8.2. Medio ambiente y desarrollo sostenible
8.2.1. Validación de la situación ambiental de partida
Index-arbitrage strategies have grown considerably in popularity since their introduction. For example, Shalen (2002) reports that in July 2002, over 8% of trading volume at the NYSE was related to index-arbitrage programs.
Of course, in reality, implementing index-arbitrage strategies is not as simple as the example above suggests. Several problems may arise. One that we have already mentioned is that the dividend level used in the calculations represents only a forecast. If we use the cash dividend formula (4.12), we must estimate M, the present value of dividends expected over the life of the futures contract. This must be done by using estimates of cash dividends expected from each of the companies in the index and summing these up.
The alternative procedure of using a dividend yield is computationally simpler but con- ceptually requires a bit more care. Since dividends tend to be bunched, there are seasonal
effects to be taken into account. That is, the average dividend yield over a year may be higher or lower than the yield over a specific shorter period. The dividend yield d used as an input into the formula must be the appropriate one given the maturity of the futures contract.
In either case, it is possible that the realized dividend rate will differ from the estimate. Thus, the profits from the strategy are uncertain and could even be negative. The use of the word “arbitrage” in this context is somewhat liberal.
A second problem is that index-arbitrage strategies require buying or selling the spot asset, which is the underlying index. Literally buying or selling the index (i.e., the basket of stocks comprising the index) will entail substantial transactions costs among other problems. In some cases, one can use traded instruments that track specific indices (for example, Standard and Poor Depository Receipts, or SPDRs, which track the S&P 500 index). If no such instruments are available, one can use a smaller basket of stocks that tracks the index closely. In many countries, the emergence of exhange-traded funds (ETFs), which track broad-market and sectoral indices, has also helped diminish the severity of this problem. Of course, a tracking error may still remain between the exact performance of the index and that of the tracking portfolio.
Other issues too may arise that are common to most derivatives arbitrage strategies. One is execution risk. In the ideal case, the two legs of the arbitrage strategy should be executed simultaneously at the observed respective prices. While electronic trading has facilitated simultaneity considerably, some room for slippage exists. For example, the uptick rule restricts when short-selling may be possible. Second, transactions costs (bid-offer spreads) and differences in borrowing and lending rates must be taken into account in calculating whether or not arbitrage opportunities exist.
Collectively, all of these factors suggest that while large deviations from the theoretical fair price cannot persist, small deviations may not represent genuine arbitrage opportunities. The data bears this out: index futures often deviate by small amounts from their theoretical levels but rarely by substantial levels (see, for example, Figure 4.1 on the percentage mis- pricing in the CBoT futures contract on the Dow Jones Industrial Average). Shalen (2002)
This figure, taken from Shalen (2002), shows the percentage mispricing in the clos- ing level of the CBoT futures contract on the Dow Jones Industrial Average. The mispricing is relative to the theoretically fair price.
FIGURE 4.1
Mispricing in the Dow Jones Industial Average Futures Contract 1.00% 0.75% 0.50% 0.25% 0.00% ⫺0.25% ⫺0.50% ⫺0.75% ⫺1.00% ⫺1.25% ⫺1.50% ⫺1.75% ⫺2.00% ⫺2.25% ⫺2.50% 10/6/19971/6/19984/6/19987/6/199810/6/19981/6/19994/6/19997/6/199910/6/19991/6/20004/6/20007/6/200010/6/20001/6/20014/6/20017/6/200110/6/20011/6/20024/6/20027/6/2002
reports that, for example, the mean absolute mispricing in the DJIA futures contract on the CBoT has been less than 0.20% since 2000 and less than 0.15% since 2001. Mispricing tends to be highly correlated with volatility of the underlying index, perhaps because higher volatility levels increase execution and implementation risk in the arbitrage strategy.
4.7 Exercises
1. What is meant by the term “convenience yield”? How does it affect futures prices? 2. True or false: An arbitrage-free forward market can be in backwardation only if thebenefits of carrying spot (dividends, convenience yields, etc.) exceed the costs (storage, insurance, etc.).
3. Suppose an active lease market exists for a commodity with a lease rate expressed in annualized continuously compounded terms. Short-sellers can borrow the asset at this rate and investors who are long the asset can lend it out at this rate. Assume the commodity has no other cost of carry. Modify the arguments in the appendix to the chapter to show that the theoretical futures price is F = e(r−)TS.
4. What is the “implied repo rate”? Explain why it may be interpreted as a synthetic borrowing or lending rate.
5. Does the presence of a convenience yield necessarily imply the forward market will be in backwardation? Why or why not?
6. How do transactions costs affect the arbitrage-free price of a forward contract? 7. Explain each of the following terms: (a) normal market, (b) inverted market, (c) weak
backwardation, (d) backwardation, and (e) contango.
8. Suppose that oil is currently trading at $38 a barrel. Assume that the interest rate is 3% for all maturities and that oil has a convenience yield of c. If there are no other carry costs, for what values of c can the oil market be in backwardation?
9. The spot price of silver is currently $7.125/oz, while the two- and five-month forward prices are $7.160/oz and $7.220/oz, respectively.
(a) If silver has no convenience yield, what are the implied repo rates?
(b) Suppose silver has an active lease market with lease rate = 0.5% for all matu- rities expressed in annualized continously compounded terms. Using the formula developed in Question 3, identify the implied repo rate for maturities of two months and five months.
10. Copper is currently trading at $1.28/lb. Suppose three-month interest rates are 4% and the convenience yield on copper is c= 3%.
(a) What is the range of arbitrage-free forward prices possible using
S0e(r−c)T ≤ F ≤ S0er T (4.14)
(b) What is the lowest value of c that will create the possibility of the market being in backwardation?
11. You are given the following information on forward prices (gold and silver prices are per oz, copper prices are per lb):
Commodities Spot One Month Two Month Three Month Six Month
Gold 436.4 437.3 438.8 440.0 444.5
Silver 7.096 7.125 7.077 7.160 7.220
(a) Which of these markets are normal? inverted? neither? (b) Which are in backwardation? in contango?
(c) Which market appears prima facie to have the greatest convenience yield? 12. Suppose the convenience yield is close to zero for maturities up to six months, then
spikes up for the forward period between six and nine months, and then drops back to zero thereafter. What does the oil market seem to be saying about political conditions in the oil-producing countries?
13. Suppose there is an active lease market for gold in which arbitrageurs can short or lend out gold at a lease rate of = 1%. Assume gold has no other costs/benefits of carry. Consider a three-month forward contract on gold.
(a) If the spot price of gold is $360/oz and the three-month interest rate is 4%, what is the arbitrage-free forward price of gold?
(b) Suppose the actual forward price is given to be $366/oz. Is there an arbitrage opportunity? If so, how can it be exploited?
14. A three-month forward contract on a non-dividend-paying asset is trading at 90, while the spot price is 84.
(a) Calculate the implied repo rate.
(b) Suppose it is possible for you to borrow at 8% for three months. Does this give rise to any arbitrage opportunities? Why or why not?
15. If the spot price of IBM today is $75 and the six-month forward price is $76.89, then what is the implied repo rate assuming there are no dividends? Suppose the six-month borrowing rate in the money market is 4% p.a on a semiannual basis. Is there a repo arbitrage, and how would you construct a strategy to exploit it?
16. The current value of an index is 585, while three-month futures on the index are quoted at 600. Suppose the (continuous) dividend yield on the index is 3% per year.
(a) What is the implied repo rate?
(b) Suppose it is possible for you to borrow at 6% for three months. Does this create any arbitrage openings for you? Why or why not?
17. A three-month forward contract on an index is trading at 756, while the index itself is at 750. The three-month interest rate is 6%.
(a) What is the implied dividend yield on the index?
(b) You estimate the dividend yield to be 1% over the next three months. Is there an arbitrage opportunity from your perspective?
18. The spot US dollar-euro exchange rate is $1.10/euro. The one-year forward exchange rate is $1.0782/euro. If the one-year dollar interest rate is 3%, then what must be the one-year rate on the euro?
19. You are given information that the spot price of an asset is trading at a bid-ask quote of 80− 80.5, and the six-month interest rate is 6%. What is the bid-ask quote for the six-month forward on the asset if there are no dividends?
20. Redo the previous question if the interest rates for borrowing and lending are not equal, i.e. there is a bid-ask spread for the interest rates, which is 6− 6.25%.
21. In the previous question, what is the maximum bid-ask spread in the interest rate market that is permissible to give acceptable forward prices?
22. Stock ABC is trading spot at a price of 40. The one-year forward quote for the stock is also 40. If the one-year interest rate is 4%. and the borrowing cost for the stock is 2%, show how to construct a riskless arbitrage in this stock.
23. You are given two stocks, A and B. Stock A has a beta of 1.5, and stock B has a beta of −0.25. The one-year risk-free rate is 2%. Both stocks currently trade at $10. Assume a CAPM model where the expected return on the stock market portfolio is 10% p.a. Stock A has an annual dividend yield of 1%, and stock B does not pay a dividend. (a) What is the expected return on both stocks?
(b) What is the one-year forward price for the two stocks? (c) Is there an arbitrage? Explain.