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Topic 4 Futures and Forwards II

Topic 4 Futures and Forwards II. Futures and Forwards II. In our second futures and forwards lecture we will: Develop pricing models for forwards when the underlying asset has cash flows associated with it during the period of the forward contract.

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Topic 4 Futures and Forwards II

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  1. Topic 4Futures and Forwards II

  2. Futures and Forwards II • In our second futures and forwards lecture we will: • Develop pricing models for forwards when the underlying asset has cash flows associated with it during the period of the forward contract. • Discuss the relationship between futures and forward prices. • Examine the relationship between forward/futures prices and the expected future spot price. • Discuss accounting issues relating to hedging.

  3. Forward Contracts on a Security that Provides a Known Cash Income • Recall that we demonstrated that a forward contract on a security that pays no dividends will have, at time 0, a delivery price of: and that we could attribute this formula to the notion of the opportunity cost that the short party faced: to induce the short party to enter the contract, the long party must pay them interest at least in the amount that the short could get if they simply sold the asset now and invested it at the risk-free rate.

  4. Forward Contracts on a Security that Provides a Known Cash Income • Recall, however, that during the period of the forward contract, if the short party physically holds the underlying asset, then they will garner any benefits that accrue to the asset during that period. • For example, if the forward contract were written on a stock, and the stock paid a dividend, then if the short party physically held the stock on the ex-dividend date, they would receive the dividend. • The short party still has no risk (ignoring credit risk) in the forward contract; as a result they should still only earn the risk-free rate for being the short party. Consequently, the benefits that accrue to holding the underlying asset would reduce the amount that the long would have to pay the short to induce them to enter the contract.

  5. Forward Contracts on a Security that Provides a Known Cash Income • To see this, consider a security that had a perfectly predicable set of cash flows that will occur between t and T. • Examples are stocks that have known dividends and coupon payments from bonds. • Denote the present value of those cash flows as I (discounting at the risk free rate). For their to be no arbitrage, the relationship between F and S must be: F = (S-I)erT

  6. Forward Contracts on a Security that Provides a Known Cash Income Example: • Assume a 10 month forward on a dividend-paying stock. Current price is $50. Assume r = 8%, and dividends of .75 in 3, 6, and 9 months. I = .75e-.08(.25)+ .75e-.08(.5) + .75e-.08(.75) = 2.162 • T-t = 10/12 = .83333 years: F = (50-2.162)e.08*.83333 = 51.136

  7. Forward Contracts on a Security that Provides a Known Cash Income Second example: • Assume a 3 month forward on a dividend-paying stock with current price of $100. Assume r = 4%, and the stock will pay a dividend of $2.00 in 1 month. I = 2.00-.04(1/12) = 1.9933 • T-t = 3/12 = .25 years: F = (100-1.9933)e.04*.25 = $98.99 • We can use this example to demonstrate the arbitrage opportunity that enforces this rule.

  8. Forward Contracts on a Security that Provides a Known Cash Income • To see this, consider what would be the case if this did not hold: • Case 1: F > (S0-I)erT • Let’s say that we saw F=101, how would the arbitrageur exploit the opportunity? • At time 0: • Short the forward contract (i.e. agree to deliver the stock in three months for $101). • Borrow $100 today at the risk-free rate and buy the stock. • At time 1 month: • Reinvest the dividend at the risk-free rate. • At time 3 months they do 4 things: • Deliver the stock into the forward contract and receive $101. • Receive $2.103 (2e.04(2/12)) from the reinvested dividends. • Repay $101.005 (100e.04(3/12)) for the $100 you borrowed at time 0. • Net time 3 cash: +101 + 2.013 – 101.005 = $2.008

  9. Forward Contracts on a Security that Provides a Known Cash Income • We can show this on a timeline: Actions 1 3 0 Cash Positions

  10. Forward Contracts on a Security that Provides a Known Cash Income • We can show this on a timeline: Actions • Short futures contract • Borrow $100 at 4% • Buy stock for $100 1 3 0 Cash Positions • 0 • +$100 • -$100 • $0

  11. Forward Contracts on a Security that Provides a Known Cash Income • We can show this on a timeline: Actions • Receive $2 dividend • Reinvest at 4% • Short futures contract • Borrow $100 at 4% • Buy stock for $100 1 3 0 Cash Positions • +$2 • -$2 • $0 • 0 • +$100 • -$100 • $0

  12. Forward Contracts on a Security that Provides a Known Cash Income • We can show this on a timeline: Actions • Receive $2 dividend • Reinvest at 4% • Receive $101 from futures delivery • Repay loan ($101.005) • Receive reinvested dividends (2.013) • Short futures contract • Borrow $100 at 4% • Buy stock for $100 1 3 0 Cash Positions • +$2 • -$2 • $0 • +$101.000 • -$101.005 • -$ 2.013 • + 2.008 • 0 • +$100 • -$100 • $0

  13. Forward Contracts on a Security that Provides a Known Cash Income • What about the opposite situation? • Case 2: F < (S0-I)erT • Let’s say that we saw F=98, how would the arbitrageur exploit the opportunity? • At time 0: • Take a long position in the forward contract (i.e. agree to buy the stock in three months at $98). • Short the stock for $100 today and invest at risk-free rate. • At time 1 month: • Borrow $2 at risk free rate. • Pay $2 to the person from whom you borrowed the stock. • At time 3 months: • Receive 101.005 (100e.04(3/12)) from $100 you invested at risk-free rate. • Buy stock for $98 via forward contract. Return stock to original owner. • Repay $2.013 (2e.04(2/12)) for the $2 you borrowed at time 1. • Net time 3 cash: +101.005 - 2.013- 98.00 = $0.9920

  14. Forward Contracts on a Security that Provides a Known Cash Income • We can also show this on a timeline: Actions 1 3 0 Cash Positions

  15. Forward Contracts on a Security that Provides a Known Cash Income • We can show this on a timeline: Actions • Go long futures contract • Short stock at $100 • Invest $100 at 4% 1 3 0 Cash Positions • 0 • +$100 • -$100 • $0

  16. Forward Contracts on a Security that Provides a Known Cash Income • We can show this on a timeline: Actions • Receive $2 dividend • Reinvest at 4% • Short futures contract • Borrow $100 at 4% • Buy stock for $100 1 3 0 Cash Positions • +$2 • -$2 • $0 • 0 • +$100 • -$100 • $0

  17. Forward Contracts on a Security that Provides a Known Cash Income • We can show this on a timeline: Actions • Receive $2 dividend • Reinvest at 4% • Receive $101 from futures delivery • Repay loan ($101.005) • Receive reinvested dividends (2.013) • Short futures contract • Borrow $100 at 4% • Buy stock for $100 1 3 0 Cash Positions • +$2 • -$2 • $0 • +$101.000 • -$101.005 • -$ 2.013 • + 2.008 • 0 • +$100 • -$100 • $0

  18. Forward Contracts on a Security that Provides a Known Dividend Yield • Some securities such as stock indices and currencies essentially have a continuous dividend yield instead of a discrete dividend. • Thus you can think of the asset as paying a continuous dividend at rate q, based on the value of the security. • Thus if q=.10, and the security price is $50, the dividends in the next small period of time are paid at the rate of $5 per year. • The same basic logic for pricing forward contracts on instruments that pay a discrete dividend applies to forward contracts on instruments that pay a dividend yield: the total earnings of the short party still should still be the risk-free rate. As a result the dividend yield reduces the rate that the long party must pay.

  19. Forward Contracts on a Security that Provides a Known Dividend Yield • Thus, the formula for determining the forward price is: F = S0e(r-q)T where q is the dividend yield, expressed in annual terms. • For example, if the S&P 500 had a dividend yield of 4% the six-month risk free rate were 3%, and the value of the S&P were 1009.37, then the 6 month forward price for an S&P forward contract would be: F = 1009.37e(.03-.04)(.5)= $1004.34 • Note that when q>r, you will find an inverted market. The next page shows CME quotes for the S&P 500 index for 9/13/06.

  20. Forward Contracts on a Security that Provides a Known Dividend Yield • What does the fact that the prices are rising with maturity indicate?

  21. Forward Contracts on a Security that Provides a Known Dividend Yield • We can develop a somewhat more formal proof of the pricing formula: • Again consider two portfolios, • One long forward contract and cash equal to Ke-r(T-t) (f+Ke-r(T-t)). • 1*e-q(T-t) units of the security with all income being reinvested in the security. Thus at time T you will once again have one unit of the security, which is worth ST. • Clearly A and B once again have the same payoffs at time T, and so once again, setting them equivalent:  f+Ke-rT = Se-qT Or f = Se-qT - Ke-rT setting f = 0 and solving for K leads to: K = F = Se(r-q)T

  22. Forward Prices Versus Futures Prices • Hull demonstrates that if interest rates are constant, then forwards and futures prices are the same. • Obviously in the real world interest rates are not constant, and so forward and futures prices are not the same. • The reason for this is the discounting of the mark to market cash flows. • One way to see this is to consider if St is highly correlated with rt - so that when r increases, S tends to increase. • When rates rise, the spot price of the asset will rise as well and so will the futures price. • A short party in the futures contract will have to raise cash to mark to market – when rates are high. • If rates fall, the spot price of the asset will fall, and so will the futures price. The short party will receive cash from the mark to market, when rates are low. • So on average the short party in the futures contract must raise funds when rates are high and invest funds when the rate is low. If instead they had used a forward contract they would not have any marking to market – so, assuming they wind up at the same closing price, the short futures contract would be more expensive than the short forward contract. • The short party, therefore, would demand a higher futures price to induce them to enter into this contract as opposed to a forward contract.

  23. Forward Prices Versus Futures Prices • There are a some empirical papers that address the differences between futures and forwards prices. • Cornell and Reinganum studied forward and futures prices on Pounds, C. Dollars, Marks, Yen and Swiss Francs and found little statistical differences between them. • For commodities and precious metals, French as well as Park and Chen, find that forward and futures prices are significantly different from each other, and that the futures prices tend to be higher than the forward prices.

  24. Stock Index Futures • A stock index tracks the changes in the value of a hypothetical portfolio of stocks. The weight of a stock in the portfolio equals the proportion of the portfolio invested in the stock. • A few tidbits about stock index futures: • The percentage increase in the value of a stock index over a small interval of time is usually defined so that it is equal to the percentage increase in the total value of the stocks in the portfolio at that time. • Cash dividends received on the portfolio are ignored when percentage changes in most indices are calculated. • If the hypothetical portfolio of stocks remains fixed, the weights assigned to each stock do not remain fixed. Consider this example of three stocks.

  25. Stock Indices • In practice, the weights for stock indices are calculated in one of two ways: • Price weighted (DJIA and Nikkei 250): Stocks are price weighted -> i.e. assume you own one share of each stock and then add prices together (adjusting for splits, changes, etc.) • Market capitalization weighted (S&P and most other indices): weighted based on percentage of market capitalization for entire portfolio:

  26. Futures Prices of Stock Indices • As mentioned earlier, most indices can be thought of as a security that pays dividends. Usually it is convenient to consider them as paying dividends continuously. Let q be the dividend yield rate, then the futures price is given by: F = Se(r-q)(T-T). • If you were uncomfortable with using the dividend yield approach (say if you were using a small index), then you could estimate individual dividends and apply the formula for a futures contract on a security paying a known dividend.

  27. Index Arbitrage • Once again we can use arbitrage arguments to prove the equation: • If F > Se(r-q)(T-T), then buy the stocks underlying the index and short futures contracts. If F < Se(r-q)(T-T), do the opposite: short the stock and buy the futures contract. • Because the transactions costs of buying the stocks are relatively high, arbitrage can sometimes be found in the market; this is known as index arbitrage. So-called Program Trading is where your computer system executes trades as soon as the arbitrage becomes possible. • Note that during the 1987 crash, there was a breakdown in the relationship between F and S. This was due to poor mechanics of trading (on the 19th) and then due to restrictions placed by NYSE on program trading (on the 20th).

  28. Hedging Using Index Futures • Consider the problem faced by the manager of a large equity portfolio: • They may wish to change the risk-profile of the portfolio, but do not want to bear the transactions costs associated with changing the actual holdings. • Consider, for example, a large mutual fund such as Fidelity Magellan, which has roughly $45 Billion in assets.* • It has a Beta of approximately 1.01 (relative to the S&P 500). So it has an almost perfect exposure to the general market risk of the stock market. • Let’s just assume for a moment that the managers of Fidelity decided that they wanted to reduce the exposure of the fund to the stock market (assuming that the prospectus of the fund would allow them to do this.) Let’s say they want to reduce the Beta of the portfolio to 0.5. *As of 9/12/2003. Source: Fidelity web site (http://personal.fidelity.com/products/funds/mfl_frame.shtml?316184100 )

  29. Hedging Using Index Futures • One way to do this would be to sell half of the fund’s assets and invest it in the risk-free rate. • This is not practical, there is no way they could unload $22.5 Billion in stocks without depressing the market. They would get terrible prices for the assets, not to mention that they would lose a lot of money on commissions as well. Further, selling the assets would take quite a bit of time. • In addition, the fund managers may only want to temporarily change the risk profile of the portfolio. They may want to hold the stocks for the long run, but want to temporarily reduce the risk of the portfolio – for say 3 months. • A relatively easy way for them to change the portfolio’s risk would be to hedge that risk using a stock index futures contract. • The next few slides discuss the general method for doing this (as outlined in Hull’s book), and then we will return to an example based on Magellan.

  30. Hedging Using Index Futures • Recall the definition of β: it is the slope of the regression line between the excess returns (i.e. returns less the risk free rate) on a portfolio of stocks and the excess return on the market (i.e. the stock market as a whole.) • When β=1, the return on the portfolio tends to mirror the return on the market: i.e. the portfolio has the same risk as the market. When β=2, the excess return on the portfolio is twice that of the market - its risk is twice as high. • Consider also that for a futures contract on large indices, such as the S&P 500, the index can serve as a proxy for the market as a whole. • Thus we can say that if a portfolio has a β=3, its expected excess return will be equal to three times the excess return on the underlying index.

  31. Hedging Using Index Futures • Hull defines the following terms: • P - the value of the portfolio • A - underlying asset value of one futures contract (if one futures contract is on m times the index, A = mF) • If you are working with a portfolio that exactly mirrors a specific index upon which futures contracts are written, and you want to completely hedge the risk of the portfolio, all you have to do is short an appropriate number of future contracts. • The optimal number of contracts to short when hedging is: N* = P/A

  32. Hedging Using Index Futures • If, however, your portfolio does not exactly match the underlying index, (but assuming that your underlying index has a Beta of 1), then the optimal number of futures contracts to completely hedge the risk is proportional to the Beta of your portfolio: N* = β(P/A) • Note that if the your index had a Beta that was something other than one, the optimal number futures contracts would be proportional to the ratio of your portfolios Beta to the Beta of the underlying index, with both of those Beta’s defined relative to the same underlying market.

  33. Hedging Using Index Futures • The previous formulas, while important, do not exactly help our Magellan fund manager. They do not want to completely eliminate the exposure to the stock market (i.e. to have a net β of 0), but rather want to reduce that exposure in half, i.e. to have a net β of 0.5. • Let β be the current portfolio position and let β* be the desired portfolio position. To get to the new portfolio position the optimal number of index futures contracts to short is: N*= (β- β*)(P/A) • Note that if β*> β, that is, you want to increase risk, you will get a negative number for N*, that means you take a LONG position (since N* is the number of contracts to short). Hull flips β and β* and then says that is the number of contract to go long, but the effect is the same!

  34. Hedging Using Index Futures • So let us return to our Magellan example. • Remember that the β of Magellan is 1.01, and that we want to reduce it to 0.50. • We will assume that we want to reduce that risk for 1 month. • We should use the S&P 500 index futures contract to do this. • There are two S&P 500 Index Futures contracts, and each trade on the CME. • The S&P Index Futures contract is based on $250 times the index amount. • The “Mini” S&P Futures contract is based on $20 times the index amount. • Obviously to hedge a $45 Billion position, we will use the larger contract.

  35. Hedging Using Index Futures • On 9/12/2006, the S&P 500 settled at 1318.07, and the S&P 500 index futures contract settled at 1317.60. The value of the portfolio was (roughly) $45,000,000,000. Thus: P = $45,000,000,000 A = $250*1317.60 = 329,400.00 β = 1.01 β*= 0.50 • So our optimal number of contracts to hedge would be: N*=(β-β*)*(P/A) N*=(1.01-0.50)*(45,000,000,000/329,400) N*=(0.51)*(136,612.02) = 69,672 contracts. • In reality, there will be a problem with this, because the largest position the CME allows is 20,000 contracts in a given month, so you would really have to spread your hedge over 6 or 7 months worth of contracts, but we will ignore this for now.

  36. Hedging Using Index Futures • So how well would the hedge work? • On 9/13/2003, the S&P is at 1318.07, and the net value of the portfolio was $45,000,000,000. • Assume that on 9/18/2006, the S&P closes at 1351.09, so it had increased in value by 2.505%. Since the β of the portfolio was 1.01, we would expect the value of the portfolio to increase by 2.505*1.01, or 2.53%. Thus, the new portfolio value would be: 45,000,000,000 * 1.0253 = 46,127,250,000.That is you have made (46,127,250,000-45,000,000,000 = 1,127,250,000) on the portfolio. • Of course, the S&P index futures would rise as well, and you are short the index. Indeed, assume that the September Futures contract on 9/18/2003 settles at 1349.61. So your hedge has moved: -69,672 * (1349.61-1317.60)*250 = -557,479,467

  37. Hedging Using Index Futures • So what is the net change in your position? • Remember, you made 1,127,250,000 on the portfolio, but lost 557,479,467 on the hedge (the S&P index futures), so net you have made: 1,127,250,000 - 557,479,467 = 569,770,533 on your net position. • This is 50.5% of what you would have made had you not hedged. • Why was it not exactly 50% or 49%? Because the β of your portfolio was probably changing when rates change. • This illustrates the notion that you will frequently make more money by not hedging, but that what you are doing is reducing risk.

  38. Hedging Using Index Futures • Indeed, the following table shows the hedged and un-hedged position for various dates in September, 2006.

  39. Hedging Using Index Futures • Hedging an Individual Stock’s exposure. • Basically you just treat the stock as a portfolio, and use its β to determine the hedge ratio. • Forwards and Futures on Currencies • You are not responsible for Forwards/Futures on Currencies

  40. Commodities Futures • Commodities futures contracts fall into two groups: those where the underlying is held primarily for investment and those where the underlying is held primarily for consumption. This is really important because if it is held for consumption some of our arbitrage arguments break down and no longer hold. • Examples: Gold is held primarily for investment, whereas wheat, oil, cattle, etc. is primarily held for consumption.

  41. Commodities Held for Investment • Some commodities, primarily gold and silver, are held by a significant number of investors solely for investment. If storage costs are zero, then they can be treated as securities that pay no income. Thus the correct formula is: F = Ser(T-t) • If there is a fixed storage cost, then let U be the present value of those storage costs incurred during the life of the contract, and treat it as negative income. This allows us to use the standard formula for securities that pay a known income: F = (S - (-U))er(T-t) = (S+U)er(T-t)

  42. Commodities Held for Investment • If the storage costs are proportional to the price of the commodity, they can be treated as a negative dividend yield: F = Se(r-(-u))(T-t) =Se(r+u)(T-t) where u is a per annum proportion of the spot price. • Consider a one year gold futures contract, and assume that it costs $2 per ounce to store gold, payment made at the end of the year. If the spot price is $450 and the risk free rate is 7% for one year, then: U = 2.00e-.07=1.865, and so the forward price would be: F = (S0+U)erT = (450+1.865)e.07(1)=484.63

  43. Commodities Held for Investment • Suppose that an arbitrageur observes a violation of the pricing rule of this type: F>(S+U)er(T-t) • As usual, they would take the following actions to exploit the arbitrage: • Borrow S+U dollars at the risk-free rate and use it to buy one unit of the commodity and to pay the storage costs. • Short a futures contract on the commodity. • Clearly, then, as arbitrageurs do this F will approach (S+U)er(T-t), so the situation will not last long.

  44. Commodities Held for Investment • Suppose next that the arbitrageur notes the following: F<(S+U)er(T-t) • If the commodity is held primarily for investment they can do the following: • Sell the commodity, save the storage costs, and invest the proceeds at the risk-free rate. • Buy the futures contract. • If, however, the majority of investors are holding the commodity for consumption, not investment, then they will not be willing to execute this second strategy. • The reason for this is that they need to be able to use the commodity. Therefore this last strategy will not apply, and as a result the lower bound will not hold. Thus, for futures on commodities not held for investment, the following is the most we can say: F£(S+U)er(T-t) • If storage costs are proportional: F£Se(r+u)(T-t).

  45. Convenience Yields • In the equation above, the difference between F and (S+U)er(T-t) is the convenience yield. Frequently it can be written as: Fey(T-t) = (S+U)er(T-t) • where y is the c. yield. If using the proportional version: Fey(T-t) = Se(r+u)(T-t) • or rewriting this in the more usual form: F = Se(r+u-y)(T-t)

  46. The General Cost of Carry Model • All of the futures and spot price relationships we have seen before have a common theme. The general name for this is the “cost of carry” model. Essentially this says that the futures price is a function of the cost of carrying (holding) the underlying asset. This is consistent with our intuition that the forward price is really a form of compensating the short for “holding” the asset for the long party. • For non-dividend paying securities, the cost of carry is r. • For a stock index it is r-q, where q is the dividend yield. • For a commodity with storage costs proportional to its price, it is r+u. • Define the cost of carry as c. The general model to use, then is: F = S ec(T-t) and for a consumption asset it is: F = S e(c-y)(T-t).

  47. Futures Pricing Models • We’ve developed quite a few pricing models, let’s make sure we enunciate them: • Security paying no income: F = S0erT • Security with known discrete dividends: F = (S0-I)erT • Security with dividend yield: F = S0e(r-y)T • Investment Commodity with fixed storage costs: F = (S0+U)erT • Investment Commodity with proportional storage costs: F=S0e(r+u)T • Consumption Commodity with fixed storage costs: F<=(S0+U)erT • Consumption Commodity with proportional costs: F<=S0e(r+u)T • Consumption Commodity with convenience yield: F=S0e(r+u-y)T • General cost of carry model: • F=S0ecT, where c = r, r-q, r+u, etc. as appropriate. • F=S0e(c-y)T, where c is as above and y is the convenience yield.

  48. Delivery Choices • Most futures contracts do not specify a delivery date, but rather a range when delivery is permissible. As a result, the short party in the contract will choose to make delivery when it is to their advantage. This presents a problem when trying to determine futures prices: what day to use for T in all of the (T-t) equations? • Consider the dividend yield model: F=S0e(r-y)T. • If F increases as T increases, the benefits from holding the asset are less than the risk-free rate - so the short party must want to get their money out as soon as possible. They will then want to make delivery as soon as possible, so you should assume T will be the soonest date possible. • If, however, F decreases as T increases, the opposite is true, and they will deliver as late as possible, or so you should assume.

  49. Futures Prices and the Expected Future Spot Price • A common question is whether the futures price is the same as the expected future spot price. That is, is the price of a wheat futures contract for delivery in three months the same as the market’s expected spot price of that wheat in three months? • I would begin by pointing out that none of our pricing equations utilize the expected spot price, so I would not expect any such relationship to hold.

  50. Futures Prices and the Expected Future Spot Price • To answer this we must look at a risk and return analysis. • From CAPM, recall that there are two types of risk. Non-systematic risk is that risk which can be diversified away, whereas systematic risk cannot. In general, the higher the systematic risk of an investment, the higher the expected return demanded by an investor.  • Consider now the risk in a futures position. • Begin by assuming that at time t the speculator puts the present value of the futures price into a risk-free investment, i.e. they invest: Fe-rT at the risk free rate (giving F dollars at time T) and they take a long position in the futures contract.

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