AMERICAN FINANCE ASSOCIATION

Stock index futures prices are generally below the level predicted by simple arbitrage models. This paper suggests that the discrepancy between the actual and predicted prices is caused by taxes. Capital gains and losses are not taxed until they are realized. As Constantinides demonstrates in a recent paper, this gives stockholders a valuable timing option. If the stock price drops, the investor can pass part of the loss on to the government by selling the stock. On the other hand, if the stock price rises, the investor can postpone the tax by not realizing the gain. Since this option is not available to stock index futures traders, the futures prices will be lower than standard no-tax models predict. ON 24 FEBRUARY 1982, the Kansas City Board of Trade began trading futures contracts on the Value Line Average stock index. During the next two months, both the Chicago Mercantile Exchange and the New York Futures Exchange also initiated trading in stock index futures. Although these contracts are actively traded, their prices have puzzled both practitioners and academics. The observed price structure, which gives the relation between the futures and spot prices as a function of the time to maturity, is much flatter than simple arbitrage models predict. In fact, the futures prices are often below the spot price. For example, on 1 June 1982, the spot price for the S&P 500 index was 111.68 and the December 1982 futures price was 110.55. In this paper, we suggest a solution to this puzzle that is based on Constantinides's [2] model of capital market equilibrium with personal taxes. The standard futures pricing models ignore an important difference between the way stock and futures contracts are taxed; futures traders in a cash settlement contract must pay taxes on all gains in the year they occur, while stockholders pay taxes only on realized gains and losses. This means that stockholders have a valuable timing option. If the stock price drops, the investor can pass part of the loss on to the government by selling the stock. On the other hand, if the stock price rises, the investor can postpone the tax by not realizing the gain. Since this option is not available to futures traders, the futures prices will be lower than the standard no-tax model predicts. In fact, this timing option can lead to the inverted price structures observed for stock index futures contracts. Throughout the paper we assume that forward and futures prices are equal. Of course, it is now well known that these prices will not be exactly equal if interest * Graduate School of Management. UCLA, Los Angeles. This research was supported by the Center for the Study of Futures Markets, Columbia University, and the Foundation for Research in Economics and Education, UCLA. We would like to thank Gordon Alexander, Fischer Black, Michael Brennan, Tom Copeland, Robert Geske, Dave Mayers, Richard Roll, Mark Rubinstein, Clifford Smith, Hans Stoll, and, particularly, George Constantinides for helpful comments. 675 This content downloaded from 134.208.96.14 on Thu, 12 Sep 2013 03:09:20 AM All use subject to JSTOR Terms and Conditions 676 The Journal of Finance rates are stochastic. Cox, Ingersoll, and Ross [5], Jarrow and Oldfield [10], Richard and Sundaresan [14], and French [8] examine the theoretical difference between forward and futures prices in a variety of contexts. Nonetheless, simulations and empirical studies by Rendleman and Carabini [13], Cornell and Reinganum [4], and Elton, Gruber, and Rentzler [6] indicate that the difference is economically insignificant.' In the remainder of the paper forward and futures prices are used interchangeably. The empirical results presented in this paper are limited to contracts on the S&P 500 index and the New York Stock Exchange composite index. The Value Line index is excluded because it is not a value-weighted average. Instead, it is based on a geometric average of the component stocks' price changes. This means that the rate of change in the Value Line index is not equal to the return one would receive from holding the component stocks. Rather than complicate the results of our study by attempting to adjust for the bias produced by the geometric averaging, we ignore the Value Line Contract. In the next section, we derive the forward price for a stock or portfolio of stocks assuming that markets are perfect and that both the dividend payout and the interest rate are constant. The prices predicted by this model are significantly higher than the prices observed for stock index futures contracts. In Section II we extend the model by introducing stochastic interest rates, seasonally varying dividends, and a simple tax structure that does not include the timing option. Although this richer theory does lead to more accurate predictions, the theoretical prices are still consistently higher than the actual prices. Section III completes the model by explicitly recognizing Constantinides's timing option. Some very preliminary results support this model. One popular explanation for the "low" level of futures prices involves the constraint traders face when selling stocks short; since these constraints are not imposed on futures traders, investors who want to go short may be attracted to the futures market and drive down the futures prices. This hypothesis is examined in Section IV. We find that short sale constraints alone will not lower the futures prices; however, these constraints play an important role when the timing option is introduced. In the last section we discuss some implications of our model and summarize the paper. I. The Perfect Markets Model In this section we develop a model of stock index futures contracts under the following simplifying assumptions: (1) Capital markets are perfect; there are no taxes or transactions costs, there are no restrictions on short sales, and assets are perfectly divisible; (2) The riskfree borrowing and lending rates are equal and constant; and (3) Dividends are paid continuously at a constant rate of D dollars per period. Suppose a trader purchases one share of stock (or a portfolio of stocks) at time ' French [9] finds a statistically significant difference between futures and forward prices for copper and silver. However, since he finds that the futures prices are larger than the forward prices, distinguishing between these prices would magnify the puzzle, rather than explain it. This content downloaded from 134.208.96.14 on Thu, 12 Sep 2013 03:09:20 AM All use subject to JSTOR Terms and Conditions Taxes and Stock Futures 677 t for a price of P(t) and then follows a trading strategy of investing all dividends in riskfree bonds. Since the dividends are paid continuously, the value of the portfolio at time T is rT V1(T) = P(T) + D er(T-w) dw = P(T) + (Dlr)[erT-t) -1] (1) where r is the continuously compounded interest rate. In other words, by investing P(t) in the stock at time t, the trader can obtain P(T) + (Dlr)[er(T-t) 1] at time T. An investor can also obtain this payoff by combining forward contracts and riskfree bonds at time t. Suppose the trader initiates one long forward contract with a price of F(t, T) and invests {F(t, T) + (Dlr)[er(T-t) l]e-r(T-t) in discount bonds. Since the forward contract has no value when it is written, the initial value of this portfolio is V2(t) = {F(t, T) + (Dlr)[er(T-t) ]e-r(T-t) (2) When the contracts mature at time T, the bonds yield F(t, T) + (D/r)[er(T t)-1] and the forward contract yields P(T) F(t, T). The total value of the portfolio is V2(T) = P(T) + (Dir)[er(Tt) 1] (3) This is exactly equal to the payoff received from investing P(t) in the first portfolio strategy. Since these two portfolio strategies have the same value at time T (and since they do not involve any intermediate cash inflows or outflows) they must have the same value at time t. In other words, the stock price must equal P(t) = {F(t, T) + (D/r)[er(T-t) 1]}e-r(T-t) (4) Equivalently, the forward price must equal F(t, T) = P(t)er(T-t) (D/r)[er(T-t) 1] (5) The two terms in Equation (5) reflect two different factors. The first term arises because payment in a forward transaction is deferred until the contract matures. For example, consider a stock that does not pay any dividends. Holding a forward contract on this stock is exactly equivalent to holding the stock, except the stock requires payment at time t while the forward contract requires payment at time T. Therefore, the forward price for a non-dividend-paying stock is equal to the deferred value of the stock price. F(t, T) = P(t)er(T-t) (6) The second term in Equation (5) reflects the fact that forward traders do not receive dividends that are paid on the underlying security. Thus, the forward price is reduced by the time T value on the dividends that are paid over the life of the contract. Under the perfect markets assumptions, all of the variables that affect the This content downloaded from 134.208.96.14 on Thu, 12 Sep 2013 03:09:20 AM All use subject to JSTOR Terms and Conditions 678 The Journal of Finance forward price are directly observable. In fact, if the stock's dividend yield is defined as the dividend flow per dollar invested in the stock at time t,