Stock Index Futures
Answers to Problems and Questions
1. Theoretically, the more securities in a portfolio, the less the unsystematic
risk. Most users of stock index futures as a hedge do so in order to
reduce systematic risk. Because of the Nasdaq’s substantial
technology/dot.com/Internet exposure, an insurance company portfolio is
likely to have more in common with the S&P 500 than with the Nasdaq
100. The S&P 500 is probably a better hedging device.
2. The deterioration of the basis does work to the advantage of the short
seller. If you are long a contract, the deterioration of basis will be to your
detriment. It seems, then, that the statement has some merit. It is
important to note, though, that the deterioration of the basis is not “free
money” to the short seller and that market movements after the first
delivery month can result in substantial losses on the other leg of the
spread. It is also dangerous to say that any investment strategy is
“usually a winner.” Price movements in the underlying index or sharp
changes in interest rates could overwhelm any gain from the basis.
3. Theoretically, this strategy makes perfect sense. Index futures and T-
bills could be combined to produce a synthetic stock portfolio, and
probably at much less commission cost. The synthetic portfolio might
also provide a more attractive income stream for some portfolio
The disadvantage lies primarily with the fact that futures contracts are
not well understood by many governing bodies that have fiduciary
responsibility for investment portfolios.
4. This is not a very accurate statement. It is true that the deterioration of
basis would work against you, but this ignores the fact that over the long
run the stock market has risen substantially. Holders of index futures
contracts would benefit from long term rises in the value of the
5. The passage of time means that the price of a stock index futures contract
will decline, everything else being equal. The short seller wants the price
Chapter Nine. Stock Index Futures
6. If the dividend yield on the stock index were to exceed the T-bill rate, the
market would be inverted. This is an unlikely event.
7. While index futures could be used as a hedge, with only five securities
there would be considerable unsystematic risk in the portfolio making it
difficult to hedge without using individual equity options.
8. Beta measures the sensitivity of a portfolio to market movements.
Portfolios that are quite sensitive need a larger hedge than portfolios with
9. No. Hedging involves reducing risk. It does not necessarily involve
removing it all. In fact, if you remove all market risk, the portfolio’s
expected return converges on that of the riskfree rate. For this reason
continuous hedging is normally not a good idea.
10. The key question is what happens to the dividend yield. If the dividend
yield does not change because the stock prices rise proportionately, then
there would be no effect on the S&P500 futures. If the dividend yield did
increase, the fair annual premium on the futures would decline, and so
would the value of the futures contract.
11. F = S e(R – D)t
= 340 e (.0588 - .0255)(63/365) = 340 e .0057 = 341.96
12. F = S e(R – D)t
= 340 e (.0563 - .0255)(120/365) = 340 e .0101 = 343.46
13. Using the September 2001 contracts,
x 0.8 402.7
1589.20 x 250
To hedge 80%, sell 80% x 402.7= 322 contracts
14. x 0.8 421.8
1517.20 x 250
To hedge 100%, sell 422 contracts
Chapter Nine. Stock Index Futures
15. x 1.1 2207.7
1494.80 x 250
To hedge 40%, sell 40% x 2207.7= 883 contracts
16. F = S e(R – D)T
1499.00 = 1475.50 e(R - .022)85/365
1.0159 = ex
ln 1.0159 = 0.0158013 x = 0.0158013
(R - .022)(85/365) = 0.0158013
R = 8.99%
17. Solve for the equilibrium futures price:
F = Se(R – D)T
= 329.83e(.0602 - .0265)(121/365) = 333.54
The actual futures price is above this, so the appropriate action would be to sell
the futures and buy the underlying basket of stock, anticipating that the basis
would return to its theoretical value.