Forward Prices, Yields, and the Cost of Carry
in
Discrete-Time Full-Carry Markets*
by
David C. Nachman
J. Mack Robinson College of Business
Georgia State University
Atlanta, GA 30303-3083
Email: __dnachman@gsu.edu__
Tel: 404-651-1696
Fax: 404-651-2630
and
Stephen D. Smith
J. Mack Robinson College of Business
Georgia State University
Atlanta, GA 30303-3083
and
Federal Reserve Bank of Atlanta
Email: sdsmith@gsu.edu
Tel: 404-651-1236
Fax: 404-651-2630
This version
September, 2003
*The views expressed here are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Atlanta or the Federal Reserve Board.
Forward Prices, Cash Yields, and the Cost of Carry
in
Discrete-Time Full-Carry markets
ABSTRACT
One of the easiest and most intuitive pricing relationships in finance is the simple cost of carry formula for the forward price of an asset or commodity. The intuition of the cost of carry story is enriched by allowing for a yield on the underlying. In discrete-time models, the resulting forward price formula depends on how the yield is specified, either as an ordinary yield or as a current yield. Both specifications are important in practice. A dividend yield may be modeled as an ordinary yield or a current yield. Commodity loan rates such as in gold leases are per force current yields. We have found no transparent textbook exposition of the pricing formula for both specifications. The key to such an exposition is the dynamics of reinvesting the periodic yield. We provide these dynamics here. In the process we clarify the assumptions on yields on which the simple cost of carry story is based and we make transparent how the “no arbitrage” arguments work.
Introduction
One of the easiest and most intuitive pricing relationships in finance is the simple cost of carry formula for the forward price of an asset or commodity. It is easy because students can see that a forward contract is one mechanism to effect possession (the purchase) of the underlying at a future date at terms, a price, agreed upon now. All that is required to implement this strategy is to save enough money to have the forward price at the future date. Since the forward price is known now, how much to save is an easy present value calculation.
The other obvious way to have the underlying at the future date is to buy it now and hold it till that date, a simple buy and hold strategy referred to in the literature on futures and forwards as the cash-and-carry strategy. Since both strategies yield possession of the underlying at the future date, to avoid arbitrage, they must both have the same cost of implementation. Equating these costs of implementation yields the simple formula for the forward price as the future value of the current spot price of the underlying. With one further specification of the relevant interest rate (for computing future and present values), one can interpret that interest rate as the opportunity cost of carrying the underlying to the future date.
This easy and intuitive story can be enriched considerably by allowing assets or commodities that have a yield. Here we intend to use the term yield in a manner to include any benefits (positive or negative) from ownership of the underlying that are realizable in cash, a cash yield, but that are not available to the long position in a forward contract. This usage of the term cash yield is in contrast to the term convenience yield. In popular usage the term convenience yield may involve cash and non-cash items. However, the traditional meaning of the term convenience yield is that it is the unspecified non-cash benefits from ownership of an asset or commodity implied in the forward price that are not obtained by the long position in a forward contract. The distinction is important here. In this traditional sense then, we treat only forward contracts on assets or commodities with zero convenience yield, the so-called full-carry markets.^{1}
Examples of a positive cash yield are dividends on a stock or a stock index, the cash value of lease rate for gold, and the foreign risk free interest rate in foreign currency transactions. Costs of storage of the underlying like rent and insurance costs are examples of a negative yield. Intuitively, a known and positive yield reduces the cost of carry and a negative yield increases the cost of carry.
In discrete-time models, the resulting forward price formula depends on how the yield is specified, either as an ordinary yield or as a current yield. Both specifications are important in practice. A dividend yield may be modeled as an ordinary yield or as a current yield. The relevance of the discussion here is underlined by the recent initiation of trade in futures contracts on individual equities, the so called single stock futures contracts, in the U. S.^{2} Understanding the role of dividend yield and its specification in the determination of futures prices is essential in trading in such equity futures. Flexibility in the specification of dividend yields may prove of some practical use there since one specification may not be right for each firm’s stock.^{3}
Commodity loans may have the principal and interest specified in units of the commodity, such as in gold leases. For such loans the cash value of the interest is evaluated at the current price of the commodity making this cash yield a current yield. Active derivatives markets in gold and other precious metals is a topic of interest in practice and in the classroom.^{4}
We have found no transparent textbook exposition of the pricing formula for both specifications. The key to such an exposition is the dynamics of reinvesting the periodic yield. We provide these dynamics here. These arguments produce exact pricing solutions and do not depend on the value of yields being certain (known) for the contract period. In the case of ordinary yields the values of yields are predictable (but not certain) and in the case of current yields they are not even predictable. The important thing that makes the arguments work is the ability to reinvest the value of the yield at the price on which this value is based.
The basic model of the paper is presented in the next section. In the following section, we present the ordinary yield specification and discuss versions of the two acquisition strategies mentioned above that establish the corresponding forward pricing formula. This is the formula presented in many investment textbooks yet there is some ambiguity created by at least one prominent presentation.^{5} In the following section, we present the current yield specification and the version of the two acquisition strategies that establish the corresponding forward pricing formula. The corresponding forward price formula is rarely presented in investment textbooks.
We interpret the forward price formulas developed here in terms of a (net) cost of carry, we compare the associated costs of carry, and discuss the financial intuition behind each of the results. Finally, the distinction between ordinary and current yields disappears when the time between the discrete dates here goes to zero and the corresponding formulas converge to the familiar one in continuous time.
It is well known that pricing formula for forward contracts have broader applications to futures prices when interest rates and spot prices are uncorrelated and they provide bounds on futures prices when interest rates and spot prices are correlated.^{6} In this paper, as in most textbook discussions of forward prices, interest rates are assumed to be deterministic and so the results here apply as well to futures contracts.
THE MODEL
Throughout the paper we will ignore the usual frictions such as indivisibilities, taxes, brokerage commissions, margin requirements and other transaction costs. Time is measured discretely. Now is date 0. Typically, contract maturity is date .
The spot price of the underlying at date is denoted by . This price is *ex* the yield at date and is in general not known prior to date . All transactions in the spot market are at this spot price and forward contracts are assumed to be settled at the *ex* price .
The known and constant yield is denoted by the symbol . Since this is expressed as a percentage of the price of the underlying, we restrict to the economically reasonable range, -1 < < 1. For ease of exposition in calculating present and future values, we assume that the risk free interest rate per period is a simple interest rate that is known and constant and is denoted by > 0.^{7}
The strategy used to acquire the underlying at date , via spot market transactions (the cash-and-carry strategy) is called Strategy 2. The strategy used to acquire the underlying by going long in the forward contract is called Strategy 1. Also for simplicity, we standardize units so that one forward contract is for one unit of the underlying.
Ordinary Yield
In this section we study the case when the cash yield at date is a known and constant percentage of the spot price at date . Thus the cash value of the yield at date is = , per unit of the underlying. An investor who owns units of the underlying at date receives at date . Consistent with common usage, such a yield is referred to as an ordinary yield.
Let denote the forward price established at date 0 for delivery at date . In this case we have that in the absence of arbitrage opportunities,
= . (1)
This is the formula familiar from investments texts. See for example formula (16.2) page 582 in Bodie, Kane, and Marcus (2004), the text we use for our undergraduate course in valuation. There the yield is a dividend and they note that “This relation is only approximate in that it assumes the dividend is paid just before the maturity of the contract.”^{8} We are puzzled by this qualification. It creates ambiguity for our teaching of the cost of carry story. We thought there was a model under which the relation is a precise exact formula, but we found no reference we could give our students that was transparent. This is the main reason for this paper.
To establish this result, consider the following strategies. Initially assume that > 0.
Strategy 1. At date 0, enter a long forward contract and save , the present value of the forward price.
Strategy 2. At date 0 buy = units of the underlying.^{9} At each date < , with = units of the underlying, borrow the value of the cash yield to be paid at the next date. The face value of this loan is = and it is known at date . Use the proceeds of the loan to buy additional units of the underlying. Use the value of the cash yield realized at date to repay the loan from the previous date.^{10} This yields exactly one unit of the underlying at date with zero interim cash flow. The details are given in the Appendix.
Since Strategy 1 and Strategy 2 both yield one unit of the underlying at date with zero interim cash flow, to avoid arbitrage, it must be that both strategies have the same initial cash flow at = 0, i. e.,
= .
Plugging in the value for = and solving for gives (1).
For the case of > 0, Strategy 2 starts out with = < 1 unit of the underlying and through reinvestment of the periodic cash yield accumulates = units of the underlying at date . Thus at date , = = 1, i. e., the cash-and-carry strategy accumulates exactly one unit of the underlying with zero interim cash flow.
This Strategy 2 works as well for the case when < 0 or when = 0. For the case when < 0, the signs of the quantities involved in implementing Strategy 2 are just the opposite for the positive case. The strategy involves starting out with = > 1 units of the underlying. At each date, the present value of the next date's cash costs is liquidated and saved for one period to pay these costs at the next date. This again gives = = 1 units of the underlying at date . This establishes (1) for < 0.
For the case when = 0, the above strategy involves no borrowing or lending and no liquidation or acquisition of units of the underlying other than the initial quantity = = 1. The resulting version of (1) is in fact the simple cost of carry formula mentioned at the beginning of this paper.^{11}
Thus (1) holds for all economically reasonable values of . The essential feature of this argument is that the quantities held of the underlying , , … , , in the cash-and-carry Strategy 2 are deterministic. To make these quantities deterministic essentially requires the ability to invest the value of the cash yield at the price on which this yield is based.
Predictable Cash Yield
We note that in the specification above, the value of the cash yield at each date is not known for certain. At any date , only the value of the cash yield at date is known. In the parlance of stochastic processes, the process , , … is predictable^{12} in that is known at date , but this process is not deterministic. The value of future cash yields depend on the evolution of the *ex* prices , …. These prices are not assumed to be known at date .
As the current yield specification that follows shows, predictability of the process , , … is not crucial, but rather is a consequence of the specification of the cash yield as an ordinary yield. What is crucial, as we argued above, is the deterministic nature of the quantities , , … , , in the cash-and-carry Strategy 2. To make these quantities deterministic essentially requires the ability to invest the value of the cash yield at the price on which this yield is based.
Current Yield
The ordinary yield specification of the previous section is the popular one in much of the investments literature. Another specification that is natural in the case of commodity loans is what may be called a current yield specification where the cash yield at date is expressed as a percentage of the current spot price at date . As an example, consider a gold lease where at each date , the lease rate must be paid in gold per ounce of gold leased. The cash value of this payment is , where is the spot price of gold at date .
In this section, we assume the value of the cash yield at date is a known and constant percentage of the *ex* price , i. e., the value of the cash yield at date is = . This specification simplifies the cash-and-carry Strategy 2. Reinvestment of the cash yield is direct, involving no borrowing or lending. In this case, we have that in the absence of arbitrage opportunities,
= .^{13} (2)
To establish this result, consider the following strategies. Initially assume that > 0.
Strategy 1. At date 0, enter a long forward contract and save , the present value of the forward price.
Strategy 2. At date 0 buy = units of the underlying. At each date , with = , reinvest the value of the cash yield = to acquire units of the underlying. This yields exactly one unit of the underlying at date with zero interim cash flow. The details are given in the Appendix.
Since Strategy 1 and Strategy 2 both yield one unit of the underlying at date with zero interim net cash flow, to avoid arbitrage, it must be that the strategies have the same cash flow at = 0, i. e.,
= .
Plugging in the value for = and solving for gives (2).
For the case of > 0, Strategy 2 starts out with = < 1 unit of the underlying and through reinvesting of the value of the cash yield accumulates = units of the underlying at date . Thus at date , = = 1, i. e., the cash-and-carry strategy accumulates exactly one unit of the underlying with zero interim cash flow.
Strategy 2 works as well for the case when < 0, as long as > 0, or when = 0. For the case when < 0, the signs of the quantities involved in implementing Strategy 2 are just the opposite for the positive case. The strategy involves starting out with = > 1 units of the underlying, at each date liquidating the value of the cash costs at that date by selling units of the underlying. For the case when = 0, the above strategy involves no acquisition and no liquidation of units of the underlying other than the initial quantity = = 1. The resulting version of (2) is again the simple cost of carry formula mentioned at the beginning of the paper.
Unpredictable Cash Yield
In contrast to the ordinary yield specifications, the specification of the cash yield as a current yield produces a stochastic process , , … that is not predictable. In the model leading to formula (2), = , i. e., the cash yield process is a scalar multiple of the *ex* price process, and this process is not predictable. In these current yield model, the cash yield inherits all the uncertainty of the spot price process , , …. We note again that this stochastic nature of the value of the cash yield is not a problem as long the quantities , , … , , in Strategy 2 are deterministic. To make these quantities deterministic only requires the ability to invest the value of the cash yield at the price on which this yield is based.
Costs of Carry
The formulas (1) and (2) for the forward price can be written in the form = , where is the net cost of carry for formula i, i = 1, 2, or just cost of carry for short. The cost of carry for formula (1) is clearly
= , (1’)
By noting that = 1 + , we have that
= . (2’)
The formulas (1) and (2) can therefore be compared directly on the basis of their costs of carry. For ease of reference, we will refer to the different yield specifications by the number of the corresponding price equation. To get a handle on the difference in the costs of carry, it may help to look at the one period case.
For ordinary yield model (1), in implementing Strategy 2, the investor acquires initially the quantity = = , when = 1. Using the proceeds from borrowing the value of the cash yield of date 1, the investor acquires additional units for a total of one unit that is carried from = 0 to = 1. On that unit, the investor foregoes the interest rate on the purchase price of the underlying but the investor earns the cash yield on this purchase price for a net cost of carry of = .
In the case of the current yield model (2), implementing Strategy 2 involves acquiring initially = = units. No additional units are acquired until the next date since the cash yield is a current yield, so units are carried from = 0 to = 1. On this number of units, the investor foregoes the interest rate on the purchase price of the underlying but the investor earns the cash yield on this purchase price. In essence, the per unit cost of carry is still , but only units are carried. So the cost of carry on the full unit price of is = .
Since the forward price in each model is of the form = , the model that has the larger net cost of carry gives the higher forward price. Recall that for simplicity and economic reality we assume that the interest rate is positive and that . Since when = 0, we also assume here that 0. When > 0 and > 0, then < . This makes economic sense. The condition is that the per unit cost of carry is positive but the number of units carried in the current yield specification is less than a full unit, ensuring that the net cost of carry for the current yield formula (2) is smaller than for the ordinary yield formula (1).
If either 0 or 0, then we have that , because 0 implies > 0. Again this makes economic sense. Either the per unit cost of carry is negative and the units carried in the current yield specification is less than one, or the per unit cost of carry is positive but the number of units carried in the current yield specification is greater than one unit. The condition that (<) 0 is therefore equivalent to the condition that (>) > 0. When the cash yield is negative, the net cost of carry for both specifications is positive and the one for an ordinary yield is the least.
Typical situations encountered in the classroom and in practice^{14} are those of contango or backwardation relative to the spot price.^{15} Contango (backwardation) relative to the spot price is the condition that the forward price is larger (smaller) than the spot price. For the ordinary yield model (1), we have contango (backwardation) is synonymous with . Similarly, for the current yield model (2), we have contango (backwardation) is synonymous with . So we have contango (backwardation) in one model if and only if we have it in the other model, and this if and only if . In the case of gold, is referred to as the gold lease rate and the quantity is called the forward rate.^{16}
Continuous Compounding
The distinction between the ordinary yield specification and current yield specification that is the essence of the comparisons above disappears with continuous compounding. By standard arguments, as the number of compounding periods per unit of time gets arbitrarily large, for given values of and constant per unit of time, formula (1) converges to
.
Similarly, formula (2) converges to
.
As in the above interpretations, the per unit net cost of carry is the same for both specifications, but the difference in units carried disappears because the units carried are changing continuously.^{ 17}
References
Bodie, Z., A. Kane, and A. J. Marcus. Essentials of Investments (2004), 5^{th} Edition, Irwin McGraw-Hill.
Cox, J. C., J. E. Ingersoll, and S. A. Ross, “The Relationship Between Forward and Futures Prices,” Journal of Financial Economics, 9 (1981), 321-346.
Hull, J. Options, Futures, and Other Derivatives, (2003), 5^{th} edition, Prentice Hall.
Kapner, K., and R. McDonough, “Doing Your Homework on Individual Equity Futures,” *Futures*, 31 (2002), 50-52.
Kolb, R. W. Futures, Options , & Swaps, (1997), 2^{nd} edition, Blackwell Publishers.
McDonald, R. L. Derivatives Markets, (2003), Addison Wesley.
McKay, P. A., “Single-Stock Futures Arrive in the U. S. with Room to Grow,” The Wall Street Journal, November 11, 2002.
Siegel, D. R., and D. F. Siegel. The Futures Markets, (1990), Probus Publishing.
Williams, D. Probability with Martingales, (1991), Cambridge University Press.
Appendix
The purpose of this appendix is to verify the statements about Strategy 2 made for the yield models in the text. We do this first for the ordinary yield model with the language for the case of > 0. This establishes the general quantity relationship.
In Strategy 2, initially buy = units of the underlying and borrow the value of the date 1 cash yield = (face value of the loan), where = . Use the proceeds from this loan to purchase = additional units of the underlying at = 0. Net cash flow at = 0 is
- + - = -,
and the total units of the underlying owned going to date = 1 are
+ = +
=
= = ,
validating the calculation of the value of the cash yield for date = 1.
At date = 1, the units of the underlying are worth and the value of the cash yield on these units is . Use this value to repay the date = 0 loan, and borrow the value of the date = 2 cash yield = , where = . Use the proceeds of this loan to buy = additional units of the underlying at date = 1. Reinvest the value in units of the underlying at = 1. Net cash flow at = 1 is
+ - + - - = 0.
Total units of the underlying owned going into = 2 are
+ = +
= = ,
validating the calculation of the value of the cash yield for date = 2.
Proceed on in this manner, following these three steps, at each date prior to contract maturity. Repay the loan of the previous date with the value of the cash yield realized at the current date. Reinvest the value of the units brought to the current date. Borrow the value of the cash yield to be realized at the next date to buy additional units of the underlying at the current date.
The net cash flow from these three steps is zero at each date after the initial date. The number of units owned going into date is = = 1. At date , repay the loan of date - 1 with the value of the cash yield for this one unit of the underlying. The value of the one unit of the underlying is . This completes the argument.
The general quantity relationship for this strategy is the following equation
= + ,
where the second term on the right is the units of the underlying purchased at date using the proceeds of the borrowed date yield. When is a constant multiple of the *ex* price , drops out of the above relationship and it can be solved recursively from the terminal condition that = 1.
We now verify the statements about Strategy 2 made in the current yield model, again with the language of the case when > 0. The details are as follows. Let = 0 and initially buy = units of the underlying at the *ex* price . Net cash flow at date = 0 is - . At date = 1, the units of the underlying are worth and the value of the cash yield is = . Reinvest this value and invest the cash yield in the underlying, purchasing = additional units. Total number of units of the underlying owned at date = 1 is = + = = , and the net cash flow is zero.
Continue in this manner, following these two steps at each date . Reinvest the value . Invest the value of the cash yield to acquire additional units of the underlying. Total units acquired by date is
= + = = ,
and the net cash flow at date is zero. At date , = , and the date value of the cash yield gives additional units for a total of = = 1 unit worth . This completes the argument.
1 See Hull ( 2003), pages 58-60, Kolb (1997), page 74, and Siegel and Siegel (1990), pages 85-90.
2 See P. A. McKay (2002). The new journal SFO: Stock Futures & Options is the official journal of single stock futures, stock index futures and stock options.
3 See Kapner and McDonough (2002).
4 For example in the case *American Barrick Resources Corporation: Managing Gold Price Risk*, Case 9-293-128, Harvard Business School, 1995.
5 See the comments following formula (1) in the section on ordinary yield below.
6 See Cox, Ingersoll, and Ross (1981).
7 The arguments we present will permit the yield and the interest rates to be time varying, but they must be deterministic and known at the initiation of the contract. For ease of exposition, we assume them to be constant through time. The practice that is common in most t*ex*ts on futures is to assume that interest rates and the value of yields are deterministic. The assumption of deterministic yields in this paper still allows for the cash value of yields to be stochastic.
8 Footnote 6 page 582, Bodie, Kane, and Marcus (2004).
9 “Adjusting the initial quantity in this way in order to offset the effect of income from the asset is called **tailing** the position.” McDonald (2003), page 124.
10 It is this feature of the argument that requires that the yield be realized in cash.
11 This is formula ( 16.1) page 581 in Bodie, Kane, and Marcus (2004).
12 The term “previsible” is also used. See Williams (1991), section 10.6 for a definition and applications to gambling strategies, and sections 15.1 and 15.2 for applications to trading strategies in finance.
13 This is the formula implicit in commodity forward prices where there is lease market for the commodity in McDonald (2003). See McDonald (2003), formula (6.12), page 172 for the effective annual lease rate and solve back for the forward price.
14 See the reference in footnote 4.
15 The meaning here is in contrast to the meaning of these terms in the discussions by Keynes and Hicks about the relation of the forward price to the expected future spot price. See Hull (2003, p. 31). We mention this because we warn our students against any interpretation of the forward price as any kind of forecast of the future spot price.
16 See __www.kitco.com__.
17 See Hull (2003), page 49, and McDonald (2003), page 124.
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