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Convenience Yields

The notion of “convenience yields” was first introduced by Kaldor (1939) as the value of physical goods, held in inventories resulting from their inherent consumption use, which accrues only to the owner of the physical commodity and must be deducted from carrying costs. Similarly, Brennan and Schwartz (1985) define the convenience yield as the flow of services that accrues to an owner of the physical commodity but not to an owner of a contract for future delivery of the commodity. These benefits of holding physical stocks often stem from local shortages, and the ability to keep the production process running (Cho and McDougall, 1990). Working (1949) showed that the convenience yield can assume various levels over time, especially for seasonal commodities like wheat. He argued that when inventory levels are high the convenience derived from holding an additional unit of the physical good is small and can be zero or even negative. On the other hand when inventory levels are low, the convenience yield can be significant.

The notion of convenience yields has become an integral part in explaining the term structure of commodity futures prices. The risk premium theory as advanced by Keynes (1923), Hicks (1938) and Cootner (1960) relates futures prices to anticipated future spot prices, arguing that speculators bear risks and must be compensated for their risk-bearing services in the form of a discount (“normal backwardation”). The theory of storage as proposed by Kaldor (1939), Working (1948, 1949), Telser (1958), and Brennan (1958) postulates that the return from purchasing a commodity at time t and selling it forward for delivery at time T, should be equal to the cost of storage (interest forgone, warehousing costs, insurance) minus a convenience yield.

An alternative expression for the futures price can be obtained by stating the storage costs and convenience yield as a constant proportion per unit of the underlying commodity: In contrast to Keynes’s risk-premium theory, the theory of storage postulates an inter-temporal relationship between spot and futures prices which could be referred to as “normal contango.” Abstracting from the convenience yield, the futures price would be an upwardly biased estimator of the spot price. In this case stores would be compensated for holding the commodity in their elevators. However, the theory of storage predicts that the higher the possibility of shortages in the respective commodity, the higher the convenience yield will be, and positive amounts of the commodity will stored even if the commodity could be sold for higher spot prices. This observation is referred to as
“inverse-carrying charge” (Working, 1948). Both theories have been subject to empirical studies. Empirical studies of Keynes’s risk-premium theory have been ambiguous.

Evidence supporting the risk-premium theory has been found by Houthakker (1961, 1968, 1982), Cootner (1960), and Bodie and Rosansky (1980). However, Telser (1958) and Dusak (1973) could not find evidence of a systematic risk premium in commodity markets. Early attempts to test the theory of storage were conducted by Telser (1958) and Brennan (1958) relating inventory data to convenience yields for several commodities. These “direct tests” suffer from the difficulty of obtaining, defining, and measuring inventory data. Fama and French (1987) propose “indirect test” strategies, building on the variation of differences in spot and futures prices (the basis). The logic of the indirect testing methodology is based on the proposition that when inventories are low (i.e. the convenience yield is high, negative basis) demand shocks for the commodity produce small changes in inventories, but large changes in the convenience yield and the interest adjusted basis. In this case, following Samuelson’s (1965) proposition, the spot prices should change more than futures prices and the basis should exhibit more variability than when inventory levels are high. Hence, negative carry is associated with low inventory levels.

Alternatively, if the variation of spot and futures prices is nearly equal when the basis is positive, it can be concluded that positive carrying costs are associated with high inventory levels. This reasoning should hold in particular for commodities with significant per unit storage costs. Fama and French (1987) find significantly differing basis standard deviations across the twenty-one commodity groups studied. Basis variability is highest for commodities with significant per unit storage costs (wood and animal products) and lowest for precious metals. This finding is consistent with the theory of storage.

More recent studies of the inter-temporal relationship of futures prices incorporating convenience yields have been carried out in the context of pricing contingent claims by arbitrage. The most prominent authors to apply continuous time stochastic models to the pricing of commodity contingent claims are Brennan and Schwartz (1985), Gibson and Schwartz (1990a and b, 1991), Brennan (1991), Gabillon (1991), and Garbade (1993). Typically the analysis starts off by assuming an exogenously given geometric Brownian motion process for the spot price relative changes of the commodity. From a theoretical point of view the derivation of the futures price under the constant and deterministic convenience yield assumption is associated with the problem that only parallel shifts in the term structure can be modeled, since both spot and futures prices in equation (5) have equal variance. This is inconsistent with Samuelson’s (1965) proposition of decreasing volatility of futures prices over time to delivery or settlement. Brennan (1991) estimates and tests alternative functions and stochastic processes for convenience yield and its dependence on price and time. Both Brennan (1991) and Gibson and Schwartz (1990a, 1991) present stochastic two-factor models of the term structure of commodity and oil futures prices respectively, incorporating an “autonomous” stochastic process for the convenience yield. The process governing the convenience yield changes is modeled as an Ornstein-Uhlenbeck process with Gaussian variance. An analysis of the time series properties of the convenience yields is presented in Gibson and Schwartz (1991) who find support for a mean-reverting pattern in the convenience yield series. Gibson and Schwartz (1990a and b) solve this partial differential equation numerically and obtain values for crude oil futures and futures options under the appropriate boundary conditions. Both Brennan (1991) and Gibson and Schwartz (1990) report that the accuracy of commodity futures pricing relative to the simple continuous compounding model can be enhanced by adding a mean-reverting convenience yield as a second stochastic variable. Although the Brennan and Gibson and Schwartz models are consistent with Samuelson’s decreasing volatility pattern, the convenience yield is specified Independently of the spot price of oil, which implies that although the spot price of oil is stable, the convenience yield tends to a long-run mean level. Based on this critical remark Gabillon (1991) proposes an alternative two-state variable stochastic model, where the system of stochastic processes consists of the current spot price of oil and the long term price of oil. Gabillon uses the ratio of current spot price to long-term price and time to maturity in order to determine the convenience yield level. According to this model the current term structure of futures prices depends on the relative level of the spot price. Garbade (1993) presents an alternative two-factor arbitrage free model of the term structure of crude oil futures prices, with the term structure fluctuating around some “normal shape” in a mean-reverting manner, abstracting from the convenience yield.

More empirical and theoretical work is necessary in order to shed light on the relative pricing efficiency of alternative models of the term structure of commodity futures prices. In particular, shortcomings in the appropriate modeling of the convenience yield process and its distributional properties as well as its relation to the spot price of oil are still unresolved.

Moreover, current research has not addressed problems associated with the assumption of constant spot price volatilities and interest rates. The assumption of constant interest rates might not be warranted, especially in the long run. Potential corporate finance applications have been discussed by Gibson and Schwartz (1991) in the case of long-term oil linked bonds.

Other useful applications might concern the valuation of long-term delivery contracts and the hedging of such commitments with respect to both price and convenience yield risk.


Resources for Convenience Yields:

Bodie, Z. & Rosansky, V. I. (1980). Risk and return in commodity futures. Financial Analysts Journal, 36, 27–39.

Brennan, M. J. (1991). The price of convenience and the valuation of commodity contingent claims. Stochastic models and option values. (Ed.) Lund, D. & Øksendal, B.
Amsterdam: North-Holland. 135–57.

Brennan, M. J. & Schwartz, E. S. (1985). Evaluating natural resource investments. Journal of Business, 58, 135–57.

Cho, D. W. & McDougall, G. S. (1990). The supply of storage in energy futures markets. The Journal of Futures Markets, 10, 611–21.

Fama, E. F. & French, K. R. (1987). Commodity futures prices: some evidence on forecast power, premiums, and the theory of storage. Journal of Business, 60, 55–74.

Gabillon, J. (1991). The Term Structure of Oil Futures Prices. Working Paper M17, Oxford Institute of Energy Studies.

Garbade, K. D. (1993). A two-factor, arbitrage-free, model of fluctuations in crude oil futures prices. The Journal of Derivatives, 1, 86–97.

Gibson, R. & Schwartz, E. S. (1990a). Stochastic convenience yield and the pricing of oil contingent claims. Journal of Finance, 45, 959–76.

Gibson, R. & Schwartz, E. S. (1990b). The pricing of crude oil futures options contracts. UCLA. Working paper.

Gibson, R. & Schwartz, E. S. (1991). Valuation of long term oil-linked assets. Stochastic models and option values. (Ed.) Lund, D. & Øksendal, B. Amsterdam: North-Holland. 73–102.

Houthakker, H. S. (1982). The extension of futures trading to the financial sector. Journal of Banking and Finance, 6, 37–47.



A convertible is a bond with an option for the holder to exchange the bond into “new” shares of common stock of the issuing company under specified terms and conditions. These include the conversion period and the conversion ratio. The conversion period is the period during which the bond may be converted into shares. The conversion ratio is the number of shares received per convertible. The conversion price, which is the effective price paid for the common stock, is the ratio of the face value of the convertible and the conversion ratio.

Convertibles almost always have a call provision built in. Special types of convertibles are mandatory convertible bonds, exchangeable bonds, and LYONS. A convertible is much like a bond with a warrant attached. However, this concept is not very useful for valuation purposes.

An important problem is that the exercise price of the warrant (the conversion price) is paid by surrendering the accompanying bond. Therefore the exercise price changes through time.

The fact that most convertibles are callable creates another valuation problem. Brennan and Schwartz (1980) have developed a model which takes all these factors into account.

Motives for the issuance of convertibles can be divided into traditional and modern motives.

Traditional motives are that convertibles are (1) a deferred sale of stock at an attractive price and (2) a cheap form of capital (Brigham, 1966). These motives are criticized by

Brennan and Schwartz (1988). The first motive is based on the fact that normally the conversion price is above the market price of the underlying stock at the issuance date.

However, the conversion price should in fact be compared to the underlying stock price at the exercise date. If the underlying stock price is higher than the conversion price, the company suffers an opportunity loss. If it is lower than the conversion price, the conversion right will not be exercised. The second motive is based on the fact that the coupon rate of a convertible is lower than the coupon rate of an ordinary bond. However, if the cost of the conversion right is taken into account it can be demonstrated that the cost of convertibles is relatively high for a numerical example). The cost of convertibles is neither a reason to issue, nor a reason to refrain from issuing convertibles. Its cost is just an adequate compensation for the risk involved in its investment.

Modigliani and Miller have demonstrated that in perfect markets the financing decision of the firm is irrelevant for its market value. Therefore, modern motives for the issuance of convertibles are based on market imperfections. Brennan and Schwartz (1988) argue that convertibles are relatively insensitive to the risk of the issuing company. If the risk increases, the value of the bond part decreases, but the value of the warrant part increases, because the value of a warrant is an increasing function of the volatility. This makes it easier for bond issuers and purchasers to come to terms when they disagree about the riskiness of the firm.

Because of the insensitivity towards risk, convertibles may result in lower agency costs between share and bondholders. Bond-holders are less concerned about the possibility that shareholders attract risky projects. Because of their conversion right they also participate in the value created if risky projects are undertaken (Green, 1984). Other motives, based on imperfections, include the reduction of flotation costs compared to the case where the firm raises debt now and equity later, and the possibility to ”polish” the company’s financial accounts by recording the convertible as debt on the balance sheet With regard to the optimal moment to call convertibles, Ingersoll (1977) has emonstrated that this moment occurs when the conversion value, this is the value of the common stock to be received in the conversion exchange, equals the call price. However, in an empirical study he finds that in practice the calls show a delay. On average the conversion value of the bonds was 43.9 percent above the call price.

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