Derivatives Soluion

8/4/2019 Derivatives Soluion

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Derivatives Soluion: Application for FinancialFuturesCase Solution for 'Peoples Federal Savings Bank' 1. Should Peoples Federal Savings have hedged its September 1 savings certificate

rollover?

Yes. The reasons are explained as below:Peoples had accumulated assets of $556m. These assets were funded by short term consumerdeposits, consisting largely of 3-month fixed rate savings certificates. These savings certificates werehighly affected by interest rate fluctuations. The long term loans provided to people generate interestearnings which are do not increase or decrease with the interest rate fluctuations. Therefore, there

was a mismatch between the interest rates earned by the bank and the interest rates that it had togive out. This caused large losses over the period 1979-1982 when interest rates rose.Table 1.1

The bank would violate the regulatory capital requirements if its losses were not controlled. The T- bill interest rates were on the rise.$400mm in savings certificates were to be rolled over on September 1. If interest rates continued torise, then these certificates would be rolled over at the prevalent high interest rates (as mentioned inthe case, the savings certificate interest rate was fixed at a spread over the T-bill interest rate).If the firm hedges itself from the interest rate fluctuations, then the loss that would be caused due tothe savings certificate rollover at a high interest rate would be offset by the futures position.

Let us look at this in detail:

From exhibit 3, Profit and Loss Statement, comparison of the interest payment expenses ( as denoted by Dividends) has increase from 1979 to 1981 by 104.3% which is attributed to the rise in T-billsinterest rates. Table 1.2

Time Period 1981 1980 1979 % Increase from 1979 to1981

Dividends (denotes interest paid or credited to

members)40,162 26781 19660 104.3%

Net Income (3125) 800 2169

Table 1.3



Time

Period

1981 Jan -

Jun

1982 Jan-

Jun

% increase in the first 6 months of 1981 as compared to the

first 6 months of 1980

Dividends 19982 22602 13.11%

Due to the above reasons, Peoples Federal Savings Bank took the right decision of hedging itself fromthe interest rate fluctuations and the effect a further rise in interest rates would have on theSeptember 1 savings certificate rollover.

2. What would you have advised Mr. Myers to do on August 6?

On 6th August I would have advised Mr Myers to hold on to the futures position even though his firmhad already paid $690000 in variation margin calls. Mr Myers took short positions on the T-billfutures contracts in order to hedge the firm against the risk of rolling over the September 1 savingscertificates at a high interest rate. Since his futures position is not a speculative position, he does notneed to worry about the margin calls on his futures contracts.The strategy used for hedging with short futures contracts was as below:If interest rates rise, the futures price would fall and a profit would be registered. At the same time,

the savings certificates would be rolled over at a higher interest rate. Therefore, the futures contract would offset his losses from the rollover at high interest rate.If interest rates fall, the futures price would rise and a loss would be registered for the futurescontracts. However, the savings certificates would be rolled over at a lower interest rate.Example: Please refer the table below to see how a rise/fall in interest rates affect the futuresposition and savings certificate roll over.Table 1.4Interest Rate for Savings Certificate Interest Payment on Savings Certificate for 3 months Interestpayment Profit/Loss Futures Price Total futures position Futures profit/loss Net Profit/LossMay-20 11.5 11500000 88.58 88580000Jun-25 13 13000000 -1500000 86.41 86410000 2170000 670000

Aug-10 8 8000000 3500000 91.1 91100000 -2520000 980000

Therefore, on 6th August, I would have advised Mr. Myers to enjoy his vacation without worryingabout the rising interest rates as his position is hedged.

3. How should Mr. Myers explain his futures losses to the board on August 27 ?

First, Mr. Myers should explain the board his reasoning on taking short futures position on 90 day T-Bill contracts. As per table 1.1, 1.2 and 1.3 in answer no. 1, Mr. Myers can establish the need to hedgeagainst the risk of increasing interest rates.Mr. Myers should then explain to the board the hedging strategy. An example of the hedging strategy is as below:Table 1.5Interest Rate for Savings Certificate Interest Payment on Savings Certificate for 3 months Interestpayment Profit/Loss Futures Price Total futures position Futures profit/loss Net Profit/LossMay-20 11.5 11500000 88.58 88580000Jun-25 13 13000000 -1500000 86.41 86410000 2170000 670000

Aug-20 8 8000000 3500000 91.1 91100000 -2520000 980000

The short position of futures essentially fixes the interest rate for September (September 1st beingthe savings certificates roll over date). The interest rate as on May 20th was 11.5% ( from exhibit 4.90-Day T-Bill Rates). In the absence of a hedging strategy, if the interest rates go up to say 13%, there

would be a loss of $1.5m on the savings certificate roll over. As we see from Exhibit 4, the interestrate on a 90 Day T Bill had risen to approximately 13%. On the other hand, if the interest rates



reduced to say 8%, then the savings roll over interest rate will be 8% + spread.

Essentially, this implies that a hedging strategy will not only limit the firm¶s downside but also thegain. However, at the contract maturity date, the firm should still be able to lock in an interest rate of 11.5% ( May 20th rate).

As per the table above, the position on the short futures contract is -$2.52mm, but the gain fromreduction of interest rate on savings certificate roll over is $3.5mm, the net gain being close to$1mm.

Therefore, Mr. Myers should be able to justify his position well with the board members. 2. Alpha Investors1. What futures position should Jim take to hedge his portfolio?

Jim should take a short future position S&P 500 stock index. This will help in hedging Jim¶s portfolioagainst the fall in the market prices. It will also lock the price of the stock portfolio at the currentprice. However, he needs to aware that the hedging position limits not only loses but also the profits.

Any profit he makes by the increase in the value of the stock portfolio will be offset by loss he will

make in his future position. Similarly, any loss he makes in the value of the stock portfolio will beoffset by the profit in futures position.He should short Dec S&P futures as he faces uncertainty till September end. This will ensure that he

will not be called to deliver the contract and is able to close out the position without settling. Heshould by 434 futures contract. Please see table below for details.N* = *P/F 1.18P 36129094Contract Size 500Spot Price of the index 196.65F 98325

N* 434

= 1.18P = 361290942. What risks can Jim eliminate by shorting the S&P 500 stock index futures contracts? How effective do you except his hedge to be?Jim can eliminate two types of risks associated with a stocks portfolio.If the hedger feels that the stocks in the portfolio have been chosen well. In these circumstances, thehedger might be very uncertain about the performance of the market as a whole, but confident thatthe stocks in the portfolio will outperform the market. A hedge using index futures removes the risk arising from the market moves and leaves the hedger exposed only to the performance of theportfolio relative to the market.

Another reason for hedging may be that the hedger is planning to hold a portfolio for a long period of time and requires short-term protection in the uncertain market situation. The alternative strategy of

selling the portfolio and buying it back later might involve unacceptably high transaction cost.The hedge will protect Jim from market fluctuation as is illustrated in the table below.

Current Value of the portfolio 36129094Spot price of index on Sept 30 180 185 191.85 193.65 196.65 199.75Futures Price of Index today 196.65 196.65 196.65 196.65 196.65 196.65Futures Price of Index on Sept 30th 182 187 193.85 195.65 198.65 201.75Gain on futures position= N*(Difference in price)*Contract size 3176017 2092052 607020.222216792.9 -433586 -1105644



Profit/ Loss on index -6.18% -3.57% 0.00% 0.94% 2.50% 4.12%Dividend Payout by index 1.04% 1.04% 1.04% 1.04% 1.04% 1.04%Retun on Market -5.14% -2.53% 1.04% 1.98% 3.54% 5.16%Risk Free interest rate 2.36% 2.36% 2.36% 2.36% 2.36% 2.36%Expected Return on portfolio ( using WACC) -6.49% -3.41% 0.80% 1.91% 3.76% 5.66%Expected portfolio value in three months including dividends (without hedge) 33785939 34897024

36419210.6 36819201 37485852 38174725Profit or loss without hedging -2343155 -1232070 290116.625 690107.2 1356758 2045631Total Value of position in three months (with hedge) 36961956 36989076 37026230.8 3703599437052266 37069081Profit or loss with hedging 832861.7 859982 897136.846 906900.2 923172.3 939986.9Return of the position 2.31% 2.38% 2.48% 2.51% 2.56% 2.60%

3. What return can Jim expect to earn during the third quarter of 1985 assuming headopts your hedging strategy?

Jim is going to short the index futures as a hedging strategy and not as a speculating strategy. SoJim¶s aim is just to ensure that his portfolio is hedged in the 3 months of market uncertainty.However, using the hedging strategy as suggested, Jim can earn the following return based on theindex futures price as on 30th September.Extracting from the table above:

Profit or loss without hedging -2343155 -1232070 290116.625 690107.2 1356758 2045631Total Value of position in three months (with hedge) 36961956 36989076 37026230.8 3703599437052266 37069081Profit or loss with hedging 832861.7 859982 897136.846 906900.2 923172.3 939986.9Return of the position 2.31% 2.38% 2.48% 2.51% 2.56% 2.60%

4. Auto StarIf you were Rob Rough, what advice would you give to Edith Cooper?

Auto Star¶s extreme leverage and reliance on variable rate financing exposed the company to anunacceptable level of financial risk. Hence hedging is the best way for it to proceed. The risks whichthe company currently faces are:1. The market fluctuation in the exchange market.2. Market fluctuations in the prime rate market.3. The company was highly levered and paid huge interest every year.However, Hedging will have its own problems as the movement in the value of the position in the



hedge can cause the company to pay huge amount of margin money.The company should1. Hedge the exchange rate movements2. Hedge the interest rate movements.In order to hedge itself from the interest rate movements the company should look at cross hedging

with either a T-Bill or CD futures contract as futures contracts on prime-based instruments were not

available.The cross-hedge needs to fulfill two criteria:1. It should have a strong co-relation to the movement of the prime rates so as to hedge prime rateexposure.2. It should also have a limited downside and hence should limit the payment in the margin account.

Analyzing the correlation between prime vs T-Bill rates and prime vs 3-month CD rates in Exhibit 2, we see that the T-Bill futures have a stronger co-relation to the prime interest rate than the CD futurecontracts. To limit the downside it should minimize the number futures contract it needs buy tocover its risk. It should find the optimum hedge ratio by calculatingh*=s/f

5. Stock Index Futures Arbitrage

1. What is the theoretical price of the MMI March ¶86 futures contract?

The theoretical price of MMI March ¶86 futures contract is

F0=S0*e(r-q) TF0=311.74*e ((0.068*23/365)-3.41/1374.5))Therefore, the theoretical price is 312.30.2. Assume that Jim is subject to a $5,000,000 position limit. What position should he take to exploitthe mispricing of the March ¶86 MMI futures?

On Feb 26th, the March 86 MMI futures ( which were to expire on March 21) were priced at $313.55.The expected futures price is $313.03.Therefore since F0 > S0*e(r-q) T, there is an arbitrage opportunity for Jim. Profits can be made by

buying the stocks underlying the index at the spot price (i.e, for immediate delivery) and shortingfutures contracts.

1) Jim has to keep 10% margin money. So Jim needs to keep $500,000 as margin money.2) Jim would borrow $5,000,000 + $500,000 = $5,500,000 at 8% interest rate.3) Jim would go long on the underlying securities of the MMI index for $5,000,000. No. of stocksthat Jim would buy for each underlying security is showed in the detailed calculation below.4) Jim would earn a dividend of $8353.36 on the underlying stocks for 23 days.5) Jim would short MMI futures contracts at $313.55.6) End of 23 days: Jim would close out his futures contracts by delivering the underlying assets.7) Jim would pay interest on $5,500,000 that he borrowed + interest of 8%.Spot Price MMI Index on 26th Feb 311.741 month Treasury bill rate 0.068

Dividend 3.41Total stock price 1374.5Dividend interest rate 0.002480902Number of days 23Expected index future price 312.3028957Index spot price after .25% impact 312.51935

Spot price of the Index 311.74



Value of 1 contract ($250*Index Number) 77935Position limit 5000000No of contracts that can be bought 63.99603737No of contracts that can be bought ( after rounding off) 64

Value of position 5000000 Value of margin 500000

Amount of the money needed to be borrowed 5500000Price of the March futures contract 313.55

Major Market Index Profile: Investment of $5000000Company Stocks (20) Stock Price Stock Price after market impact Stocks purchased Dividend pershare Total Dividend received

American Express 64 64.16 3897 0 AT&T 22.5 22.55625 11083 0Chevron 37.875 37.9696875 6584 0Coca-Cola 92 92.23 2711 0.78 2114.28Dow Chemical 48.75 48.871875 5115 0Du Pont 70.5 70.67625 3537 0

Eastman Kodak 55 55.1375 4534 0Exxon 54.875 55.0121875 4544 0General Electric 75.5 75.68875 3303 0.58 1915.74General Motors 78.25 78.445625 3187 0IBM 158.125 158.5203125 1577 0International Paper 57 57.1425 4375 0Johnson and Johnson 48.375 48.4959375 5155 0Merck 150.75 151.126875 1654 0.9 1488.8153M 97.25 97.493125 2564 0Mobil Oil 30.175 30.2504375 8264 0Philip Morris 101.175 101.4279375 2465 1.15 2834.525Procter and Gamble 67 67.1675 3722 0Sears 42.875 42.9821875 5816 0

U.S. Steel 22.625 22.6815625 11022 0Total 1374.6 8353.36

3. What rate of return can Jim expect to earn on his position?

Jim would earn a return of without considering interest payment.

Collect dividend on stocks 8353.36delivery on futures contract 5016800Final cash flow not considering interest 5025153Investment made 5000000Return on Investment 7.96%

Jim would lose money in this position if interest is considered. The calculations are as below:



Time Action Cash-Flow Now Short index futures contract 0Borrow @ Spot price of index (considering 10 % margin) at 8% 5500000Buy stocks in index -5000000Pay margin -500000

After 23 daysCollect dividend on stocks 8353.360002Cash flow from delivery of assets underlying futures contract 5016800Pay back loan + interest -5527796.03Receive the margin back 500000

Net Cash flow -2642.669757

Though the entire process appears somewhat simple, it is actually more complicated. Borrowing andlending rates are never the same as we see in this case. Thus there is an imperfect market here.Investors have to pay a commission for buying and selling index portfolios, though Jim here did nothave to pay any commission.1) The risk free rate is 6.8% but Jim can borrow only at 8%.2) Buying the underlying stocks impact the market.3) There is the requirement of margins in the stock index futures market.4) The different between the expected and the actual futures price have to be large enough toaccommodate the imperfect market conditions.

All the four aspects have to be considered to arrive at the net benefits accruing from index futuresarbitrage.

Thus due to the imperfect market conditions, Jim would not be able to make a profit on thisarbitrage opportunity.4. Who is addition to securities dealers, would you expect to engage in index-futures arbitrage?

1) Asset managers of mutual funds and hedge funds2) Those who already own the underlying assets.3) Speculators (Eg., Portfolio managers)

5. Why do index futures often trade at a premium or discount to their theoretical values? How do youexpect the pricing efficiency of broader market index futures, like S&P 500, to compare to the pricingof MMI futures?If we assume that CAPM is true, the relationship between the futures price and the expected futuresspot price depends on whether the return on the asset is positively or negatively correlated with the

return on the stock market.

F0=E (ST) e(r-k) T where k is the investor¶s required rate of return, r is the risk free return, ST is the price of the asset attime T, F0 is the futures price today and E denotes the expected value.Positive correlation will tend to lead to a futures price lower than the expected future spot price.k>r, so that F0 < E(ST). This shows that when the asset underlying the futures contract has positivesystematic risk, we should expect the futures price to understate the future spot price. An example of an asset that has positive systematic risk is a stock index.



Negative correlation will tend to lead to a futures price higher than the expected future spot price.K E(ST). Thus, when the asset underlying the futures contract has negative systematic risk, weshould expect the futures price to overstate the expected futures spot price.

Only when the correlation is zero will the theoretical futures price be equal to the expected futuresspot price. i.e., k=r. Therefore, the futures price is an unbiased estimate of the expected future spotprice when the return from the underlying asset is uncorrelated with the stock market.

A broader market index will ensure that the return from the underlying asset is uncorrelated with thestock market and therefore the theoretical price will tend to be equal to the expected futures spotprice. Therefore, there would be greater pricing efficiency.

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Derivatives Soluion