topic 4 — variable annuities and other structured products...

Fina556 – Structured Products and Exotic Options

Topic 4 — Variable annuities and other structured products

Variable Annuities

• Insurance companies have created a variety of products that

enable their policyholders to participate in bull markets while

also providing downside protection in bear markets.

• Variable annuity policies exist to attract investment dollars to

insurance companies from the mutual fund investments that

they resemble.

• The main benefit from a VA policy is the accumulated invest-

ment; additional benefits come from policy “rider” – secondary

features attached to an insurance policy.

1

• A variable annuity is a contract between you and an insurance

company, under which the insurer agrees to make periodic pay-

ments to you, beginning either immediately or at some future

date. You purchase a variable annuity contract by making either

a single purchase payment or a series of purchase payments.

• A variable annuity offers a range of investment options. The in-

vestment options are typically mutual funds that invest in stocks,

bonds, money market instruments, or some combination of the

three.

2

Although variable annuities are typically invested in mutual funds,

variable annuities differ from mutual funds in several important ways:

1. Variable annuities let you receive periodic payments for the

rest of your life (or the life of your spouse or any other person

you designate).

The feature offers protection against the possibility that, after

you retire, you will outlive your assets.

3

2. Variable annuities are tax-deferred.

• You pay no taxes on the income and investment gains from

your annuity until you withdraw your money.

• You may also transfer your money from one investment op-

tion to another within variable annuity without paying tax at

the time of the transfer.

• When you take your money out of the variable annuity, how-

ever, you will be taxed on the earnings at ordinary income

tax rates rather than lower capital gains rates.

• In general, the benefits of tax deferral will outweigh the costs

of a variable annuity only if you hold it as a long-term invest-

ment to meet retirement and other long-range goals.

4

3. Variable annuities have a death benefit.

• If you die before the insurer has started making payments

to you, your beneficiary is guaranteed to receive a specified

amount – typically at least the amount of your purchase pay-

ments.

• Your beneficiary will get a benefit from this feature if, at

the time of your death, your account value is less than the

guaranteed amount

Variable annuities are not suitable for meeting short-term

goals because substantial taxes and insurance company charges

may apply if you withdraw your money early. Variable annu-

ities also involve investment risks, just as mutual funds do.

5

Accumulation phase and payout phase

• During the accumulation phase, you make purchase payments,

which you can allocate to a number of investment options. The

money you have allocated to each mutual fund investment op-

tion will increase or decrease over time, depending on the fund’s

performance.

• In addition, variable annuities often allow you to allocate part

of your purchase payments to a fixed account. A fixed account,

unlike a mutual fund, pays a fixed rate of interest. The insurance

company may reset this interest rate periodically, but it will

usually provide a guaranteed minimum (e.g. 3% per year).

6

Example

You purchase a variable annuity with an initial purchase payment of

$10,000. You allocate 50% of that purchase payment ($5,000) to

a bond fund, and 50% ($5,000) to a stock fund. Over the following

year, the stock fund has a 10% return, and the bond fund has a 5%

return. At the end of the year, your account has a value of $10,750

($5,500 in the stock fund and $5,250 in the bond fund), minus fees

and charges.

7

• During the accumulation phase, you can typically transfer your

money from one investment option to another without paying

tax on your investment income and gains, although you may be

charged by the insurance company for transfers.

• If you withdraw money from your account during the early years

of the accumulation phase, you may have to pay “surrender

charges”. You may have to pay a 10% federal tax penalty if you

withdraw money before the age of 591/2.

8

Payout phase

• You may receive your purchase payments plus investment income

and gains (if any) as a lump-sum payment, or you may choose

to receive them as a stream of payments at regular intervals

(generally monthly).

• Under most annuity contracts, you can choose to have your

annuity payments last for a period that you set (such as 20

years) or for an indefinite period (such as your lifetime or the

lifetime of you and your spouse or other beneficiary).

9

• During the payout phase, your annuity contract may permit you

to choose between receiving payments that are fixed in amount

or payments that vary based on the performance of mutual fund

investment options.

The amount of each periodic payment will depend, in part, on

the time. Some annuities do not allow you to withdraw money

from your account once you have started receiving regular an-

nuity payments.

10

Guaranteed Minimum Withdrawal Benefit (GMWB)

• This provides the policyholder with an option to withdraw a

certain fixed percentage (seven percent is typical) of the initial

deposit every year until the entire principal is returned.

• Assume an investor invests $100, 000 in a contract with this

feature. This amount is placed in an investment account that

behaves like a mutual fund.

• Assuming a seven percent withdrawal allowance, the policy-

holder could withdraw $7,000 each year until the total with-

drawals reach $100,000. This would take just over 14 years.

Note the policyholder can withdraw the funds irrespective of

how the investment account performs.

11

Numerical example

• Suppose the investment account earns ten percent in the first

two years but earns returns of minus sixty percent in each of the

next three years. The situation would like this:

Year Rate

earned

during

the year

Fund after

withdrawals

Fund after

withdrawals

Amount

withdrawn

Guaranteed

remaining

balance

1 10% 110,000 103,000 7,000 93,0002 10% 113,300 106,300 7,000 86,0003 −60% 42,520 35,520 7,000 79,0004 −60% 4,208 7,208 7,000 72,0005 −60% 2,883 0 7,000 65,000

• At the end of year five before any withdrawal the value of the

fund, $2,883, is not enough to cover the withdrawal payment

of $7,000.

12

At this stage the guarantee kicks in: the value of the fund is set

to zero and the policyholder’s ten remaining withdrawal payments

are financed under the insurance company guarantee. The poli-

cyholder’s income stream is protected irrespective of the market

performance.

• If the market does really well the policyholder participates in

this growth. Suppose the investment account grows at a com-

pound annual rate of ten percent the policyholder would have

an account value of $183,925 after fourteen years.

13

What sort of option our investor has acquired under this rider?

To simplify matters, assume the policyholder starts taking the with-

drawal benefit annually from the end of the first year. The package

can be viewed as a guaranteed annuity – certain of $7,000 per

annum for 14 years plus a 14-year European call option on the in-

vestment account.

• Recall that the account is depleted by the periodic withdrawals

so there is a chance it will be wiped out.

• The strike price on this call option is zero, because the policy-

holder is entitled to the full account balance after the fourteen

years.

• If the market does well there will be funds left at the end whereas

if performance is bad the account balance will have shrunk to

zero before the principal is repaid and will remain there.

14

How the benefit is funded?

• A common way is using a percentage deducting from the ac-

count balance.

• For a contract with a seven percent withdrawal allowance, a

typical charge is around 40 to 50 basis points. (This can be

charged against the account value or the guaranteed remaining

balance.)

• The insurance company receives this fee for providing the guar-

anteed withdrawal option.

• So if the fund earned ten percent in the first year and the fee

was fifty basis points the effective return to policyholder is 9.5

percent.

• The insurance company receives this fee as long as there is a

positive balance in the account. Should the account go to zero

the fee income stops.

15

Put option

• The assets in the account balance are being held by a third party

such as a mutual fund; they are not owned by the insurer. So we

can also regard the benefit provided by the insurance company

in terms of a put option.

• If the investment account stays positive there is no payment

under the put option.

• The option is exercised automatically when the account balance

first becomes zero. As soon as this happens the insurer is on

the hook for the remaining stream of withdrawal payments. So

we see this is a put option with a random exercise time.

• In a perfect world the market value of this put option should be

equal to the market value of the contributions. We have noted

that these contributions continue as long as the put option is

extant.

16

Static withdrawal model – continuous version

• The withdrawal rate G is fixed throughout the life of the policy.

• When the personal account value Wt ever reaches 0, it stays at

this value thereafter (absorbing barrier).

τ = inf{t : Wt = 0}, τ is the first passage time of hitting 0.

Under the risk neutral measure Q, the dynamics of Wt is gov-

erned by

dWt = (r − α)Wt dt + σWt dBt − G dt, t < τ

Wt = 0, t ≥ τ

W0 = w0

policy value = EQ

[∫ T

0Ge−ru du

]+ EQ[e−rT WT ].

17

Consider the modified unrestricted stochastic process:

dW̃t = (r − α)W̃t dt − G dt + W̃t dBt

W̃0 = w0.

Lemma If W̃T > 0, then W̃t > 0 for any t < T .

Once the process W̃t becomes negative, it will never return to the

positive region. This is because when W̃t = 0, only the drift term

−G dt survives, which always pulls W̃t into the negative region.

Note that

WT = W̃T1{τ>T} = W̃T1{W̃T>0}= max(W̃T ,0).

WT = 0 if and only if τ ≤ T .

18

Solving for W̃t, we obtain

W̃t = Xt

(w0 − G

∫ t

0

1

Xudu

)

where

Xt = e

(r−α+σ2

2

)t+σBt

.

The terminal payoff

max(W̃T ,0) = GXT

(w0

G−∫ T

0

1

Xudu

)+

, x+ = max(x,0).

Defining A =1

T

∫ T

0

1

Xudu and observing T =

w0

G, we obtain

EQ[W̃+T ] = w0EQ[XT (1 − A)+].

G du

Xuis the number of units withdrawn over (u, u+du), and its value

at maturity T isGXT

Xudu.

19

Equity-linked notes (Hong Kong examples)

• Callable dual accrual cash or share security

• Early redemption equity-redeemable warrants

• Super certificate (lookback minimum and lock-in level)

• Target redemption notes

• Guaranteed equity bonds

• Guaranteed annuity options

20

24 Month Callable Dual Accrual Cash or Share Security

On Wal-Mart Stores, Inc and Intel Corp.

issued by Merrill Lynch (Feb. 9, 2006)

Payment/delivery on the maturity date

• If the settlement prices of BOTH the underlying stocks are higher

than or equal to the respective exercise price, each warrant holder

will receive 100% of the notional amount per warrant held.

• If either one of the settlement prices is lower than the respective

exercise price, each holder will receive per warrant physical

delivery of a number of the “Worst performing” stock equal to

Notional amount / exercise price of worse performing stock

It is a forced conversion when the share prices decline (opposite

effect to that of a convertible bond).

21

Issue size: 10,000,000 warrants

Minimum subscription: 100,000 warrants

Notional Amount: USD 1 per warrant

Issue Price: 100% of the Notional Amount

Trade Date: Feb. 9, 2006

Issue Date: Feb. 23, 2006

Valuation Date: Feb. 11, 2008

Maturity Date: Feb. 19, 2006

22

Underlying stocks (uncorrelated)

Reference price Exercise price

Wal-Mart Stores Inc. USD 45.48 USD 39.5676

Intel Corp USD 20.77 USD 18.0699

• Exercise price = 87% x reference price

Terminal payoff = min (1, min(S1(T)/S1*,S2(T)/S2*))

= 1- max(1 - min(S1(T)/S1*,S2(T)/S2*), 0),

where A1* and S2* are the reference prices of asset 1 and asset 2.

The investor shorts a put on the minimum of two assets.

23

Additional coupon (accrual feature)

Unless the warrants have been called, over each observation period (3-month

period), the holder receives

4.075% x n /N of notional amount

where N = number of New York Business Days in the period in the applicable

Observation Period;

n = number of New York Business Days in the applicable Observation

Period on which the closing prices of BOTH the Underlying Stocks

are at or above the respective Exercise Price.

This is like an accrual note with the underlying index being the

minimum of two share prices. The accrual feature can be viewed as

a series of daily binary options which pay

4.075%/N x notional amount

when

min(S1(T)/S1*,S2(T)/S2*) > 1.

24

Overall description

• The investor believes that the prices of BOTH underlying

shares at maturity will remain at a level above or equal to

their respective Exercise Prices, earning an enhanced yield.

• The warrant pays out a fixed 4.075% coupon for the first

quarter.

• The coupon received would depend on the trading path of

BOTH underlying stocks due to the accrual feature.

25

• Issuer’s Call: On any of the Observation Date, provided that

BOTH underlying stocks are greater than or equal to the

reference prices, the issuer can call by paying 100% of the

Notional Amount. This occurs when the value of the

embedded put is less than the present value of the enhanced

yield over the remaining period. This “call” right given to

the issuer is like a Bermudan put option.

26

Risks

1. Market risks – underlying shares

2. Credit risk – default of Merrill Lynch

3. Liquidity risk – will not be listed on any securities exchange

and do not expect a trading market with only Merrill Lynch as a

possible buyer.

4. Interest rate risk – bond component: par plus coupons and

issuer’s call.

warrant = bond (series of binary options – accrual feature)

- European put on minimum of two uncorrelated stocks

- issuer’s call (Bermudan put)

27

2-Year JPY Early Redemption Equity-Redeemable

(“ER”) Warrants Linked to a Basket of Japan

Equities (Japan Basket ER Warrants)

Type of investor

• He holds the belief that over the next two years the prices of

all of the shares in the Japan Share Basket will not decline

by more than 12.00% from their respective Reference Price.

• He must be willing to take delivery of the worst two

performing shares if any of the shares in the basket falls

below their respective Strike Price at the Valuation Date.

This product is not principal-protected.

28

Japan Share Basket : A basket made up of the 5 shares as shown in

the table below:

NameBloomberg

Code

Reference

Price

Strike

Price

Trigger

Price

Mitsubishi Estate Co. Ltd. 8802 JT JPY 1,146 JPY 1,008.48 JPY 1,123.08

Sumitomo Realty 8830 JT JPY 1,185 JPY 1,042.80 JPY 1,161.30

Nippon Building Fund Inc. 8951 JT JPY 987,000 JPY 858,560 JPY 967,260

Japan Real Estate Investment Corp. 8952 JT JPY 890,000 JYP 783,200 JPY 872,200

Japan Prime Realty Investment Corp. 8955 JT JPY 322,000 JYP 283,360 JPY 315,560

29

Issue Size : 1,000,000,000 warrants

Minimum Subscription : JPY 10,000,000

Notional Amount : JPY 1,000,000,000

Issue Price : 100%

Trade Date : 14 June 2005

Issue Date : 28 June 2005

Maturity Date : 28 June 2007, subject to the following

business day convention

Periodic payment : Payable quarterly in arrear on each

Periodic Payment Date and accruing on

1 30/360 basis at the Periodic Payment

Rate

30

Reference Price : Executed price of each Share in the Share Basket

on Trade Date

Settlement Price : Closing price of each Share in the Share Basket

on the last Observation Date, as determined by

the Calculation Agent

Strike Price : 88.00% of the Reference Price of each Share in

the Share Basket

Trigger Price : 98.00% of the Reference Price of each Share in

the Share Basket

31

Periodic Payment Rate : For the first period, from the issue date to

the first Periodic Payment Date, 10.00%

p.a. fixed in the first period

Thereafter, 10.00% p.a. if the closing

prices of all the Shares in the Share

Basket are at or greater than their

respective Strike Prices on an

Observation Date. Otherwise, the

Periodic Payment Rate is deemed to be

1.00 p.a..

Early Redemption by

Issuer :

If the closing prices of all the Shares in

the Share Basket are at or greater than

their respective Trigger Prices on an

Observation Date, the Warrants will be

redeemed in full at 100% of the Notional

Amount together with accrued interest

on the related Periodic Payment Date.

32

Worst Performing Share : The Share in the Share Basket which

has the lowest value on Valuation Date

according to the following formula:

(Settlement Price / Reference Price) 1

Redemption at Maturity

Date :

On the last Observation Date:

(1) If the Settlement Prices of ALL the

Underlying Stocks are higher than or

equal to their respective Strike Price,

each holder of the ER Warrant will

receive 100.00% of the Notional

Amount per warrant held.

33

(2) If the Settlement Price of ANY of

the Underlying Stocks are lower than

their respective Strike Price, each

warrant holder will on the maturity

Date receive per Warrant physical

delivery of the Worst Performing

Share equal to:

Any fraction of a trading lot of the

Shares to be delivered shall be paid

out in JPY cash at a price calculated

using the relevant Settlement Price.

Stock"PerformingWorst" theofPriceStrike

AmountNotional

Settlement Currency : JPY

34

Payout at any Observation Date

Performance

on relevant

Observation Date

Periodic Payment

(p.a.)

Redemption

prior to Maturity

25% 10.00% Yes, Redeems Par





-1% 10.00% Yes, Redeems Par

-2% 10.00% Yes, Redeems Par

-3% 10.00% No, Warrant Continues





35

Payout at Maturity if not Early Terminated (i.e. Warrant is held to

Maturity Date)

Performance

on relevant

Observation Date

Periodic Payment

(p.a.)

Redemption

prior to Maturity

25% 10.00% Par in Cash





-1% 10.00% Par in Cash




-10% 10.00% Par in Cash

-25% 1.00% 85.23% in Shares

-50% 1.00% 56.82% in Shares

36

Summary

• Juicy coupons for potential short life if stock prices stay

above the trigger prices (98%). The juicy coupons in the

early coupon represents the premium of the put option sod

to the issuer.

• When the stock prices fall below the strike prices (88%), the

coupon reduces to 1% pa. In addition, the investor receives

the worst performing stock at maturity.

37

Issuer : BNP Paribas (AA/Aa2)

Issue Amount : USD3,000,000

Determination Date : 21 June 2007 (The Valuation Date)

Maturity Date :

Issue price : 100% of Note Denomination

Coupon : Zero

28 June 2007 (5 Busines Days after

Determination Date)

2-Year USD Super Certificate Plus

(with lookback minimum and lock-in Level)

Linked to Basket

38

I Name RIC S i ,0 K i S Barrier, i S Lock-in, i

1 Lloyds TSB Group Plc. LLOY.L GBP4.735 GBP4.735 GBP3.3145 GBP5.2085

2 Altria Group, Inc. MO.N USD66.85 USD66.85 USD46.795 USD73.535

3China Petroleium and

Chemical Copr. (Sinopec)0386.HK HKD2.90 HKD2.90 HKD2.03 HKD3.19

Share Basket : A basket made up of the 3 shares

(each being a "Share") as shown

in table below.

Note Denomination

(ND): USD50,000 face value

Particpation Rate

(PR): 320%

39

Initial Spot Price of

ith

Share (S i ,0):

Reference Price of

ith

Share (K i ):

Barrier Price of

ith

Share (S barrier, i ):

Lock-in Levels :

The market price of the ith

Share as shown in the table

above.

100% of Initial Spot Price of the ith

Share in the

Share Basket.


Share in the

Share Basket.


Share in the

Share Basket.

40

Final Spot Price of

ith Share (S i,f ) :

Final Reference Exchange

Rate for GBP (FX f ) :

Final Reference Exchange

Rate for HKD (FX f ) :

Hong Kong Time on the Valuation Date.

(Determination Date).

The price of the ith Share

on the Valuation Date.

The mid-market USD:GBP exchange rate

as per Reuters page :GBP-" at 17:30

London Time on the valuation Date.

(Determination Date).

The mid-market USD:HKD exchange rate

as per Reuters page "HKD=" at 16:00

41

Worst Performing :

Share value (the "Worst Performance") on the

Valuation Date (Determination Date)

according to the following formula.

Mathematically, Worst Performance is

defined by the following formula

Means the Share that has the lowest value

%10010,

,

i

fi

S

S

%1001ePerformanc0,

,

i

fi

worstS

S

42

Share Amount : If the Worst Performing Share is denominated GBP (HKD),

Share Amount shall mean a quantity of the Worst

Performing Share equal to

(a) an amount in GBP (HKD), equal to the Note

Denomination converted into GBP using the Final

Reference Exchange Rate for EUR, divided by

(b) the Reference Price of the Worst Performing Share;

rounded to the nearest integer.

If the Worst Performing Share is denominated USD, Share

Amount shall mean a quantity of the Worst Performing

Share equal to the Note Denomination divided by the

Reference Price of the Worst Performing Share; rounded to

the nearest integer.

43

Monitoring Period : The period from and including the Launch Date

to and including the Valuation Date.

Barrier Event : A Barrier Event is deemed to have occurred if

the price of at least one Share at the Valuation

Time is at or below its corresponding Barrier

Price on any Exchange Business Day during the

Monitoring Period.

Look-back Period : The period from and including the Launch Date

to and including 21 December 2005.

Call Strike Level ( ) : Means the lowest daily closing price level

observed as compared against the

corresponding Initial Spot Price during the

Look-back Period, subject to a minimum of

90% and a maximum of 100%.

44

where

and where is the lowest daily closing

price observed in respect to the ith Share during

the look-back Period.

MinimumLookback,3to1

Min ii

S

Minimum

iS

%90,Max100%,Min0,

Minimum

MinimumLookback,

i

ii

S

SS

45

Lock-in Event : A Lock-in Event is deemed to have occurred if

the prices of all Shares, at the Valuation Time

on an Exchange Business Day during the

Monitoring Period, are at or above their

corresponding Lock-in Prices in respect of a

particular Lock-in Level; such Lock-in Level is

then deemed to have been reached. For the

avoidance of doubt, more than one Lock-in

Event can occur during the Monitoring Period.

Actual Lock-in Level : The highest Lock-in Level reached among the

Lock-in Levels reached in respect of all Lock-in

Events where applicable.

(PerformanceLocked)

46

Redemption Amount : (Case 1): If the at least one lock-in Event has

occurred during the Monitoring Period, the

Issuer shall pay the Note holder the following

amount in respect of each Note held on

Maturity Date.

Case 1 occurs when at least one share has increased by more than 10%.

This is called a lock-in event. In this case, it is principal protected plus

extra percentage based on the stock performance.

at maturity

,Min,1ePerformancMax%1000,

,

3to1Locked

i

fi

i S

SPRND

47

(Case 2): If Lock-in Event has not occurred

during the Monitoring Period and that

(Case 2a): Barrier Event has not occurred

during the Monitoring Period, the Issuer

shall pay the Note holder the following

amount in respect of each Note held on

Maturity Date.

,Min,0Max%1000,

,

3to1i

fi

i S

SPRND

48

(Case 2b): at least one Barrier Event has

occurred during the Monitoring Period.

(Case 2b-i): and if Performanceworst

0, that is the Worst Performance is

greater than or equal to zero, the

Issuer shall pay the Note holder the

following amount in respect of each

Note held on Maturity Date.

,,0Max%1000,

,

3to1i

fi

i S

SMinPRND

or

49

(Case 2b-ii): and if Performanceworst<

0, that is the Worst Performance is

less than zero, the Issuer shall pay to

the Note holder the Share Amount

(fractional entitlement will be subject

to cash settlement) AND shall pay the

Note holder the follow amount (if the

amount is greater than zero) per Note

in respect of each Note held on

Maturity Date.

,,0Max0,

,

3to1i

fi

i S

SMinPRND

where

%1001MinePerformanc0,

,

3to1worst

i

fi

i S

S

50

1. To the note holder, it is most desirable to have

(a) A small value of . This occurs when there are drops in the share prices

during the lookback period.

(b) A higher performanceLocked value (which can be greater than 10%). It is easier

to be achieved when the share prices are more correlated.

2. Cases 2a and 2b-i are principal protected.

3. When knock-out event occurs and performanceworst < 0 [Case 2b(ii)], the note holder

acquires the share plus some cash compensation. In this case, the note is not

principal protected.

Remark

Complexity almost for the sake of complexity. The structures are “opaque” for both

parties – cannot really work out where the value lies. Product controllers have trouble

marking to market.

51

Target redemption notes

Example 7.5% USD Target Redemption Index Linked Deposit (issued

by Bank of East Asia, 2004)

Selling points - Enjoy potentially higher returns with Index Linked

Deposit

• 100% principal protection plus 7.5% guaranteed coupon return over

a maximum of 5-year investment period.

• 1st year annual coupon is guaranteed at 6.5% (very juicy), payable

semi-annually.

52

• The remaining coupon rate of 1% will be based on the LIBOR

movement. The inverse floater formula is

However, the total coupon received will not shoot beyond the

target rate of 7.5%. If the coupon payment accrued during the

deposit period is less than the target rate, then the remaining

amount will be paid at maturity.

max{7% 2 6-month LIBOR (in arrears), 0}

Early termination

Once the accumulated coupon payment reached the target rate, the

deposit will be terminated automatically.

53

Worst scenario

The deposit is held for 5 years until maturity so that the annual return for

the deposit is only 1.5% per annum.

Market background

The US Fed policy makers voted unanimously to keep the Fed Fund Rate

unchanged at 1% on 28 October 2003, the lowest level in the past 45

years. They had indicated that the interest rate would remain at a low

level for a considerable period.

Potential risk

If the 6-month LIBOR rises beyond 3.5% one year afterwards and never

come down again. The deposit is then held for 5 years until maturity.

54

Equity target redemption notes

SG Product

• 10-year fund that is 100% capital guaranteed

• Pay a juicy fixed coupon of 10% in the first year

• For Year Two, the coupon payment is referenced to the average

performance of the 6 worst stocks in a basket of 24 blue-chip stocks.

• From Year Three onwards, the investor gets the better of the previous

year’s coupon or the payout formula.

• Once the aggregate coupon payments reaches or exceeds 20%, the

fund terminates with full payment of the coupon for that year.

max{0,10% + 0.5 average performance of the 6 worst stocks}

55

Worst scenario: 10-year fund with total coupon of 20%

Blending equity and rates

• Design products that have both equity and fixed-income risk.

• The equity and fixed-income markets typically offset each other

during economic downturns, therefore hedging the investor

against excessive downside in one market.

56

Guaranteed equity bonds (GEBs)

• The issuer (usually an insurance company) guarantees a

stated interest rate and some protection from loss of

initial capital, and provides an opportunity to earn

additional interest based on the performance of an

equity market index (say, Standard and Poor’s 500

Composite Stock Price Index).

• GEBs credit interest using a formula based on changes

in the index to which it is linked. It enables investors to

achieve potential capital appreciation by participating in

the positive performance of the index but also provide

investors with a guaranteed minimum return of their

investment at maturity.

57

Term

The index term is the period over which index-linked interest is

calculated. Interest is credited to the investor at the end of a term.

Participation Rate

The participation rate decides how much of the increase in the index

will be used to calculate index-linked interest. For example, if the

calculated change in the index is 9% and the participation rate is 70%,

the index-linked interest rate for the contract will be 6.3% (9%

70%=6.3%).

• The company usually guarantees the participation rate for a specific

period, from one year to the entire term.

• When that period is over, the company sets a new participation rate

for the next period.

• Some contracts guarantee that the participation rate will never be set

lower than a specified minimum or higher than a specified

maximum.

58

Cap Rate

Some contracts may put an upper limit, or cap, on the index-linked

interest rate. This is the maximum rate of interest the contract will

earn.

Floor

The floor is the minimum index-linked interest rate that will be paid.

The most common floor is 0%. A 0% floor assures that even if the

index decreases in value, the index-linked interest that can earn will be

zero and not negative.

Guaranteed interest compounding

Some contracts pay simple interest during an index term. That means

index-linked interest is added to the original premium amount but does

not compound during the term. Others pay compound interest during

a term, which means that index-linked interest that has already been

credited also earns interest in the future.

59

Dividends

Depending on the index used, stock dividends usually are not included

in the index value. For example, the S & P 500 is a tock price index

and only considers the prices of stocks. It does not recognize any

dividends paid on those stocks.

Early withdrawal

In most cases, investors cannot take all or part of the money out of

contract at any time during the term. There will be a cost and the

index-linked interest on the amount withdrawn will not be paid.

Indexing method

The approach used to measure the amount of change, if any, in the

index.

60

Point-to-Point

The contingent claim C(t) in year t for one unit of GEB can be

represented as followed

.1)0(

)(),)1(),)1(,1max(min()(

S

tSRgRtC t

tt

t

• While subject to the maximum cap rate that can be earned under

this design, the first random term allows the investors to have a

participation rate in any potential upside gain in the equity

market.

• In the event of an adverse market environment, the downside risk

is constrained to the minimum guarantee floor component, that is,

(1 + g)t.

• The presence of the cap rate, although placing an upper bound on

the rate of return of the contract, could reduce the cost of such

design substantially.

61

The payoff in year t for one unit of GEB is given byt

s

t

s gRtC1

)1(),1),1,1max(min(max)(

where the random variable RS again measures the appreciation of the

referenced index fund in year s.

• RS is solely determined by the index levels at the beginning and the

end of year s:.1

)1(

)(

sS

sSRs

• The interest is credited each year for the annual reset GEBs. The

credited interest cannot be lost even if the index subsequently

goes down.

• The index level used to determine the appreciation of the index is

reset annually. This `lock in’ feature can be extremely valuable,

particularly in a more volatile market.

62

High Water Mark (Lookback)

where),)1(),)1(,1max(min()( tt

t gRtC

.1)0(

)(max],0(

S

sSR

ts

t

63

Point-to-Point with Barrier

To increase the participation rate, we apply an up-and-in

barrier option to the point-to-point GEBs. An up- and-in

barrier GEB provides purchasers with the greater of the

index return times the participation rate and a minimum

guaranteed return if the index rises above a barrier for the

monitoring period and offers the minimum guaranteed

return otherwise.

else)1(

)(if)1(,)1(1minmax)(

t

tt

t

g

BsSgRtC

where .1)0(

)(

S

tSRt

64

Annual Reset with Barrier

In each period, the GEBs will provide customers with the

greater of either the annual index return times the

participation rate or a minimum guaranteed rate if the index

value for the monitoring period rises above a barrier.

Otherwise, the GEBs will credit to the policyholder the

minimum guaranteed rate as annual return.

else)1(

)(if)1(,)1,1min(maxmax)(

1

t

tt

t

t

s

g

BsSgRtC

65

Average

ere wh,)1(),)1(,1min(max)( tt

t gRtC

.1)0(

)(1

S

sS

R

t

st

66

0.15Initial volatility

0.045Initial interest rate

9000Barrier level

5800Initial index level

100Number of simulation

1.03Guaranteed annual return

1.3Cap rate

Particip

ratePoint-point Ratchet

Point-to-

Point with

barrier

Ratchet

with

barrier

Average

150 0.956431 1.298844 0.984182 1.000207 0.934207

130 0.973574 1.288121 0.997821 1.018307 0.947832

110 0.968545 1.217861 0.985055 1.004369 0.943764

90 0.946537 1.126441 0.966122 0.960818 0.921269

70 0.957623 1.107188 0.970936 0.981477 0.924691

50 0.918295 1.026742 0.926781 0.933261 0.898979

In general, we can see that the change of participation rate does not place great effect to

the point-to-point, lookback, and average contracts until 50%.

67

Impact of cap rate Participation rate = 0.8

Cap Rate

(%)

Point-to-

PointRatchet Lookback

Point-to-

Point with

barrier

Ratcjet

with

barrier

Average

No Cap 1.015731 1.145941 1.053112 0.976734 1.000487 0.935367

50 0.963433 1.116318 0.993219 0.929419 0.962648 0.914589

40 0.975355 1.140566 1.004696 0.938573 0.978622 0.934108

30 0.956171 1.133052 0.973526 0.926027 0.977767 0.928259

20 0.928098 1.123112 0.933465 0.917057 0.981053 0.919391

10 0.911383 1.075609 0.911383 0.911383 0.956211 0.911383

• The decreasing cap rate will lower the price of the contract.

• For the average design, the price does not change so much when

the cap rate decrease.

• For ratchet design, the price also does not change so much when

the cap rate decrease until 10%.

68

Annual Guaranteed Interest

Some contracts pay guarantee interest during an index term if the

index-linked return is less than the guarantee return or negative.

Annual

Guarantee

Return

Rate

Point-to-

PointRatchet Lookback

Point-to-

Point with

barrier

Ratchet

with

barrier

Average

6 1.050396 1.310486 1.050396 1.050396 1.126012 1.050396

5 1.006993 1.244254 1.009694 1.003458 1.073681 1.001639

4 0.966467 1.148995 0.975274 0.953988 0.998599 0.952589

3 0.963088 1.136211 0.977767 0.930989 0.991021 0.928912

2 0.914577 1.065549 0.944937 0.872169 0.914411 0.881798

1 0.927131 1.053979 0.965791 0.854578 0.893834 0.883096

No 0.916407 1.025659 0.951387 0.820625 0.859302 0.872415

The increasing guarantee interest will increase the price of the contracts.

69

NameGuaranteed

Equity Bond

Guaranteed FTSE

Bond 4-5 yr

Guaranteed

FTSE Bond 5 yr

Guaranteed

Capital Bond 5 yr

Participation rate 112% 105% 105% 110%

Minimum return 100% 110% 100% 10%

Term 5 years 5 years 5 years 5 years

Minimum investment £1,000 £5,000 £2,500 £500

Average

Five days initial

average and six

months final

daily averaging

Monthly readings

over last years of

investment

Monthly

readings

over last year

of

investment

Monthly readings

over last year of

investment

Source: Structured Retail Products (April 2006)

70

Guaranteed annuity options

• In the United Kingdom, the GAO guarantees a minimum conversion

rate of lump sum of annuity. Typically, guarantees of £111 annual

annuity per £1,000 maturity lump sum have been offered for male

policyholders, and around £91 annuity per £1,000 maturity lump sum

for females. The conversion rate is known as the guaranteed annuity

rate of GAR.

• Let g be the guaranteed annuity rate (e.g., g = 1/9 for a rate of 111

annuity per 1000 lump sum), and let ax(t) be the market price at t of a

whole-life annuity of 1 per year payable immediately to a life aged x.

The value of separate fund account at t is Ft.

• The payoff under the GAO at the maturity of the separate fund account,

t = n (which is the annuity vesting date), for a life age 65 at vesting, is

)0,)(max( 65 nn FnagF

This option is in-the-money when a65(t) is greater than 1/g and out-of-the-

money otherwise.

71

topic 4 — variable annuities and other structured products...

Documents