force of interest for borrowing and lending differs, with rb > rl . We assume all transaction costs to occur at time 0, and none at time t. Consider an arbitrageur who sells the forward and buys the stock (assume no dividends for simplicity, or consider the stock without the dividend). The arbitrageur will have an upfront cash cost of k plus Sa + k( ). We assume he/she borrows to finance that position. At time t the payoff is


! Sa+ 2k( ) " er

bt

Repayment of borrowing

! "## $##+ F0,t ! St( )

Value of short forward

! "# $#+ S

t= F0,t ! S

a+ 2k( ) " er

bt

Denoted by F+

! "## $##.

The quantity F+ is the upper bound for current forward price so that the arbitrage is not profitable. In the same fashion, the lower bound F! below which arbitrage is profitable is given by the formula F!

= Sb! 2k( ) " erl "t . It should be noted that this analysis may

actually underestimate all costs involved. Quasi-arbitrage is a substitution of a lower-yielding position by a higher-yielding position. If a company can borrow at 8.5% and lend at 7.5%, clearly there is no arbitrage possible. But if a company is already lending at 7.5%, and it is possible to arrange for a cash-and-carry with implied repo rate of 8%, an arbitrage becomes possible. Does the forward price predict the future price? When you invest in stock, you expect to earn the risk-free rate plus the risk premium. But when buy enter into a forward, you put down no money, so you should not get the risk-free rate, just the risk premium (as you are still exposed to the risk of the underlying). Consider a one-year forward. Let r be the risk-free force of interest for that year. Let ! be the expected force of return of the stock. Then we have F

0,1= S

0! e

r and E S

1( ) = S0 ! e".

This tells us that E S

1( ) = F0,1 ! e" # r.

The expression ! " r is the risk-premium for the underlying. This tells us that the price of a forward is a downward (assuming positive risk premium) biased predictor of the future price of the underlying, and the degree of the bias is determined by the risk-premium of the asset. Recall that a forward can be replicating by a long position in the underlying (earning the risk-free rate plus the risk-premium) and a short position in a zero-coupon risk-free bond (earning the risk-free rate). As the risk-free rate is paid, only the risk-premium accrues to the forward. More on Futures Forward and futures prices may differ. The reason is that with the futures contracts, interest is earned on any mark-to-market proceeds, and lost on any required margin deposits. The required margin during the life of a contract (called the maintenance margin) is generally lower than the initial margin, but is nevertheless required. We will illustrate this in the exercise now. Exercise 5.1 Consider a futures contract on the stock market index in a hypothetical country called Cuba Libre, such index being called Viva Cuba Libre, or VCL for short. Suppose that you enter into a contract with a notional value of 1 million libretas (currency of Cuba Libre). The margin required is 7%, or 70,000 libretas. Each contract correspondes to


1000 units of VCL index, and that 1000 is used as the contract multiplier. Futures are initially trading at 1000 libretas. Calculate the dollar-weighted rate of return over a week on this contract, assuming the following: • The maintenance margin is 5%. • Risk-free interest rate is 0.02% per day. • When additional margin deposit is required, you pay it in. • When funds become available to withdraw, you take them out immediately. • Prices at the close of the days of the week for the futures on the VCL index are: Monday: 990 libretas; Tuesday: 1025 libretas; Wednesday: 1075 libretas; Thursday: 920 libretas; Friday: 1000 libretas. • Position is costless to enter into and to close (other than the margin deposit), and it is closed at the end of the week. Compare the net cash flow from the futures contract to that for a costless forward entered at the beginning of the week, for the purchase at the end of the week. Solution. We have the following history of this position Futures Price Margin price change balance Week beginning 1000 --- 70000 Monday close 990 – 10 70000 !1.0002 "1000 !10 = 60014 As of Monday close, maintenance margin required is 5% of 990,000, i.e., 49,500, so that 10,514 libretas are withdrawn and 49,500 libretas remain in the margin account

Tuesday close 1025 35 49500 !1.0002 +1000 ! 35 = 84509.9 As of Tuesday close, maintenance margin required is 5% of 1025,000, i.e., 51,250, so that 33,259.90 libretas are withdrawn and 51,250 libretas remain in the margin account Wednesday close 1075 50 51250 !1.0002 +1000 !50 = 101260.25 As of Wednesday close, maintenance margin required is 5% of 1075,000, i.e., 53,750, so that 47,510.25 libretas are withdrawn and 53,750 libretas remain in the margin account Thursday close 920 – 155 53750 !1.0002 "1000 !155 = "101239.25 As of Thursday close, maintenance margin required is 5% of 920,000, i.e., 46,000, so that 147,239.25 libretas are deposited and 46,000 libretas are in the margin account at the end of the day Thursday Friday close 1000 80 46000 !1.0002 +1000 !80 = 126009.20 As of Friday close, the position is liquidated and the margin balance of 126009.20 libretas is collected


Therefore, the dollar-weigthed rate of return over the trading week (five days) was:

!70000 +10514 + 33259.90 + 47510.25 !147239.25 +126009.20

70000 "5

5!10514 "

4

5! 33259.90 "

3

5! 47510.25 "

2

5+147239.25 "

1

5

=

=54.10

52076.61# 0.1039%.

Let us now compare this to a forward contract. We enter a forward at the beginning of the week. Assuming that transaction is costless, the price at which we set the purchase is the same as the initial futures price, i.e., 1000, if we take the futures-spot parity F = Se

! t and the forward-spot parity formula F = Se

! t to hold identically for both contracts. Then at the end of the week the price is 1000, and the transaction results in a zero return. This is quite interesting: we have a net positive cash flow of 54.10 from the futures contract, but a zero return from the forward. The futures part is more interesting: you had several cash flows for the week. Those cash flows end up netting to a positive number, 54.10, and this is what created your return. The structure of the futures contract and changes in the price in this problem forces you to adopt the strategy of buying when the price is down, and selling when the price is up, creating a positive return, in a sense, out of nothing. Good for you! One important practical implication of the fact that in general futures tend to benefit from volatility of the underlying, as in the long run most volatility in on the upside, when we hedge with futures, fewer futures contracts than equivalent forwards are commonly used. Nevetheless, most of the time we will assume the same pricing formula for forwards and futures. This is illustrated by the following exercises. Exercise 5.2 Consider a May futures contract that calls for delivery of 1000 ounces of gold in July (July 1) of the same year. Suppose that the current quoted spot price is $680 per ounce, and the current annual risk-free continuously compounded interest rate is 4.75%. Assume that gold carrying cost is zero. What is the current futures price assuming that the fundamental relation between cash and future prices holds? Solution.

The fundamental relation is F = Sert. We are given that r = 0.0475, t = 2

12, S = $680,000

(1000 oz.) so that

F = Sert= 680,000 ! e

2

12!0.0475

" 685,404.70. Exercise 5.3 Suppose that the value of the S&P 500 stock index is currently $1,300. If one year Treasury Bill interest rate is 6%, and the expected dividend yield on the S&P 500 Index is 2%, both of these interest rates expressed as annual effective rates of return, what should the one-year maturity futures price be?


Solution. The spot-futures parity relationship with adjustment for dividends, assuming annual effective risk-free interest rate, and annual effective dividend yield, is (approximately):

F0= S

01+ r

F! d( ),

where d is the dividend yield of the underlying. The exact formula would be

F0=

S0

1+ d! 1+ r

F( ),

Substituting problem data we get: F0 = $1,300 1+ 0.06 ! 0.02( ) = $1,300 "1.04 = $1,352.

You might wonder whether we can use the expected dividend yield in this exercise, if the actual dividend yield is uncertain. If you do so wonder, you are most certainly raising a valid point. This formula assumes that the dividend yield is known with certainty, and in real life applications you would have to be more careful about using such a formula. One more question about the above is whether the Treasury-Bill rate is appropriate for the risk-free rate used in the calculations above. The problem with the Treasury-Bill rate is that it tends to be relatively lower than other short-term interest rates, and some believe it to be unnaturally so. One possible explanation is that the margin in transactions such as buying on margin, short-selling, option writing, or purchases of futures, can be posted in cash, in which case it generally does not earn interest, or in Treasury-Bills, which earn interest automatically, as they are discounted zero-coupon bonds of short maturity (up to one year). Thus traders may buy up Treasury-Bills, bidding up their prices, and bidding down their yields. For that reason, some prefer using the short-term LIBOR (London Inter-Bank Offer Rate) index for pricing forwards and futures, as those instruments are also nearly completely free of credit risk. Quanto index futures The Nikkei 225 futures contract traded on the Chicago Mercantile Exchange is quite peculiar. Its values are derived from the Nikkei 225 index, but the currency in which it is expressed is the U.D. dollar, not the Japanese yen. The size of that futures contract is $5 times the numerical reading of the Nikkei 225 index. It is cash-settled based on the opening Osaka quotation of the Nikkei 225 index on the second Friday of the expiration month. A contract of that nature is called a quanto contract: referring to an index in one currency, but traded in another currency. What are the uses of stock index futures? • Asset allocation. This can mean switching from stocks to Treasury-Bills or the other way around. But using longer term Treasury-Bond futures together with stock index futures we can also construct allocations between stock index and a bond portfolio. If we own a diversified stock portfolio and short S&P 500 futures ot reduce stock allocation, while going long Treasury-Bond futures, the first transaction converts stocks into T-Bills, and the second one converts T-Bills into T-Bonds. This combined use of futures is called futures overlay. You can also use futures to convert a bond-portfolio manager into a stock portfolio manager. Suppose that you have found a bond portfolio manager who can beat the Treasury-Bond index consistently, but you want to invest in stocks. You can hire that


manager, while simultaneously shorting Treasury Bond futures (thus converting the manager to T-Bill plus this manager’s outperformance of the index) and going long S&P 500 futures (thus converting T-Bill return into stock return). • Cross-hedging. This refers to using a futures contract on a different security than the one we are hedging. It is quite common to hedge diversified stock portfolios with S&P 500 futures, after adjusting for the size of the portfolio in relation to the size of the contract, and for the beta of the portfolio. Currency futures and forwards These instruments are used to hedge against changes in currency exchange rates. The simplest instrument is a currency prepaid forward. Suppose that in one year you want to have 1 liberta. Suppose that the risk-free force of interest in libertas is r

For and the risk-

free force of interest in U.S. dollars is rDom. To obtain 1 liberta in one year, we must have

e!r

For today. Suppose that the exchange rate at time t is Et dollars needed to purchase

one liberta. We conclude that in order to assure a purchase of one liberta a year from now we need F

0,1

P= E

0! e

"rFor dollars. This is the price of a prepaid forward.

In general, the currency forward price (assuming a costless contract to be entered an time 0 and realized at time t) is F

0,t= E

0! e

"rFor !t ! erDom !t

= E0! e

rDom " rFor( )t.

Covered interest arbitrage is a transaction consisting of borrowing in domestic currency, lending in a foreign currency, and entering into a forward transaction to purchase the foreign currency. The principle behind this is that a position in foreign risk-free bonds, with the currency risk hedged, pays the same return as domestic risk-free bond. Eurodollar futures The Eurodollar contract is based on a $1 million 3-month deposit earning LIBOR. There is also a 1-month contract, handled similarly. Suppose that the current LIBOR is 1.25% over 3 months. By convention, this is annualized by multiplying by 4, to the quoted annualized LIBOR rate of 5.0%. The Eurodollar futures price at expiration of the contract is 100 – Annualized 3-month LIBOR. The settlement is based on then current LIBOR (i.e., for the future three months counting from the date of settlement). This futures contract is used for hedging interest rate risk. 6. Swaps Swaps are generally derivatives that trade cash flows between counterparties based on two pieces of underlying. Such a contract is written for a specific term, called the swap term, or swap tenor. An unusual, and simplest type of a swap is an exchange of a single payment for multiple payments (or, possibly, multiple deliveries of some form of underlying) in the future. This is called a prepaid swap. Swaps are commonly settled financially, i.e., currency payments are exchanged. But it is also possible to create a physical delivery swap. For example, one could arrange for a


delivery of an ounce of gold every year for the next twenty years in exchange for a fixed payment of $600 every year. Note that this transaction can be actually decomposed into a series of forwards (and that is not a coincidence). Such physical delivery swap can also be settled financially, of course, by the side delivering the gold paying the other party the difference between $600 and the price of an ounce of gold at the time of delivery. It is rare that the two counterparties of a swap are both market participants. Swaps are arranged as private transactions and not traded on exchanges. They are typically arranged by a dealer, for a client of that dealer. The dealer ends up holding one side of the swap deal, but usually seeks to make another swap arrangement that would at least partially offset that position. That second transaction is commonly called back-to-back transaction, or matched book transaction. The dealer can also seek to hedge the exposure using traded derivative instruments. It is generally difficult, however, to find an exact hedge in the market, and the dealer may have to look at the dealer’s entire portfolio exposure and hedge pieces of it, combining exposures from various transactions. The market value of a swap When a swap is entered into, it is standard that no payments change hands and future payments committed to by each party have the same market value. As time passes, the value of each counterparty’s position changes. Because what one party pays, the other one receives, the total of the values remains zero. This may create credit risk for the party, which has a positive value of the swap. Interest rate swaps A “plain vanilla” (i.e., the simplest kind, as in the “plain vanilla” ice cream) interest rate swap is a contract between two counterparties, requiring them to make them interest payments to each other over the term of the contract, based on different types of underlying bonds or interest rate indices. The long party (fixed-rate payer) pays interest to the second at a fixed rate, while the short party (floating-rate payer) pays interest to the first at a rate that changes (“floats”) according to a specified index. This kind of a swap is also called a pay-fixed swap, because the long party pays fixed. The actual cash payments are determined by multiplying the relevant rate of interest by a face amount, or principal, which is called the notional principal, or just the notional. For example, consider a semiannual swap with a notional of $10 million: Long: Pay 4%, nominal compounded semiannually, and receive 6-month LIBOR, Short: Pay 6-month LIBOR and receive 4%, compounded seminannually. If the 6-month LIBOR on the first payment date were 3% (nominal compounded semiannually), then:

Fixed-rate payer pays floating-rate payer $10M !0.04

2= $200,000,

Floating-rate payer pays fixed-rate payer $10M !0.03

2= $150,000.

This would be actually settled by the fixed-rate payer paying the floating-rate payer $50,000 rather than the two parties paying offsetting amounts.


Swaps are commonly used to manage interest rate risk exposure. An insurance company can use swaps to produce an income stream that better matches its liability structure. Suppose that you are managing a company that issued fixed-rate liabilities that credit 5.50% (e.g., GIC or SPDA), and backs them with a bond that pays 3-month LIBOR + 1%. There is a mismatch creating interest rate risk exposure. Now suppose that this company enters into a swap where it pays 3-month LIBOR and receives 5.50%. This would result in a net 1% profit to the company, without any interest rate risk. This, of course, is an idealized example, but you should understand how it works. The company used a swap to convert its floating-rate asset to a fixed-rate asset that better matched its liabilities; in doing so, it locked in a spread of 100 bps. In practice, most commonly, life insurance companies seek to convert their fixed coupon bonds income into floating rate income. A swap in which a party receives a floating rate in exchange for fixed payments on bonds that this party already holds is called an asset swap. Swap rate A short (pay-variable) swap position is equivalent to buying a fixed coupon bond with funds borrowed at the swap’s floating rate. If an interest rate swap is arranged in such a way that no payment is made upfront, and in exchange for fixed rate payment, the long side receives the market floating rate, the resulting fixed interest rate is called the swap rate. The swap rate turns out to be simply the coupon rate on a par coupon bond, with that bond’s maturity equal to the swap term. The swap curve Thanks to Eurodollar futures on 3-month LIBOR, 3-month LIBOR forward rates can be found for up to 10 years. The swap rates established against the floating rates equal to those LIBOR forward rates constitute the swap curve. The swap spread is the difference between a given swap rate and a Treasury Bond yield for the corresponding maturity. The swap’s implicit loan balance At inception of an interest rate swap, the swap has zero value to both sides. As time passes, and interest rates change, the value to each counterparty changes. Even in the absence of interest rate changes, the value will change if the yield curve is not flat, as one side makes a fixed rate payment, and the other one pays, implicitly, based on forward rates. The present value of future payments is an asset to one party, and a liability to the other party. We can call it the implicit loan balance. Deferred swap A deferred swap is a swap that begins at some date in the future, but for which the swap rate is agreed upon today. Why do firms use interest rates swaps? Short-term borrowing is generally only available to firms with very good credit rating. Even such high quality firms may have a hard time issuing large amounts of short-term debt. Longer-term (five, possibly to ten years) debt is generally easier to issue for firms with lower credit rating, and can be issued in larger amounts. Longer-term fixed interest rate debt is easier to handle for a borrower, as payments are more predictable. Floating


rates can change significantly, and having a large amount of debt issued at floating rates can be quite risky, even for a firm with good operating results. Given this situation, some firms issue large amounts of debt in five to ten year fixed coupon bonds and then swap them to floating, hoping that they will save in interest expenses in the long run, as the yield curve is typically upward sloping. Amortizing and accreting swaps A swap in which the notional amount declines over time is called an amortizing swap. A swap is which the notional amount increases over time is called an accreting swap. Currency swaps In a currency swap, the notional amounts are specified in different currencies. This is handled by exchanging the notional amounts at the beginning of the swap and returning them at the end. Currency swaps are used to manage foreign exchange exposures. Consider, for example, a British company that wants to issue debt in Japan, but finds the process of registration of securities and issuing them in Japan too much to handle. Instead the company can issue floating debt in Great Britain, and enter into a swap to pay fixed long-term Japanese Yen rate, and receive British Pound floating. The relationship used for pricing currency swaps is the Covered Interest Rate Arbitrage. Let: • E

0be the current exchange rate in dollars per unit of foreign currency,

• Et is the foreign currency exchange rate

• rDom

be the U.S. dollar-denominated (domestic) risk-free force of interest, • r

For be the risk-free force of interest in foreign currency,

• F0,t

be the forward exchange rate in dollars per one unit of foreign currency at time t. Take one unit of foreign currency now. You can then do one of these two things: • Convert it immediately into U.S. dollars and invest in the U.S. risk-free asset; this would give you E

0! e

rDom !t foreign currency units at time t. • Invest in the foreign risk-free asset and enter a forward contract to convert to dollars at time t. This would give you erFor !t !F

0,t at time t.

In the absence of arbitrage, your final payoff is the same, and the prices of the two instruments should be the same. The resulting relation is: E

0! e

rDom !t

= erFor !t !F

0,t.

This is, of course, the same relationship we discussed previously. A differential swap is a swap in which payments are made based on the difference in floating interest rates in two different currencies, with the notional in a single currency. Commodity swaps A swap in which the payments (or at least one side payments) are expressed in terms of a commodity is called a commodity swap. Those swaps may be arranged with varying quantities and prices, depending on seasonal factors.


Swaptions An option to enter into a swap is called a swaption. A payer swaption is a right, but not an obligation, to enter into a swap to pay fixed and receive floating. A receiver swaption is a right to enter swap to pay floating and receive fixed. A total return swap is a swap in which one party pays the realized total return (dividends plus capital appreciation) on a reference asset or index, and the other side pays some floating rate, e.g., LIBOR, possibly plus a spread. For example, total return on S&P 500 can be swapped into LIBOR plus a spread. Total return swaps can be used to invest in a stock index in a country requiring withholding taxes on dividends. A default swap is an instrument in which one side pays regular payments, similar to insurance premiums, and the other party makes a payment (or payments) when a specified credit event (or specified credit events) happens (happen). 7. No-Arbitrage Pricing Models Arbitrage: Investment requiring no capital outlay, but never losing money, and producing positive payoff with positive probability. In a one-period market, assuming a finite number of states of the world !

1,!

2,...,!

M, the

condition of the market being arbitrage-free means that any security, which makes only zero or positive payments in each state of the world, with positive payments made with positive probability, cannot have a zero or negative price. The simplest type of such a security is an Arrow-Debreu security

!ei making the following payments:

!1! 0

!2! 0 …

!

i! 1 …

!

M! 0

We cannot be certain that all Arrow-Debreu securities are available for trading in the market, but we do know this: linear combinations of securities (portfolios) generally (assuming short selling is allowed) are also securities. This implies that some Arrow-Debreu securities can be replicated with securities traded in the market. How many – that depends on the dimension of the vector space spanned by the securities traded in the market. Can this space be not of maximum (M) dimension? Yes, we do not know, in general, that we can place bets on all possible states of the world. The way we constructed this one-period model, all securities are elements of the M-dimensional Euclidean vector space. And their prices? It is important to note that if we write P for the price of a security, then for a linear combination of securities S

1 and S

2,

we have: P !S

1+ "S

2( ) = !P S1( ) + "P S

2( ). Therefore, price is a linear functional on the space of securities. We can work on infinite-dimensional generalizations of this, but practical applications have only finite-dimensional Euclidean spaces. If the dimension of the space spanned by traded securities is equal to the number of the states of the world, then we can find prices of all Arrow-Debreu securities – in this case


the market is called complete (assuming it is arbitrage-free). If all of their prices are positive, the market is arbitrage-free, and the vector of those prices is called the state-price vector. If the dimension of the space spanned by the traded securities is less than the number of the states of the world, then we can always extend the linear functional of existing prices to the whole space. Since the space is finite dimensional, so we can just do this step-by-step, adding a price value in each missing dimension, by pricing an appropriate Arrow-Debreu security, until we reach a functional defined on the whole space. This also shows that if the initial model was arbitrage-free, we can make the final model also arbitrage-free, by making sure that the prices of Arrow-Debreu securities we are adding are positive. Once all Arrow-Debreu securities are priced by the market and have positive prices, the market must be arbitrage-free and complete. In such a complete and arbitrage-free market a security, which is a sum of all Arrow-Debreu securities becomes available for trade. This security pays the same amount at the end of the period regardless of the state of the world. Such a security is called a risk-free security. We will denote its price by v and the value of a unit amount 1 invested in that security S

1 at the end of the period by 1 + i. Given that the Arrow-Debreu securities

form a basis for the vector space of all securities, the price of a security Sj paying Sj 1,! k( ) = Sj ! k( ) in the state ! j is:

P!ek( )!" #$ % Sj & k( )!" #$k=1

M

= Sj &1( ) %P!e1( ) + Sj &2( ) %P

!e2( ) +…+ Sj &M( ) %P

!eM( ) =

= Sj &1( ) %' 1+ Sj &2( ) %' 2

+…+ Sj &M( ) %' M .

In particular,

1 = P S

1( ) = P!ek( )!" #$ % 1+ i[ ]

k=1

M

= 1+ i( ) %&1+ 1+ i( ) %&

2+…+ 1+ i( ) %&

M.

If we define

P!ek( ) ! 1+ i( ) =" k ! 1+ i( ) = qk ,

then the qk ’s can be considered probabilities, the so-called risk-neutral probabilities. Fundamental Theorem of Asset Pricing. The single period securities market is arbitrage free if, and only if, there exists a state price vector, and this is equivalent to existence of a risk-neutral measure. Assume frictionless market for securities in which trades occur only at the times t = 0,1,2... Let i

t be the one-period risk-free interest rate, short rate. Let Vj ,t be the j-th

primitive security value (ex-dividend) at time t, and Cj ,t be the dividend (cash-flow payment) at time t for it. Assumption of no arbitrage in the market is equivalent to the existence of risk-neutral probability measure under which:

Vj ,t = Et

Vj ,t+1 + Cj ,t+1

1+ itVk ,t ,it ,1 ! k ! N

"

#$%

&'.

Going back by induction:


Vj 0( ) = ECt+1

1+ i j( )k=0

t

!t"0#

$

%

&&&&

'

(

))))

The model should be calibrated to existing default-free multi-period bonds. You can rewrite the above formula as:

Vj 0( ) = Pr !( )!"#$ Ct+1

1+ i j !( )( )k=0

t

%t&0$

'

(

))))

*

+

,,,,

Each ! "# represents a possible interest rate scenario (in the risk-neutral world). So … what’s the point? The point is: you must run interest rates scenarios, and scenarios of the future in general, modeling your financial intermediary via simulation, in order to properly establish valuation of derivatives issued by you. Exercise 7.1 The market model consists of 2 securities and a bank account. The bank account pays interest of 10% per year and is risk-free. The price of each security today is 100. There are three possible scenarios for the prices of the securities in 1 year:

Security X Security Y Scenario 1 220 0 Scenario 2 55 0 Scenario 3 0 250

Calculate the state price vector for this securities market model, if one exists. If there is no state price vector, explain why. Solution. We have the following system of equations implied by the existing price structure:

110!1+110!

2+110!

3= 100,

220!1+ 55!

2+ 0 "!

3= 100,

0 "!1+ 0 "!

2+ 250!

3= 100.

#

$%

&%

This solves to: !1=24

55, !

2=4

55, !

3=2

5. The state price vector is:

!1

!2

!3[ ] =

24

55

4

55

2

5

"#$

%&'.

Note that

q1

q2

q3[ ] =

24

55!1.10

4

55!1.10

2

5!1.10

"

#$%

&'=

=24

50

4

50

22

50

"

#$%

&'.


Exercise 7.2 You are given a securities market described by:

S 0( ) = 1 1[ ], S 1( ) =1 1.20

1 0.90

!

"#

$

%&.

Calculate the arbitrage-free price of a European put on the second asset with the strike (exercise) price of 1.10. Solution. We find the state price vector from the system of equations:

1 !"

1+1 !"

2= 1,

1.2 !"1+ 0.9 !"

2= 1.

#$%

This gives

!1=

1 1

1 0.9

1 1

1.2 0.9

="0.1"0.3

=1

3,

!2=

1 1

1.2 1

1 1

1.2 0.9

="0.2"0.3

=2

3.

#

$

%%%%%

&

%%%%%

The solution is

!1

!2[ ] =

1

3

2

3

"#$

%&'.

The put described in this problem has cash flows 0

0.20

!

"#

$

%& at time 1, and its price equals

1

3!0 +

2

3!0.2 =

0.4

3=2

15.


S 0( ) = 1 1[ ], S 1( ) =1 1.50

1 0.90

!

"#

$

%&.

Find a state price vector. Solution. Let !

1!2[ ] be the state price vector. Then we must have:

!1+!

2= 1,

1.5!1+ 0.90!

2= 1.

"#$


Hence 0.60!2= 0.50, so that !

2=5

6 and !

1=1

6. The state price vector is:

!1

!2[ ] =

1

6

5

6

"#$

%&'.

We also have

q1

q2[ ] = !

1" 1+ i( )

=1

!!2" 1+ i( )

=1

!

#

$%%

&

'((=1

6

5

6

#$%

&'(.

Exercise 7.4 You are given securities market described by:

S 0( ) = 1 1[ ], S 1( ) =1.10 1.20

1.10 0.90

!

"#

$

%&.

Find the risk-neutral probabilities. Solution. We can see that the risk-free rate is 10%. To find state-price vector, we write

1.1!

1+1.1!

2= 1,

1.2!1+ 0.9!

2= 1.

"#$

The solution

!1=

1 1.1

1 0.9

1.1 1.1

1.2 0.9

="0.2"0.33

=20

33,

!2=

1.1 1

1.2 1

1.1 1.1

1.2 0.9

="0.1"0.33

=10

33.

#

$

%%%%%

&

%%%%%

Using the notation: u = value of a unit invested in the risky instrument in the “up” state, d = value of a unit invested in the risky instrument in the “down” state, risk-neutral probabilities are:

q1= 1+ i( ) !" 1

=1+ i( ) # d

u # d=0.20

0.30=2

3,

q2=!

2" 1+ i( ) =

u # 1+ i( )

u # d=0.10

0.30=1

3.


S 0( ) = 1 2[ ], S 1( ) =2 3

2 1

!

"#

$

%&.


Is this market arbitrage-free? Is it complete? Solution. The risk-free security doubles, while the risky security appreciates by 50% or depreciates. Thus the risk-free security always beats the risky security, the market is not arbitrage-free. Because it is not arbitrage-free, it is not complete. You can also solve this by solving the system of equations for the state price vector, and you will see that the solution contains negative entries:

2!

1+ 2!

2= 1,

3!1+!

2= 2,

"#$

so that

!1=

1 2

2 1

2 2

3 1

=3

4,

!2=

2 1

3 2

2 2

3 1

= "1

4.

#

$

%%%%%

&

%%%%%


S 0( ) = 1 1[ ], S 1( ) =

1 0

1 2

1 0

!

"

###

$

%

&&&

.

This market is not complete. Find an Arrow-Debreu security, and its price, such that adding it to this market makes the market complete, while keeping it arbitrage-free. Solution.

Note that adding this vector 0

0

1

!

"

###

$

%

&&&

as a third column makes the matrix S(1) invertible.

Therefore it would be best to price this e3 Arrow-Debreu security. State price vector

must satisfy the equations !1+!

2+!

3= 1 and 2!

2= 1. This implies that !

1+!

3=1

2

and !2=1

2. Therefore we must have 0 <!

1<1

2, !

2=1

2, and 0 <!

3=1

2"!

1. The


price of 0

0

1

!

"

###

$

%

&&&

must equal !3. In fact, we can pick !

3 to be any number between 0 and

1

2, and then let !

1=1

2"!

3. We can, for example pick !

3=1

4, and the market

becomes

S 0( ) = 1 11

4

!

"#$

%&, S 1( ) =

1 0 0

1 2 0

1 0 1

!

"

###

$

%

&&&

,

clearly complete and arbitrage-free. 8. Option pricing Assume the underlying is a stock, although the reasoning works for any underlying. Let t be the time till expiration of an option, and assume that the stock price’s probability distribution over the period is a discrete distribution, defined as follows:

St= Se

ut with probability q, and St= Se

dt with probability 1! q. We assume that d ! r ! u , where r is the risk-free force of interest, and that if the stock price goes up, the option is worth fu , and if the stock price goes down, the option value is worth fd . We would like to construct a portfolio consisting of stock and risk-free asset (Treasury Bills), whose payoffs will equal the payoffs of the option in both the up and the down scenario. Since there are no arbitrage opportunities, the price of the option would have to equal the price of such portfolio. Let n be the number of shares of stock bought and B be the dollar amount of the risk-free asset, assumed to be short position. Then the price of the portfolio is nS ! B. In the up scenario, the stock part of the portfolio is worth nSe

ut, and in the down scenario, the stock part of the portfolio is worth nSedt . This gives

us the following system of equations: nSe

ut! Be

rt= fu , nSe

dt! Be

rt= fd .

By solving it we get the portfolio value (and thus the option value) as: nS ! B = e

!r"# qfu + 1! q( ) fd( )

where q = ert! e

dt

eut! e

dt. This is the basic binomial option pricing formula. It can be

generalized to the trinomial case (with three possible prices) following the same reasoning. If the interest rate is stated as an annual rate, the notation used is somewhat different: u denotes then the value of a dollar invested in the underlying in the “up” state, and d denotes the value of a dollar invested in the underlying in the “down” state, with i being the risk-free interest rate. Then the risk-neutral probability of the “up” state is


q =1+ i ! d

u ! d.

Black-Scholes formula Assumptions: All put-call parity assumptions, plus: • The continuously compounded rate of return on the underlying asset over the period of time of length t has normal distribution with mean µt and variance of ! 2

t. • Returns over disjoint time intervals are independent (this is also called the random walk model). Then the price of a European call with exercise price K and maturity time t

c = Se!"tN d

1( ) ! Ke!rtN d2( )

and the price of a European put with the same exercise price and maturity time is p = Ke

!rtN !d

2( ) ! Se!"tN !d1( )

where

d1=

lnS

K

!"#

$%&+ r '(( ) +

) 2

2

!"#

$%&t

) t, d

2= d

1! " t ,

and ! is the dividend force of interest on stock, r is the risk-free force of interest, N is the CDF of the standard normal distribution, and S is the current stock price. For a stock without a dividend you can also write the formulas for d’s in this nice form:

d1=

lnS

Ke!r"

#$%

&'(+1

2) 2t

) t=

lnS

PV K( )

#

$%&

'(+1

2) 2t

) t,

d2=

lnS

Ke!r"

#$%

&'(!1

2) 2t

) t=

lnS

PV K( )

#

$%&

'(!1

2) 2t

) t.

Exercise 8.1 A multi-period securities market model follows the binomial stock price model. You are given that u = 1.2, d = 0.92, i = 0.04, and the initial stock price is $20. Compute the arbitrage-free price of a European call option on the stock that expires in two period and has a strike price of $21. Solution. One period risk-neutral probabilities are:

qu =1+ i ! d

u ! d=0.12

0.28=3

7,

qd =u ! 1+ i( )

u ! d=0.16

0.28=4

7.

Over two periods, the stock can go:


• Up-up to 1.22 !$20 = $28.80, with call payoff of $7.80, and probability 37!3

7=9

49.

• Up-down or down-up to 1.2 !0.92 !$20 = $22.08, with call payoff of $1.08, and

probability 2 ! 37!4

7=24

49.

• Down-down to 0.922 !$20 = $16.93, with call payoff or $0.00, and probability 1649.

The expected present value, which is the arbitrage-free price of the call, equals 1

1.042

9

49!$7.80 +

24

49!$1.08 +

16

49!$0.00"

#$%&'( $1.81.

Exercise 8.2 You are given the following binomial model for the value of the short-term interest rate, risk-free over a one-year period: • One year from now this short-term interest rate is either r

1

u= r

01+ !( )with probability

0.50, or r1

d=

r0

1+ ! with probability 0.50, where r

0 in the initial short-term rate, and ! is

a parameter. The probabilities given are risk-neutral probabilities. • Annualized volatility of this short-term interest rate is ! = 25%. You are also given that in this binomial model u = e! t (the value of a dollar invested in the underlying in the “up” state) and d = e

!" t (the value of a dollar invested in the underlying in the “down” state). • The current value of the short-term rate is 4%. A 2-year European (meaning that it pays only if the short-rate breaches the floor at the end of two years) interest rate floor with a 3.5% strike level and a notional amount of 100 is issued. This derivative security will pay the difference between 3.5% and then current short-term interest rate as calculated for the 100 notional amount, if such difference is positive. (a) Calculate the value of this interest rate floor. (b) Now assume that the floor only pays the difference between 3.5% and the current short-term rate at the end of the first year, if such difference is positive. What is the value of such a one-year floor? Solution. (a) When t = 1, we have u = e! and d = e

!". In terms of the notation of this problem,

r1

u

r1

d= e

2!= 1+ "( )

2

.

We are given ! = 25%. This means that 1+ ! = e

"= 1.28402542.

Given this, the short-term rate will be in two years: 4% ! 1.28402542( )

2

= 6.59488508% with probability 0.25, 4% with probability 0.50,


and 4% ! 1.28402542( )

"2

= 2.42612264% with probability 0.25. Only the third outcome produces positive cash flows from the floor, which is then worth

100 ! 3.50% " 2.42612264%( ) = 100 !1.07387736% = 1.07387736. This cash flow is paid with probability 0.25, and its expected present value is the price of the floor. We use the risk-free rate over the next year as the rate for discounting for that year. Therefore the value of the floor is:

1.07387736

1.04 ! 1+0.04

1.28402542

"#$

%&'

!0.25 = 0.25034485.

(b) At the end of the first year, the short rate is: 4% ! 1.28402542( ) = 5.13610167%with probability 0.50,

and 4% ! 1.28402542( )

"1

= 3.11520313%with probability 0.50. Only the second outcome gives rise to a payment by the floor, such payment being 0.38479687 on the 100 notional amount. Its expected present value equals:

0.38479687

1.04!0.50 = 0.1849985.

If the floor were a two-year floor, inclusive of the first and second year, its total value would be 0.25034485 + 0.18653868 = 0.43688353. Exercise 8.3 You are given the following information about a call option on a certain stock: current stock price: S

0= 100, exercise price: X = 95, interest rate: r = 10%, continuously

compounded (i.e., force of interest), time to expiration: t = 0.25, standard deviation of the rate of return of the underlying stock: ! = 0.50. Find the price of this call option using the Black-Scholes formula. Solution. We have

d1=

ln100

95e!0.25"0.10

#$%

&'(+ 0.25 "

0.502

2

0.50 0.25) 0.43,

d2=

ln100

95e!0.25"0.10

#$%

&'(! 0.25 "

0.502

2

0.50 0.25) 0.18,

so that N d1( ) = 0.6664, and N d

2( ) = 0.5714. Thus the value of the call option is: c = 100 !0.6664 " 95e

"0.25!0.10!0.5714 = $13.70.

Exercise 8.4 On January 1, 2013, $10,000 is invested in a Cuba Libre stock selling at 50 libretas per share. The exchange rate is 2 libretas per dollar. At the same time, a contract is entered


into to deliver 20,000 libretas for dollars on December 31, 2013, at the forward rate of 1.90 libretas per dollar. On December 31, 2013, the stock is selling at 60 libretas, and the exchange rate is 1.80 libretas per dollar. Ignoring transaction costs, calculate the dollar-denominated rate of return, and the libretas-denominated rate of return. Solution. One important thing to remember is that forward contracts do not require a cash outlay, except for a commission, if any. To calculate the rate of return, we need to concentrate on cash flows. Initial $10,000 investment bought 20,000 libretas, and this purchased 400 shares. At the end of the year, these 400 shares were worth 24,000 libretas. The investor will deliver 20,000 libretas of that amount as fulfillment of the forward contract, and receive for it $10,526.32 (at the exchange rate of 1.90). The remaining 4,000 libretas will have to be exchanged at them prevailing rate of 1.80, giving $2,222.22, and the total dollar amount received will be $12,748.54. On a $10,000 investment, this represents a total rate of return of 27.49%. For the second part, January 1, 2013, libretas cash outlay was 20,000. The investor bought 400 shares of the stock, as it was selling for 50 libretas. On December 31, 2013, there is a libretas cash outlay of 20,000 as required by the forward contract. The investor also receives the payment in dollars for that delivery, at the exchange rate of 1.90, which means that $10,526.32 is paid to the investor, and at the current rate of 1.80, this is worth 18,947.37 libretas. There is also a cash flow of 24,000 libretas from the proceeds of the sale of the stock, so the net cash flow in libretas is: 18,947.37 + 24,000 – 20,000 = 22,947.37. The rate of return is 22,947.37/20,000 – 1 = 14.73684%. This is very different than the rate of return in U.S. dollars above, and that is understandable, because libreta has appreciated sharply during the period under consideration. Exercise 8.5 The following two securities coexist in the market with $1000 Treasury Bill maturing one year from now with 6% annual yield, and have the same price as that T-Bill: • A European call option on 10,000 shares of stock in Company ABC at a strike price of 10 with expiration date exactly one year from now. The risk-neutral probability of stock price being 10.50 at expiration date is p

1. Otherwise, the stock price will be 10 or less.

• A one-year forward on 2500 bushels of wheat that will enable purchase at 30 per bushel at that date. Analysts expect that the price of wheat will be at 37 with probability p2

or at 31 with probability p1. Otherwise, the price of wheat will be at 28.

Calculate the value of p1.

Solution. $1 invested in Treasury Bill will be worth $1.06 a year from now no matter what, so that1+ i = 1.06 . If on the exercise date, the price of the stock is $10.50, the call on 10,000 shares will be worth 10,000($0.50) = $5,000. If the price of the stock is $10 or less, call will expire worthless. Thus a $1 invested in the call will mature with the value of $5 in the up-state, and $0 in the down state.


p1=1+ i ! d

u ! d=1.06 ! 0

5 ! 0= 0.212.

Exercise 8.6 The following two securities have the same current price of 1000 as a Treasury Bill maturing one year from now with 6% annual coupons and a face amount of 1000. • A European call option on 10,000 shares of stock in Company ABC at a strike price of 10 with expiration date exactly one year from now. The risk-neutral probability of stock price being 10.50 at expiration date is p

1. Otherwise, the stock price will be 10 or less.

• A one-year forward on 2500 bushels of wheat that will enable purchase at 30 per bushel at that date. Analysts expect that the price of wheat will be at 37 with probability p2

or at 31 with probability p1. Otherwise, the price of wheat will be at 28.

Calculate the value of p2.

Solution.

We know from the previous problem that p1=53

250= 0.212. The forward contract will be

worth • $17,500 with probability p

2,

• $2,500 with probability p1,

• – $5,000 with probability 1 – p1 – p

2.

We have

1000 = p2

17,500

1.06+ p

1

2,500

1.06! 1! p

2! p

1( )5000

1.06.

Therefore, substituting the value of p1

6060 = 22,500p2+ 7,500p

1= 22,500p

2+ 7,500

53

250= 22,500p

2+1590,

resulting in

p2=4470

22,500= 0.1987.

9. Insurance Pricing and Valuation All insurance contracts, in one form or another, transfer the risk of the financial consequences of a future uncertain event from one party (the insured) to another party (the insurer), in exchange for some form of financial consideration. Examples include:

- Life insurance, where a fixed payment is made upon death of the insured. - Life annuity, where a prescribed series of payments is made until the death of the

insured. - Property insurance, e.g., automobile insurance or home insurance, where

reimbursement for losses is made upon the occurrence of accidental events. - Liability insurance, e.g. auto bodily injury liability or medical malpractice

insurance, where the insurer covers the financial consequences of at-fault events.


- Reinsurance, where the insured party (the ceding company) itself is an insurance firm, and the insurer (reinsurer) is a firm in business of providing insurance coverage to other insurers.

The business of insurance is viable only if the risk transfer resulting from the existence of the insurance contract brings about some reduction in the severity of the financial consequences of future events. But as the events themselves are unchanged by the existence of insurance contracts (unless the contract changes the participants’ incentives to prevent losses, which in fact is a serious practical issue in the business of insurance), the reduction occurs in the variance of the financial position of the insured. In fact, the reduction is achieved by changing the relationship of the consequences of risk to the participants’ capacity to bear it. The insurer removes the unknown consequences, or at least a part of them, from the insured party’s future for a known price in the present. The insurer combines the risk exposures of numerous parties insured, and collects their payments (premiums), thus making the risk more predictable (because of combination of many, mostly uncorrelated, risks) and more bearable (because of the combined financial resources from collected premiums). There are also more consequences in the longer run. Because of less exposure to unexpected financial consequences of risk, the insured parties can now assume a more risky posture in their business and personal activities (this is commonly referred to as moral hazard when the altered activity affects the probability of loss, sometimes deliberately: in the case of fraud), thus undertaking more projects and projects of larger scale. This, in turn, can benefit the entire society if it expands the set of opportunities in terms of production output, or other desirable activities, but it may also create more risk, because of less restraint on the part of the insured participants. The science of mathematical models applicable to insurance, and financial risk modeling in general, is known as actuarial science. Actuaries are financial professionals who are typically employed by insurance companies, and other entities involved in financial consequences of risk (e.g., consulting companies, governments, etc.). Actuaries perform the following key duties:

- Setting the premiums for insurance products. While the process of pricing insurance is a dynamic one, involving many parties, including the market for insurance products, insurance company management and marketing divisions, insurance regulators, and others, actuaries create the core portion of this process by balancing the premium income of insurance firms and the payments made by the insurance firms for benefits, claims, expenses, and distribution of profits to the firm owners.

- Setting insurance reserves. Reserves are defined as expected future payments on policies already underwritten and currently in force. They are the liabilities of insurance companies that represent claim and expense payments, or benefits promised to the insured parties, when the events named in the insurance policies, occur.

- Assuring solvency of the insurance firm, in both short-term context and in the long-run. This solvency requirement does not just mean the standard ability to make scheduled payments, which applies to all firms, but also an additional requirement of appropriate level of surplus (or capital), defined as the excess of the firm’s assets over the liabilities, and needed to assure the payment of all


insured obligations with high probability. Insurance firm’s liabilities consist nearly entirely of reserves. When an insurance premium is collected by an insurance company, a portion of it is used to pay expenses, such as marketing, or administrative, or salaries of actuaries (this is a sizable expense, as actuaries are consistently among the highest paid professionals) and other personnel, while the rest is placed in reserve to pay future claims or benefits. In addition to making certain that those reserves are large enough to cover actual promised payments, actuaries have to assure that assets and liabilities of the firm are managed properly, so that the difference between them (the surplus) remains at a level required by insurance regulators, or higher. This form of management of an insurance firm (or any other entity involved in risk management) is termed asset-liability management, and is a part of a larger field of enterprise risk management, a field of study of proper management of risky activities in which any business entity can and should be engaged.

Insurance Pricing Fundamentals The key principle of private insurance contracts is a form of a “law of conservation”: the totality of funds that are paid out to the insured parties must be originally obtained from the same insured parties in the form of premium payments. While the owners of the insurance firm do provide capital to start the company, and possibly additional capital for its continuing functioning, they do expect the opportunity to earn a return on their investment comparable with that available from other sources in the markets for capital (capital markets) of similar risk. Thus insurance consumers as a whole cannot expect to be subsidized by the insurance firm, but rather should expect to pay for their own claims and expenses and, additionally, for the cost of the capital supplied to the insurance firm by its owners. Actuaries begin the process of pricing insurance by forecasting the future costs of claims in property, casualty and liability insurance, or benefits, in the case of life, disability, or health insurance, and life annuities. For example, in the case of life insurance, this begins with the study of the future lifespan of a given insured. If we denote by T the future length of life (a real number) of an insured aged x, then the cost of a life insurance policy in a specified amount is the present value of that amount discounted from the moment of payment (upon death or just immediately following it), T years in the future (or a number close to T) to the present moment. This cost is likely to be increased by all policy expenses, such as marketing, administrative, and settlement expenses. Furthermore, starting an insurance enterprise, or issuance of new policies, requires an outlay of capital by the enterprise owner or owners, and that capital must be paid for. This cost of capital becomes an additional expense added to the cost charged to the insured party. If the insurance premium is set as the expected value of the random variable describing future payment of benefits/claims and expenses, this method of pricing is called the Equivalence Principle. This most basic principle of pricing insurance calls for the premium to be set at the level equal to the expected value (or mean) of future payouts, modeled as random variables. The process of diversification of many risks combined from various insured parties makes the average payout approximately equal to the theoretical expected value,


as a consequence of limit theorems from the probability theory, such as the Central Limit Theorem, and the Law of Large Numbers. But this diversification may not always be enough, and some provision for what actuaries call the adverse deviation (deviation of the actual amount of total claims or benefits from the expected amount, in a manner detrimental to the insurance company) must be made by appropriately increasing the insurance premium to cover this risk to the insurer’s capital. Thus the premium is typically set as the expected value of claims or benefits, adjusted for discounted value of money, plus the expected value of all expenses and taxes, plus a provision for risk of adverse deviation, commonly called the risk loading, covered by the proper estimate of the cost of capital commensurate with the total risk of the insurer. The key part of the calculation of the insurance premium is the estimate of the future losses or benefits. In the case of life insurance, since the amount to be paid is set in advance, the uncertainty is only twofold: unknown time of death and unknown rate of return that the firm will earn on the premium or premiums collected. While the simplest method of calculation of life insurance would use a one amount paid upfront (single premium), real life policies are nearly always paid for with a series of periodic (annual, quarterly, or monthly) premiums. Those periodic premiums are paid for a set period of time (e.g., five or ten years) but only if the insured is alive, or for the entire remaining life of the insured person. This means that in the calculation of premium not only the time of payment of the death benefit is uncertain (modeled as a random variable T, remaining time until death), but so is the length of time over which the premium will be paid. Luckily, the underlying random variable, T, is the same for both phenomena, although its practical implications on the present value of the death benefit, and the present value of the remaining premium payments, are different. Ideally, estimates involving the random variable T would be based on data concerning exact length of life of all people in the population. But historically, accurate data about the exact length of life has not always been easy to collect, and instead annual data (expressing the length of life in full years) has been common. A table starting with a given population of newborn persons in a given year, and then showing the population alive at any future age is called a mortality table. Population alive at age x is denoted by l

x and number dying between ages x and

x +1 is denoted by dx. The ratio qx =

dx

lx is a natural estimate of the probability of dying

between ages x and x +1, and px= 1! q

x is the natural estimate of the probability of

surviving that year. The first mortality table is generally attributed to Sir Edmund Halley, who in 1683 created it for the city of Breslau (now Wroclaw, in Poland). In the United States, commonly used tables are generally created by the Society of Actuaries, usually utilizing data collected in the National Census. Insurance companies commonly undertake their own mortality studies, in order to better understand the risks of the populations they insure, and subsequently modify the published tables. The other uncertain element in pricing of life insurance (or other products related to human mortality) is the interest rate that will be earned on the insurance company investments. Modeling of that rate of return is complicated enough if the period over which the return is considered is fixed, e.g., one year. One period models of rates of


return are usually derived based on an equilibrium approach, or arbitrage-free approach. The equilibrium models take into consideration consumption and risk preferences of the participants in the economy, and derive expected rates of return corresponding to varying levels of risk. Its crowning achievement is the Capital Asset Pricing Model (CAPM), widely utilized, but derived under severely restrictive assumptions, and thus limited in its applicability. CAPM is basically a one-period model, and thus of very limited applicability to business of life insurance, life annuities and pensions, which are, by nature, long-term and multi-period. CAPM states that the expected rate of return on a stock is given by the formula E R( ) = r

F+ ! " E R

M( ) # rF( ), where R is the random rate of return of the stock under consideration, R

M is the random

rate of return of the overall market of all risky assets, and the coefficient beta is

! =Cov R

M,R( )

Var RM( )

.

The second approach to modeling rates of return, the no-arbitrage approach, is rooted in the idea that capital markets would not allow arbitrage to exist, at least not in any persistent fashion. Arbitrage is defined as creation of an investment portfolio, which does not require any outlay of funds, yet allows positive returns with positive probability, and never loses any money. In other words, arbitrage is a “free lunch.” Given the no-arbitrage condition, two investment portfolios, which generate the same cash flows in the future, must have identical prices today. The no-arbitrage approach starts with observed prices and rates of return of assets traded in capital markets, and attempts to derive appropriate rates of return for other assets of comparable risk not directly priced by the market. These two methodologies are of significance to insurance, because insurance products are not continuously traded in the markets, but they are priced by insurance firms, and their prices must relate to risk tolerance and other preferences of market participants, as well as prices of similarly risky capital assets traded in the markets. In other words, an actuary deriving an insurance premium must be aware of the value placed on the insurance product by the firm’s customers, and prices of capital assets that can be possibly used to replicate some, or even all, of the features of the insurance product under consideration. The actuary must make certain that the estimate of the interest rate used in the calculations of present values of future cash flows of the policy under consideration correspond realistically to the interest rate that will be earned on the company’s investments, after consideration for possible additional investment expenses and taxes. Methodologies of life insurance and life annuities are quite naturally extended to the area of pricing and planning of retirement. Retirement is usually expected to be funded by a combination of government social insurance pension, employer-sponsored pension or savings plan, and private savings. The problem of providing appropriate amounts of savings for the purpose of obtaining desirable level of income upon retirement, is a natural actuarial model problem, but compounded not just by the uncertainty of the length of life, but also the length of period of employment, date of retirement, desired level of income replacement (in relation to pre-retirement income) upon retirement, as well as


additional complications of possible disability, and provision for the spouse and the survivors of the individual under consideration. Life insurance models do not, however, extend naturally to insurance coverages of property, protection against liability, group insurance including group health insurance policies, or workers compensation insurance. Life insurance policies are typically issued for long-time periods, possibly the entire life of the insured party. In contrast with that approach, automobile insurance or homeowner’s insurance, is usually issued for a relatively short period of time to individuals (personal) and to corporations (commercial), half a year to a year, and require a different modeling approach. Actuarial models for these forms of insurance covering accidental and at-fault events forecast the random variable counting the number of claims per insured received in the period of insurance, termed frequency, and separately the probability distribution of the size of those claims, termed severity. The two random variable so modeled are then combined in a collective risk model of the firm

S = X

1+ X

2+…+ X

N,

where S is the aggregate claims random variable, while N is the claim frequency random variable, and each X

i represents individual claim severity. Examples of commonly used

probability distributions describing N are: the Poisson distribution, the binomial distribution, and the negative binomial distribution. If the random variables X

i are

assumed to be identically distributed and independent, the resulting distribution of S is termed a compound distribution, derived from combining the distribution of N and the distributions of X

i.

Estimation of the frequency and severity distributions is an integral part of the work of an actuary in the areas of property, casualty, liability insurance, and other similar forms of insurance. These estimates are continuously updated based on the claim data, as well as other data collected by insurance enterprises. The challenge is additionally complicated by the fact that not all losses are covered by insurance contracts, and even those covered are typically not covered in full (with the use of deductibles, i.e., amounts paid by the insured party before insurance coverage starts, or co-insurance, requiring the insured party to share in the payment for the losses), thus the actuary does not always have full access to the data describing the losses. Furthermore, the cost of items or events insured changes continuously. This is due to inflation, but also due to changes in relative prices in items or events insured. In the United States, for example, health insurance industry struggles with increases in costs of health care well in excess of overall inflation, as well as with nearly continuous introduction of new medical technologies and new prescription drugs, which may have not been considered in historical models used for pricing of health insurance. Liability insurance, especially policies covering general (pain and suffering) and punitive damages, are not tied closely to general inflation but, rather, to current and future laws and their interpretations by the court. Thus the actuary must consider not just the estimates for frequency and severity based on historical data, but also adjustments to those estimates for the trend, or changes, of the cost of coverages provided. This requires development of forecasting methodologies for projecting future costs of claims. Standard forecasting methodologies are typically based on either


regression or time series, both created within probability theory. Basic linear regression models assume that a predicted random variable Y is related to a predictor variable X via a linear model of the form Y = a + bX + !, where a and b are parameters derived in the model estimation, and ! is a residual random variable, typically assumed to be normally distributed with mean zero and relatively small standard deviation. If empirical values of the predictor variable X are x1, x

2,…, x

n and the corresponding value of the predicted variable Y are

y1, y2,…, y

n then

the standard methodology for estimation of parameters a and b is to minimize the Euclidean distance or mean square error

yi ! axi + b( )( )2

i=1

n

" .

This approach dates back to Gauss. The relationship between the predictor and the predicted variable can be generalized to allow either X or Y to be replaced by functions of them. For example, if we know that Y is expected to grow exponentially with X then it would be natural to consider a model of the form lnY = a + bX + !. One can also have a general multivariate model of the form

Y = a + b

1X1+ b

2X2+…+ b

mXm+ !.

Note that in this general model, given values X1= x

1, X

2= x

2, …, X

m= x

m, the

predicted value of Y given those predictor variables values is

y = a + b

1x1+ b

2x2+…+ bmxm ,

as the expected value of the residual is zero. Thus this model gives the predicted value as the mean of the probability distribution of Y given that X

1= x

1, X

2= x

2, …, and

Xm= x

m. One can, in fact, generalize this approach to regression analysis to non-

parametric regression models, under which the distribution of the residuals is allowed to be completely arbitrary, instead of the normal distribution, and if that arbitrary distribution can be somehow estimated or theoretically established, the predicted value of Y is the mean of its conditional distribution. The second set of methodologies deals with the situation when the variables modeled are time-dependent, so that their historical observations do not constitute independent observations of the same random variable. Time series analysis takes into consideration the time structure of data, and it accounts for phenomena such as autocorrelation, trend or seasonal variation, all of which are common in real life insurance data. For example, an autoregressive (AR) model, which assumes a regression-type relationship of the value of a variable X

tat time t to the preceding values:

Xt= 1! "

i

i=1

p

#$

%&'

()µ +"

1Xt!1 +…+"

pXt! p + *t .

One additional set of prediction methodologies has been created with the arrival of the fuzzy set theory. A fuzzy set !E is defined by its membership function µ

E in the universe

of consideration U, so that for every element u of U, there is a value µEu( ), where


0 ! µEu( ) ! 1. Prediction methodologies typically utilize the concept of a fuzzy number,

for which U is the set of all real numbers, and use special generalizations of arithmetic operations developed for fuzzy numbers. A vitally important area of actuarial science for both life and non-life insurance is risk classification. If two insured parties have significantly different risk profiles in relation to expected claims, yet are charged the same premium for insurance, one of them (the low risk party) effectively subsidizes the other one (the high risk party), and if the insurance contracts are voluntary, the low risk party will avoid obtaining insurance or minimize its amount, while the high risk party will seek to maximize the coverage. If the insurance coverage is desirable for public policy reason, this adverse selection is commonly resolved by making the purchase of the insurance contract compulsory, and often administered by a government entity. But in private voluntary markets, when competition of insurance providers is present, the insured parties must be classified in reasonably homogeneous risk classes, within each of which the twisted incentives of inequitable premiums are no longer present. This requires that actuaries collect data concerning potential risk classes, and classify insured parties accordingly. Various approaches have been developed to address this problem. Bayesian methodologies adjust the premium based on observed experience. For contracts that bundle insured events (perils), such as auto liability, damage to vehicles, and theft, the diversification benefit (low cross correlations) becomes an important risk class pricing variable. Credibility theory treats the premium rate for a given insured party (usually a group) as a weighted average of a premium derived based on that party’s experience, and of a premium rate derived for a general population. The credibility weight assigned to the party’s experience is reflective of the accuracy of the empirical sample mean as a predictor of the true mean. Classification methods can also be derived from methodologies used in other areas of mathematics. General and fuzzy clustering, principal component, and kernel smoothing algorithms have been proposed and used for risk classification in insurance. Reserves Once an insurance contract is in places, and premium is collected, some portion of that premium must be placed in reserve for the purpose of payment of future benefits, claims and expenses. Life insurance, life annuities, and pensions, as well as long-term health insurance contracts (such as disability insurance, and non-statutory private health insurance in Germany, which is a type of contract not in existence in North America) all have a long-term nature, with risks generally increasing with age, but with premium set in advance, and rarely changed over time, and even when changed, generally not changed as rapidly as the increase in risk occurs. Such contracts must effectively have a level of premium, which is too large in relation to risk in the early part of the policy, and too small in the later part. As a result, a reserve must account for this divergence between the premiums and the payments made by the insurance company. In actuarial terminology, the expected value (i.e., the probability mean) of the present value of future cash flows (i.e., accounting for the time value of money and the risk of adverse development) is termed the actuarial present value. The most standard formula for the reserve in all forms of long-term insurance contracts is the difference between the actuarial present value of the future benefits, claims, and expenses to be paid, and the actuarial present


value of future premiums to be collected. Of course, if the future premiums were sufficient to pay future benefits, claims and expenses (including the cost of capital) at all times, reserves would not be needed. The only long-term contract, which does not use such approach to reserving, is the deferred annuity contract in the accumulation phase in the United States, which requires the reserve to be the highest possible present value (not adjusted for any probability of occurrence) of guaranteed future account balances under the contract. The Commissioners Annuity Reserve Valuation Method required for deferred annuities by the insurance regulators in the United States is a unique exception in the actuarial methodology of reserving: it actually never uses any probability concepts. The process of calculation of reserves for long-term contracts is commonly called valuation. The interest rate used in the process of calculation of present values in that process is called the valuation interest rate and the mortality table used is the valuation mortality table. In combination, the interest rate and mortality table form the valuation basis. It would seem natural that the judgment concerning the mortality table and the valuation rate belongs with the actuarial professional. This is generally the approach adopted in Great Britain, Australia and Canada. However, in many countries, including the United States (until 1980, and, to lesser degree, still so), this decision is taken away from the actuary, or even from the insurance firm management, and instead, the mortality and interest rate parameters are prescribed by law. The process of valuation based on the methodology prescribed by law is called the statutory valuation and is required in the United States of all insurance companies for the purpose of submission of their financial statements to the insurance regulators. Insurance is regulated in the United States separately in each state, and statutory valuation reports must be submitted to each state in which an insurance firm is engaged in business of insurance. Interestingly enough, the valuation methodology required of insurance companies, which issue their shares for trading in public stock exchanges in the United States (e.g., the New York Stock Exchange) is different, and prescribed by the Generally Accepted Accounting Principles (GAAP). To make things even more complicated, and possibly to create more employment opportunities for actuaries, the accounting rules for the calculation of the income tax due to the federal Government of the United States are different than the statutory, or GAAP rules, and prescribed separately in the tax laws and their interpretations by the tax agency, Internal Revenue Service. These peculiar regulatory and accounting complexities in the insurance industry in the United States are perceived by some as barriers to entry for foreign insurers. Private pensions in the United States are, however, regulated by the federal Government (Department of Labor) and generally use a valuation basis chosen by the pension plan actuary, on the basis of that actuary’s professional judgment. The GAAP rules for pension plans are, however, different than the valuation for regulatory purposes. The last quarter century has witnessed increased interest in making the reserving methodologies less command-based (with formulas and valuation bases prescribed by law) and more principle-based. This has been especially important in view of dramatic changes in the level of interest rates experienced in the 1970s and 1980s, as well as improvements in longevity of general population, making the older mortality tables, still


used for older policies (as the valuation basis for a long-term policy in the United States is assigned to a policy based on the date of its issue, and remains unchanged unless a change would result in a increase in statutory reserves), quite obsolete in many cases. The principle-based approach to reserving usually requires a complex long-term model of the insurance company. Since 1991 in the United States, such long-term models are effectively required for most long-term contracts companies, i.e., life insurance, life annuities, etc., and the process of creating them is called cash flow testing. In the long-term model, the set of cash flows generated by an insurance firm is generally treated as a stochastic process (i.e., a series of time-dependent random variables), with financial outcomes, such as payments of benefits and claims, payments of expenses, profits generated, and the level of surplus held (assets minus liabilities) modeled in each realization of the stochastic process generated. The stochastic process under consideration is influenced the most by the future scenarios of interest rates, but it is also affected by random outcomes of mortality or other basis for benefit payments (e.g., payments of any amounts, known as nonforfeiture amounts) upon policy termination. All of these phenomena must be modeled by the valuation actuary. The resulting set of generated scenarios of the future creates an empirical distribution of financial outcomes describing company’s solvency under all of the scenarios. Actuaries say that the company passes a scenario if it remains solvent during its entire duration. The minimum period modeled is ten years, although the frequency of cash flows considered during those years need not be very high (quarterly cash flows can be acceptable, and hourly, or even daily or weekly, cash flows are generally not required). Long-term insurance firms are required to pass seven scenarios of the future prescribed by the regulators (those scenarios are termed the New York 7, because they originate from seven scenarios considered in the Regulation 126 in the State of New York), and an overwhelming majority (e.g., 95%) of random scenarios generated by the insurance company internal model. In effect, the regulators want the insurance company to remain solvent with 95% probability, based on the large random sample of the probability distribution describing future financial situation of the company. This regulatory approach has created significantly increased demand for applications of stochastic processes to the modeling of insurance firms. For the short-term insurance policies, such as automobile insurance or homeowner’s insurance, or workers’ compensation, the emphasis in reserving is not on the long-term discrepancy between payouts to be made and premiums collected, but rather on the claim payments that must be made within the remaining short-term of the policy. Because for short-term policies the premium is typically paid upfront (e.g., for a six-month automobile policy the premium payment occurs at the beginning of the six-month period), payments to be made cannot be offset by any future premiums. The emphasis is therefore on forecasting the payments. Those payments come in two major categories:

- Payment yet to be made on claims that have already occurred, and have been reported to the insurance company, and

- Payments yet to be made for claims that have already occurred, and have not yet been reported (commonly called Incurred But Not Reported, or IBNR).

If a claim has already been reported to the insurance company, and reasonably well evaluated, it is generally not necessary to use any probability-based methods to estimate


their values. But for a claim that is entirely unknown to the company, some form of estimation must be made. The traditional approach to this problem has been completely deterministic (i.e., devoid of any probability applications). An actuary establishing IBNR considers the period of time since a theoretical claim has occurred and, based on the company’s own data (from historical experience) estimates how long it will take for that claim to be reported and fully paid, and what portion of it will be paid at what moment in time. Then the actuary applies the knowledge so obtained to all data about claims already known and claims not yet known, assuming that the estimates can be applied to the current situation. By applying these estimates, the actuary projects what the ultimate full amount will be paid, and compares it to the amount that has already been paid. The difference of the two represents the amount that will be paid in the future on the claims already in existence. That difference is the IBNR reserve, the largest liability item in a typical property/casualty insurance company financial statement. The process of paying the claim from the date when the claim is incurred to the date when it is fully settled is called development (in property/casualty insurance) or completion (in health insurance). The last quarter century has witnessed a gradual increase in interest in applying probability-based methodologies to estimation of the IBNR reserves and loss development in general. In such probability-based approaches, the final amount to be paid is typically modeled as a random variable dependent on, at the very minimum, time, and then typically in addition to time, other variables describing the process of development, or completion, the entity insured, the nature of the claim, etc. Regression-based models are most common. In property-casualty insurance reserving, models often use hundred of variables, including interactions of those variables, and all such variables must be always very carefully examined for multi-colinearity, i.e., dependence of the variables on each other, which reduces or even eliminated model’s predictive power. Asset-Liability Management The cash flow testing models required in life insurance and life annuities in the United States for the purpose of establishing statutory reserves are an example of the expansion of sophisticated asset-liability management models that gradually have entered insurance practice in the last quarter-century. The practice of asset-liability management began in response to increased volatility of assets held in insurance companies’ portfolios in the 1970s and 1980s. Varying interest rates were initially the greatest concern. Insurance companies have traditionally provided long-term guarantees of interest rates paid on policies used for accumulation of wealth for retirement, but those guarantees were at relatively low levels. When interest rates rose, those policies became unattractive, and were abandoned by their owners in the process of disintermediation, i.e., flight from low-interest rate insurance and bank products to higher-return investment products, or from low credit quality intermediaries to those perceived as higher credit quality. Insurance companies responded by offering higher rates of return, and pursuing higher returns themselves by investing in riskier bonds and mortgages. High rates of return in the stock markets have been countered by offering new variable annuity products tied to the performance of the stock market. But some of those strategies of the insurance firms have resulted in a significant increase of their risk exposure, and this was a new type of exposure: not to diversifiable risk of insuring individuals or firms against death, or perils, but to nondiversifiable risk of the bond market (both in the form of the risk of changing


interest rates and the risk of default of risky bonds) and the stock market. In 1991, the United States life insurance industry experienced two insolvencies of large and established insurance firms: Executive Life and Mutual Benefit. Both of these companies had sizable portfolios of risky assets that resulted in a panic of withdrawals from their retirement-type policies when risky investments declined in value and rumors of insolvency spread. Those developments illustrated the dangers of any divergence between the value of insurance firm’s assets and liabilities. The difference between assets and liabilities, the surplus (or capital) is carefully watched by insurance regulators. Since the mid-1990s, the level of surplus required is a function of risks undertaken by the company, in its asset portfolio, in the structure of the insurance products it offers, and in the interaction of its assets and liabilities. While early models used for asset-liability management called for elimination of risks of divergence of assets and liabilities, by matching assets and liabilities cash flows, in a process called immunization, or matching the values of assets and liabilities under changes of interest rates, recent developments in this area are more significantly based in probability models. One particularly important area of significance for understanding of asset-liability management is the study of options embedded in the insurance contracts and the relationship between the insured party and the insurer. Life insurance policies, as well as life annuities, traditionally contain minimum interest rate guarantees. When interest rates fall, such guarantees are equivalent to the option to purchase a bond with a coupon at the level of minimum interest rate guarantee, regardless of the current level of interest rates. Life insurance policy or disability insurance policy can be viewed as an option to receive a certain monetary value when the human capital (the ability to generate income through work) of the insured person disappears due to death or disability. But most importantly, the insurance company creates an option by the very process of issuance of insurance contracts. The policyholder is promised certain monetary values upon occurrence of insured events. This promise will be kept only if the insurance company is still in business. If the company is not in business, the insured can only make a claim on company assets. In effect, the insurance company holds an insolvency put (an option to sell at a predetermined price) on its own assets. If the value of the assets exceeds the obligation to the policyholder, the insurance company has an incentive to make good on that obligation. But as soon as the value of the assets falls below the value of the obligation, under limited liability the insurance company can just walk away from the obligation and let the policyholders take and divide the assets instead. This is, of course, the very reason for the existence of insurance regulation. Government regulates insurance firms, and prescribes their (minimum) level of surplus, because as soon as that surplus becomes negative, the insurance firm has little financial incentive to serve the best interests of its policyholders. Practical evaluations of various options embedded in insurance contracts generally follow some variation of the Black-Scholes methodology, or its simplified version, the binomial model, using a stochastic process derived from the binomial probability distribution. This


is also the underlying theoretical justification of the cash-flow testing methodologies used by long-term insurance companies, and dynamic solvency testing models used for long-term models of companies issuing short-term insurance contracts. Property/Casualty Insurance Pricing Models Insurance pricing begins with the fact that both the cost and the expected return (profit) are not known with certainty. Rather, the ultimate cost derives from a stochastic process that commences when the insurance is purchased and is resolved when final claim, expense and tax payments are made. The ultimate profit too results from an interrelated stochastic process, which depends on the ultimate costs, the premium charged at the time of sale, and the results of invested premiums and capital. Thus, the determination of an appropriate premium by the actuary or available in the market, prior to sale, is key to the opportunity to earn a profit consistent with the risk of adverse development. Of course, in complete or workably competitive markets, where equilibrium supply and demand prices are equal, those prices may or may not agree with the actuary’s calculation of expected profits. We turn next to two comprehensive pricing models developed for insurance in the past thirty to forty years and illustrate their use in the property-casualty context. Policyholder Demand Side Model A policyholder purchases insurance to trade risky and uncertain future adverse financial events for the almost (because of insolvency potential) certainty of a premium payment. In decision theory, the premium would be the certainty equivalent of the uncertain future liabilities. Insureds should be willing to pay a present value of premiums equal to the present value of expected future claims, expenses and taxes, adjusted in the discount rate for the risk of adverse deviations from the expectations. The key to the demand model is then the identification of all expected payments to (claims), or on behalf of (expenses and taxes), the policyholder. Proper policyholder demand pricing models should not directly include consideration of the insurer’s invested assets and expected investment income above the risk-free rate. Policyholders are generally unwilling, except in explicit investment-insurance linked products such as variable annuities, to trade uncertain adverse accidental events for certain premiums plus uncertain insurer investment returns. Policyholder demand models assume the presence of a supplier at the policyholder demand price. The paradigm demand model, developed at MIT by Stewart C. Myers and Richard A. Cohn, posits that the appropriate premium P at the beginning of the policy equals the net present value (NPV) of losses, L, expenses E, and taxes T (because of double taxation in the U.S., i.e., taxation of profits of corporation, and then taxation of distributions of those profits when received by the company’s shareholders) PV L + E + T( ) where PV incorporates a negative adjustment to the risk-free rate to provide the necessary profit incentive for the insurer assuming the risk. In practice, those risk-adjustments have been difficult to model and calculate from empirical data.


Shareholder Supply Side Model Insurers offer policies precisely to capitalize on the diversification benefits of pooling uncertain adverse future financial event consequences from personal and commercial risks. Insurers count on low levels of adverse event correlations among risks to provide a substantial risk spreading benefit across the insured population. A certain premium is traded by the insurer for the risk of collective adverse events in excess of the premiums and investment income from premiums over the life of the policy. If the certain premium, combined with the after-tax investment income from the insurer’s asset portfolio, is expected to provide returns to shareholders (investors of capital in the insurer) commensurate with the combined underwriting and investment risk, then the insurer offers the insurance contract at the price that produces expected returns equal to the cost of capital for those risks. The paradigm supply model assumes that the flow of invested capital and the return of that capital with realized profit, if any, should be expected to have a net present value of zero when discounted at the cost of capital of the insurance enterprise. Nominal policyholder premium, loss, and expense flows and company invested asset flows with after-tax returns are used to estimate • The size and timing of the shareholder investment to back the outstanding liabilities, • The size and timing of the return of that invested capital to shareholders as liabilities are resolved and paid, and • The size and timing of any assets to be returned as income to shareholders. There are two principal methods for estimating cost of capital for the entire firm, CAPM and the Gordon Growth (presented in the notes previously as Dividend Discount Model) model. At this point in time, the use of empirical data and the simple CAPM discussed above suffers from an omitted or confounding variable problem. The extended three factor CAPM model of Fama and French developed in the 1990s included an important variable omitted in the simple formulation, the size of the insurer. This is well known in finance as the size effect on stock market returns: smaller capitalization stocks need to earn higher percentage rates of return than large cap stocks. Recent research shows that once the omitted CAPM variables are introduced, the market beta for property-liability companies is about one; i.e., P&C companies are about average risk. The Gordon Growth Model (GGM), also known as Dividend Discount Model, is built on the common assumption that the current per share price P is equal to the present value of all future dividend payments D, discounted at the same cost of capital rate. In a simple GGM formulation, the growth rate of dividends and the cost of capital are assumed constant in perpetuity leading to a simple estimation equation:

k =D

P+ g

where D/P is the current dividend rate, k is the cost of capital, and g is the dividend growth rate. The dividend growth rate g is, of course, key and many ways of estimation have been used. Most often, the growth rate is estimated as an average of some historical rate and some forecasted rate.

dr. krzysztof ostaszewski, fsa, cfa, maaa actuarial...

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