Topic 4 Variable annuities and other structured products

Transcription

1 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 investment; additional benefits come from policy rider secondary features attached to an insurance policy. 1

2 A variable annuity is a contract between you and an insurance company, under which the insurer agrees to make periodic payments 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 investment options are typically mutual funds that invest in stocks, bonds, money market instruments, or some combination of the three. 2

3 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

4 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 option to another within variable annuity without paying tax at the time of the transfer. When you take your money out of the variable annuity, however, 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 investment to meet retirement and other long-range goals. 4

5 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 payments. 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 annuities also involve investment risks, just as mutual funds do. 5

6 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 option 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

7 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

8 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

9 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

10 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 annuity payments. 10

11 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 policyholder could withdraw $7, 000 each year until the total withdrawals 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

12 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 Fund after withdrawals Fund after withdrawals Amount withdrawn Guaranteed remaining balance the year 1 10% 110, , 000 7, , % 113, , 300 7, , % 42, , 520 7, , % 4, 208 7, 208 7, , % 2, , , 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,

13 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 policyholder 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 compound annual rate of ten percent the policyholder would have an account value of $183,925 after fourteen years. 13

14 What sort of option our investor has acquired under this rider? To simplify matters, assume the policyholder starts taking the withdrawal 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 investment 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 policyholder 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

15 How the benefit is funded? A common way is using a percentage deducting from the account 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 guaranteed 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

16 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

17 Static withdrawal model continuous version The withdrawal rate G is fixed throughout the life of the policy. When the personal account value W t ever reaches 0, it stays at this value thereafter (absorbing barrier). τ = inf{t : W t = 0}, τ is the first passage time of hitting 0. Under the risk neutral measure Q, the dynamics of W t is governed by dw t = (r α)w t dt + σw t db t G dt, W t = 0, t < τ t τ W 0 = w 0 [ T policy value = E Q 0 Ge ru du ] + E Q [e rt W T ]. 17

18 Consider the modified unrestricted stochastic process: d W t = (r α) W t dt G dt + W t db t W 0 = w 0. Lemma If W T > 0, then W t > 0 for any t < T. Once the process W t becomes negative, it will never return to the positive region. This is because when W t = 0, only the drift term G dt survives, which always pulls W t into the negative region. Note that W T = W T 1 {τ>t } = W T 1 { WT >0} = max( W T,0). W T = 0 if and only if τ T. 18

19 Solving for W t, we obtain where The terminal payoff W t = X t (w 0 G X t = e ( r α+ σ2 2 ( w0 T max( W T,0) = GX T G 0 Defining A = 1 T T 0 t 0 ) 1 du X u ) t+σb t. ) + 1 du, x + = max(x,0). X u 1 du and observing T = w 0, we obtain X u G E Q [ W + T ] = w 0E Q [X T (1 A) + ]. G du X u is the number of units withdrawn over (u, u+du), and its value at maturity T is GX T X u du. 19

20 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

21 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

22 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,

23 Underlying stocks (uncorrelated) Reference price Exercise price Wal-Mart Stores Inc. USD USD Intel Corp USD USD 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

24 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 when 4.075%/N x notional amount min(s1(t)/s1*,s2(t)/s2*) > 1. 24

25 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

26 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

27 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

28 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

29 Japan Share Basket : A basket made up of the 5 shares as shown in the table below: Name Bloomberg Reference Strike Trigger Code Price Price Price Mitsubishi Estate Co. Ltd JT JPY 1,146 JPY 1, JPY 1, Sumitomo Realty 8830 JT JPY 1,185 JPY 1, JPY 1, Nippon Building Fund Inc JT JPY 987,000 JPY 858,560 JPY 967,260 Japan Real Estate Investment Corp JT JPY 890,000 JYP 783,200 JPY 872,200 Japan Prime Realty Investment Corp JT JPY 322,000 JYP 283,360 JPY 315,560 29

30 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

31 Reference Price Settlement Price Strike Price Trigger Price : Executed price of each Share in the Share Basket on Trade Date : Closing price of each Share in the Share Basket on the last Observation Date, as determined by the Calculation Agent : 88.00% of the Reference Price of each Share in the Share Basket : 98.00% of the Reference Price of each Share in the Share Basket 31

32 Periodic Payment Rate : Early Redemption by Issuer : 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.. 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

33 Worst Performing Share : Redemption at Maturity Date : The Share in the Share Basket which has the lowest value on Valuation Date according to the following formula: (Settlement Price / Reference Price) 1 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 % of the Notional Amount per warrant held. 33

34 (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: Notional Amount Strike Price of the "Worst Performing"Stock 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. Settlement Currency : JPY 34

35 Payout at any Observation Date Performance on relevant Observation Date Periodic Payment (p.a.) Redemption prior to Maturity 25% 10.00% Yes, Redeems Par 10% 10.00% Yes, Redeems Par 5% 10.00% Yes, Redeems Par 3% 10.00% Yes, Redeems Par 0% 10.00% Yes, Redeems Par -1% 10.00% Yes, Redeems Par -2% 10.00% Yes, Redeems Par -3% 10.00% No, Warrant Continues -5% 10.00% No, Warrant Continues -10% 10.00% No, Warrant Continues -25% 1.00% No, Warrant Continues -50% 1.00% No, Warrant Continues 35

36 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 10% 10.00% Par in Cash 5% 10.00% Par in Cash 3% 10.00% Par in Cash 0% 10.00% Par in Cash -1% 10.00% Par in Cash -2% 10.00% Par in Cash -3% 10.00% Par in Cash -5% 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

37 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

38 2-Year USD Super Certificate Plus (with lookback minimum and lock-in Level) Linked to Basket Issuer : BNP Paribas (AA/Aa2) Issue Amount : USD3,000,000 Determination Date : 21 June 2007 (The Valuation Date) Maturity Date : 28 June 2007 (5 Busines Days after Determination Date) Issue price : 100% of Note Denomination Coupon : Zero 38

39 Share Basket : A basket made up of the 3 shares (each being a "Share") as shown in table below. 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 GBP GBP Altria Group, Inc. MO.N USD66.85 USD66.85 USD USD China Petroleium and Chemical Copr. (Sinopec) 0386.HK HKD2.90 HKD2.90 HKD2.03 HKD3.19 Note Denomination (ND) Particpation Rate (PR) : USD50,000 face value : 320% 39

40 Initial Spot Price of i th : Share (S i,0 ) Reference Price of i th : Share (K i ) Barrier Price of i th : Share (S barrier, i ) Lock-in Levels : The market price of the i th Share as shown in the table above. 100% of Initial Spot Price of the i th Share in the Share Basket. 70% of Initial Spot Price of the i th Share in the Share Basket. 110% of Initial Spot Price of the i th Share in the Share Basket. 40

41 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 ) : 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 Hong Kong Time on the Valuation Date. (Determination Date). 41

42 Worst Performing : Share Means the Share that has the lowest value value (the "Worst Performance") on the Valuation Date (Determination Date) according to the following formula. S S i, f i, % Mathematically, Worst Performance is defined by the following formula Performanc e worst S S i, f i, % 42

43 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

44 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 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

45 S where Min i1to3 S i,lookbackminimum i,lookbackminimum Min100%,Max S Minimum i S i,0,90% S Minimum i and where is the lowest daily closing price observed in respect to the i th Share during the look-back Period. 45

46 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 : (Performance Locked ) The highest Lock-in Level reached among the Lock-in Levels reached in respect of all Lock-in Events where applicable. 46

47 Redemption Amount : at maturity (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. ND 100% PR Max i, f Performance 1, Min, Locked i1to3 S S i,0 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. 47

48 (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. ND 100% S, Max 0,Min, 1 to3,0 i f PR i Si 48

49 (Case 2b): at least one Barrier Event has occurred during the Monitoring Period. (Case 2b-i): and if Performance worst 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. ND 100% or S, Max 0,, 1 to3,0 i f PR Min i Si 49

50 (Case 2b-ii): and if Performance worst < 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. S, i f ND PR Max0, Min, i1to3 Si,0 where Performance worst S Min i1to3 S i, f i, % 50

51 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

52 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. 1 st year annual coupon is guaranteed at 6.5% (very juicy), payable semi-annually. 52

53 The remaining coupon rate of 1% will be based on the LIBOR movement. The inverse floater formula is max{7% 2 6-month LIBOR (in arrears), 0} 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. Early termination Once the accumulated coupon payment reached the target rate, the deposit will be terminated automatically. 53

54 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

55 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. max{0,10% average performance of the 6 worst 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. 55

56 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

57 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

58 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

59 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

60 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

61 Point-to-Point The contingent claim C(t) in year t for one unit of GEB can be represented as followed t t S( t) C( t) max(min(1 Rt,(1 ) ),(1 g) ), Rt 1. S(0) 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

62 The payoff in year t for one unit of GEB is given by t C( t) max max(min(1 R s,1 ),1),(1 g) s1 where the random variable R S again measures the appreciation of the referenced index fund in year s. R S is solely determined by the index levels at the beginning and the end of year s: S( s) R s 1. S( s 1) 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. t 62

63 High Water Mark (Lookback) C( t) t max(min(1 R t,(1 ) ),(1 g) t ), where R t max S( s) s (0, ] t S(0) 1. 63

64 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. t t max min 1R (1 ),(1 g) if S( s) B C ( t t) (1 g) S( t) S (0) where 1. R t t else 64

65 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. C max (1 g) ( t) s1 t t max min(1 R t,1 ) t,(1 g) t if else S( s) B 65

66 Average C( t) max t t min(1 R,(1 ) ),(1 ) t g, where R t t S( s) s1 S(0) 1. 66

67 Cap rate Guaranteed annual return Number of simulation Initial index level Barrier level Initial interest rate Initial volatility Particip rate Point-point Ratchet Point-to- Point with barrier Ratchet with barrier Average 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

68 Impact of cap rate Participation rate = 0.8 Cap Rate (%) Point-to- Point Ratchet Lookback Point-to- Point with barrier Ratcjet with barrier Average No Cap 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

69 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- Point Ratchet Lookback Point-to- Point with barrier Ratchet with barrier Average No The increasing guarantee interest will increase the price of the contracts. 69

70 Name Guaranteed 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, 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

71 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 a x (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 max( gfna65( n) Fn,0) This option is in-the-money when a 65 (t) is greater than 1/g and out-of-themoney otherwise. 71