BRUCE HONIBALL’S INVENTION Principles of Corporate Finance 7th Edition Richard A. Brealey and Stewart C. Myers MEMORANDUM To: Bruce Honiball From: Sheila Cox Re: Gibb River Bank Equity-Linked Deposits Bruce, thank you for your memo. I think you may be onto a winner with the equity-linked deposits, though my calculations suggest that we can’t afford to be quite as generous as you propose. Spotting the option. Think of it this way. Whatever happens to Australian share prices, depositors under your scheme get back their initial investment of $A100 at the end of the year. If share prices rise by y percent, they also receive a bonus of . 5y ( $A100. For example, if prices rise by 10 percent, the bonus is . 5 ( . 10 ( $A100 = $A5. If share prices fall, depositors do not receive any bonus. Thus Share prices fallShare prices rise by y% Repayment of deposit$A100$A100 Bonuszero. 5y ( $A100 The key questions are: Is this a good deal for Gibb Bank depositors? If it is, can we afford to offer it to them? We can answer these questions by considering an alternative investment strategy that generates the same esults. Valuing the option. Suppose an investor buys a 1-year call option on the market index with an exercise price equal to its current level. Let’s call the current level of the index 100. The payoff on this call option is: Share prices fallShare prices rise by y% Payoff on callzeroy (( $A100 In other words, the bonus payment on the equity-linked deposit is exactly half the payoff on an option to buy the market index at its current level of 100. Now it is easy to see how to value the equity-linked deposit. Value is equal to the present value of $A100 received at the end of the year plus half the present value of a call option. To value the payment of $A100, we simply discount at the current interest rate. For example, if the interest rate is 5 percent, the value of the promised payment is 100/1. 05 = $95. 24. Valuing the call option could be harder. Fortunately, Black and Scholes have devised a formula that does the trick.  We need the following inputs: Current level of index100 Exercise price of call option100 Interest rate5. 0% Time to maturity1 year Volatility of index25% The only number that I had to estimate is the volatility of the Australian equity index. I simply calculated the standard deviation of market returns over the past 20 years, which was about 25 percent. You can put in your own figure if you feel that market volatility is likely to change. Using my inputs, the Black-Scholes formula says the call option is worth $A12. 28.  Remember that the value of an equity-linked deposit is equal to the present value of the promise of $A100 plus half the value of a call option. Therefore the value of the deposit is 95. 24 + . 5 ( 12. 28 = $A101. 38. If this estimate is right, we would be taking in $A100 from depositors and offering them in exchange an investment worth $A101. 38. In other words, each $A100 of deposits would have a net present value for us of + 100 – 101. 38 = – $A1. 38. It appears that the upside on the equity linked deposits is just a bit too generous. However, suppose we cut the bonus to $A3 for each 10 percent rise in the index. Then the value of the deposit falls to 95. 24 + . 3 ( 12. 28 = $A98. 92. This gives NPV of + 100 – 98. 92 = + $A1. 08, which should cover administrative costs and provide a modest profit. How could we reduce risk? That depends on how we invest the depositors’ money. Australian government debt could provide a safe 5 percent return, but would not protect us if share prices take off and oblige us to pay out large bonuses to depositors. Suppose instead that we invest $95. 24 at 5 percent. We would then be sure that we could pay back the initial deposit (95. 24 ( 1. 05 = $A100). At the same time, for each $A100 of deposits we could buy . 3 call options on the market index. That would guarantee that we could also pay any bonus. We would then have a completely hedged, risk-free position. What if equity-linked deposits sell like hot cakes and it becomes difficult or expensive to buy sufficient call options on the index? There is another way to hedge our risks. Black and Scholes showed that a call option can be replicated by a mixture of delta shares and borrowing. In my example, delta works out at . 6256. So to replicate one call, the bank would need to invest . 6256 ( $A100 in the market index and borrow $A50. 28.  The net cost of replicating one call would be 62. 56 – 50. 28 = $A12. 28. Of course, to replicate . 3 calls, the bank would need to invest only . 3 ( 62. 56 = $A18. 7 in the market index and borrow . 3 ( 50. 28 = $A15. 09.  So, if necessary, we can hedge the equity-linked deposits by borrowing to buy the market.  As time passes and share prices change, we would need to adjust the amounts that the bank borrows and invests in the market index. Bear Market Deposits. I have also looked briefly at your idea of bear market deposits. Again it is useful to break the payoffs into a fixed payment and a bonus element. Suppose we stick with your suggestion to pay a bonus of $A5 for each 10 percent fall in the index. Then the payoffs from a bear market deposit are as follows: Share prices fall by y%Share prices rise Repayment of deposit$A100 $A100 Bonus. 5y ( $A100 zero Consider a 1-year put option on the market index with an exercise price equal to its current level (100). The payoff on this put option is: Share prices fall by y%Share prices rise Payoff on puty ( $A100 zero In other words, the bonus payment on the bear market deposit is exactly . 5 times the payoff on an option to sell the market index at its current level of 100. I have also calculated the value of this downside bonus. Using exactly the same inputs, I get a value for the put option of $A7. 2.  So the value of a bear market deposit is: PV(fixed payment) + PV(bonus) 100/1. 05 + . 5 ( 7. 52 = $A99. 00 We could make a small profit on the bear-market deposit as long as it does not cost too much to administer. We can get rid of all the risk of offering such a deposit by investing the money in a mixture of a straight 1-year loan and a fraction of a put option. We can also replicate the put by investing in a mixture of a straight loan and delta shares. Since delta in this case is negative, that involves selling shares in the market index. Bruce, I hope this gives you what you need to take the idea forward. Let me know if I can be of further help. Sheila March 21, 2000 MEMORANDUM To: Sheila Cox From: Bruce Honiball Re: Gibb River Bank Equity-Linked Deposits Sheila, thank you for your memo with your interesting calculations. Let’s push ahead with the idea for equity-linked deposits. The next step is develop a paper that we can present at month’s board meeting. Have a go at reworking your paper, leaving out all that higher mathematics and concentrating instead on the guts of the proposal. Get it all down in a couple pages that anyone can understand. Thanks again, Bruce ———————–  To replicate Shiela’s calculations, follow the procedures given at pp. 601-605.  Sheila’s calculations are close, but not quite right. Part of the value of the market index is comes from the present value of dividends that investors expect over the coming year. Holders of equity-linked deposits miss out on these dividends. Therefore, when valuing the option component of these deposits, she should reduce the value of the market index by the present value of next year’s dividends. Section 6 of Chapter 21 includes an example showing how to value a call option on a dividend-paying stock.  Of course, Gibb can’t actually buy a market index. It can, however, buy a portfolio of stocks that closely tracks the index.  The amount borrowed is N(d2) times the PV of the exercise price.  Gibb could “borrow” the money by simply investing $A15. 09 less in the 1-year loan.  Sheila should have added a qualification at this point. She is assuming that delta has been estimated correctly. But if the Australian market turns out to be more or less variable than in the past, delta will not be . 6256 and Sheila’s strategy will not replicate the call option exactly.  Sheila used put-call parity (see pp. 570-572): Value of call + present value of exercise price = value of put + share price Sheila calculated the value of a call as $A12. 28. Share price is 100, and the present value of the exercise price is $A95. 24. The value of a put is therefore 12. 28 + 95. 24 – 100 = $A7. 52.  The delta for a put equals the delta for a matching call minus 1. In this case delta (put) = . 6256 – 1 = – . 3744.
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