lecture 2-1-2012 fall

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Computational Finance

Lecture 2

Part 1Fixed Income Securities: Bonds

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Main Contents Interest theory:

Simple and compounding interests

Discount

Bond Details:

Pricing and yields

Default Risk and Credit Ratings

Credit Default Swaps

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2.1 Interest Theory

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Time Value of Money $1 today is worth more than $1 in 1 year

Interest ()as a measure of time value of

money: simple interest ()

compounding interest ()

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Simple Interests Suppose that you invest $1 for years with

interest per annum. After years, under the

rule of simple interest you will have:

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Discretely Compounding Interests Compound interest means interest on interest.

Suppose that you invest $1 for years with interest

per annum, but under a compound scheme After 1 year, you will have:

After 2 years, you will have

After years:

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Compounding at Various Intervals The compounding periods of interest rates may

not be the same as the periods that they are quoted.

e.g., compounding frequency is semiannually andthe interest rate is 10% per year.

After 1 year,

After 2 years,

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Compounding at Various Intervals In general, if an interest rate is per year and the

rate is compounded times per year, the total

amount will be

after n years if we invest $1 now.

Nominal interests () and effectiveinterests ()

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Continuously Compounded Interests As compounding more and more frequent, i.e.,

as ,

where

Continuously compounded interest:

Suppose that you invest $1 with continuouslycompounded interest rate. After years, you

will have

We use continuously compounded interests only in the class

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Present Values Interest rates facilitate the comparison of

(deterministic) cash flow at different time.

Now 1 year

$1000 $1100Discounting

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Present Values of Cash flow Streams If an amount is to be received in 1 year and the

interest rate is , then the present value is

A cash flow stream:

0 1 2 3 4

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2.2 Bond Pricing and Yields

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Fixed Income Securities

Fixed income securities are financial instruments

that promise to the holder a fixed stream of

income or a stream of income that is determinedaccording to a specified formula over a span of

time.

They are an important fund-raising tool for

governments, corporations, and even individuals.

Bonds are a typical example of such securities.

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U.S. Credit Market Instruments as of

2008 Quarter 3

U.S. Stock Market (Common):$19,648 billion

U.S. Credit Market Debt: $51,796bnDebt by Selected Major Borrowers (Not Exhaustive List):

U.S. Government Securities (Includes Agency & GSE):

$13,850bn (27%)

Corporate & foreign bonds: $11,262bn (22%)

Municipal bonds: $2,669bn (5%)

Mortgage: $14,720bn (28%)

%s are percent of Total U.S. Credit Market Debt, source is Federal Reserve Flow of Funds

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Bond Characteristics

Bonds () are a kind of financial instruments

issued by government or corporations to borrow

money from the public. Bonds usually obligates the issuer to make

periodic interest payments to the bondholder,

called coupon () payments.

At the maturity () of a bond, the issuer

repays the principal, called the par value or the

face value (), to the bondholder.

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Bond Example

To illustrate, suppose that you bought a 2-year HK

government bond with a par value of $1, 000 and

a coupon rate of 8%. It costs you $1,000. That means, you just lent $1,000 to HK

government for 2 years. In return, the government

is obligated to repay you $80 per year for the

stated life of the bond. At the end of the 2-year lifeof the bond, the government needs to repay $1,000

to you.

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Another Example

HK government started to issue 3-year inflation-

index bonds () in July 2011. The par

value of one bond is $10,000. It pays couponssemiannually and the coupon rate depends on the

inflation rate of the latest six month.

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Bond Pricing

For investors, bonds are an important investment

vehicle because they are traded frequently in the

market. We need to know how to determine its value when

trading.

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Bond Pricing

Interest rate: per annum

Face Value:

Coupon rate:Principal

Coupons

Coupon

0 1 2 3 4 n

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Example

Suppose that a 2-year bond with a par value of

$100 provides coupon at the rate of 6% per annum

semiannually. The interest rate is 5% per year.What should the bond be sold at?

See attached excel file.

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Bond Prices, Interest Rates, and Time-to-

Maturity

Bond price and market interest rate is inversely

related. Keeping all other factors the same,

the bond price will rise when interest rate falls bond price will fall when interest rate rises.

The longer the maturity of the bond, the greater

the sensitivity of its price to fluctuations in the

interest rate.

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Bond Prices at Different Interest Rates

Use one numerical example to illustrate the

previous slide: 8% coupon bond, coupon paid

semiannually, face value $1,000Time-to-maturity 2% 4% 6% 8% 10%

1 Year $1059.11 $1038.83 $1019.13 $1000 $981.41

10 Years 1541.37 1327.03 1148.77 1000 875.38

20 Years 1985.04 1547.11 1231.15 1000 828.41

30 Years 2348.65 1695.22 1276.76 1000 810.71

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Bond Yields

Bond A:

Therefore, the yield is 5% per year.

This equation can be solved using a trial and

error procedure.

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2.3 Default Risk

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Default Risk and Bond Pricing

Although bonds generally promise a fixed flow of

income, that income stream is not riskless. If the

issuing company or government goes bankrupt,the bondholder will not receive all the payments

they have been promised. We refer this

uncertainty as the default risk.

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This default risk affects the price of a bond (or

equally, the YTM of a bond).

Price Coupon Rate Maturity YTM

Treasury Note 105.74 5.125% 2016 4.846%

Emerson Elec co. 101.80 5.125% 2016 5.035%

(Yahoo! Finance, Sept. 16, 2007)

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YTM difference in the US bond market:

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YTM difference in the Canadian bond market:

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Default Risk Premium

To compensate for the possibility of default,

corporate bonds are usually sold at a YTM higher

than Treasury bonds. The difference is known asdefault premium.

If the firm remains solvent and actually pays the

investor all of the promised cash flows, the investor

will realize a higher yield. If, however, the firm goes bankrupt, the corporate

bond is likely to provide a lower return than the

government bond.

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Credit-Rating Agencies

In the financial world, some investment advisory

firms provide credit quality information of

corporate and municipal bonds to the public. Theyare known as credit-rating agencies.

Several famous agencies:

Standard & Poors ()

Moodys ()

Fitch Rating ()

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Credit-Rating Agencies

Some grades used by credit rating agencies:

Moodys S&P Fitch Descriptions

Aaa AAA AAA Prime Maximum Safety

Aa AA AA High Grade High Quality

A A A Upper Medium Grade

Baa BBB BBB Lower Medium Grade

Ba BB BB Non Investment GradeB B B Speculative

Caa CCC CCC Substantial default risk

C - - Highly Speculative

- D D Default

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2.4 Credit Default Swaps

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A credit default swap (CDS, ) is an

insurance policy on the default risk of a bond or

loan. The seller of the swap collects an annual premium (and

sometimes an upfront fee) from the swap buyer.

The buyer of the swap collects nothing unless the bond

issuer or loan borrower defaults, in which case theseller of the swap essentially pays the drop in value

from par to the swap buyer.

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Cash flow illustration:

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Pricing ofCDS

CDS was originally designed to allow lenders to

buy protection against losses on sizable loans. The

natural buyers would be large bondholders orbanks that had made large loans.

An investor holding a bond with a BB rating could

in principle raise the effective quality of the debt

to AAA by buying a CDS. Therefore, the fair CDSpremium should be the yield spread between BB

and AAA-rated bonds.

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CDS Premium in the Crisis

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CDS as a Speculation Tool

CDSs can be used to speculate on financial health

of firms.

Swap buyer need not hold the underlying bond or loan.Someone in August 2008 wishing to bet against the

financial sector might have purchased CDS and have

profited as CDS premium spiked in September.

At their peak there were reportedly $63 trillion worth of

CDS; US GDP is about $14 trillion.

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CDS, Systemic Risk, and The Credit

Crisis of 2008 Transactions of CDSs are accomplished largely in

the Over-the-Counter (OTC,) market.

Therefore, this product lacks of transparency. As the subprime-mortgage market collapsed in

2008, the potential obligations on these contracts

ballooned to such levels that the ability of CDS

sellers to honor their commitments was in doubt. AIG, counterparty risk and systemic risk.

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New Regulations onCDS

New regulations on CDS will be implemented

CDS contracts will be required to be traded on an

exchange with collateral requirements to limit risk. Exchange trading will increase transparency of

positions of institutions.

lecture 2-1-2012 fall

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