Eduardo S. Schwartz

Evaluating Natural Resource Investments

Notwithstanding impressive advances in the theory of finance over the past 2 decades, practical procedures for capital budgeting have evolved only slowly. The standard technique, which has remained unchanged in essentials since it was originally proposed (see Dean 1951; Bierman and Smidt 1960), derives from a simple adaptation of the Fisher (1907) model of valuation under certainty: under this technique, expected cash flows from an investment project are discounted at a rate deemed appropriate to their risk, and the resulting present value is compared with the cost of the project. This standard textbook technique reflects modern theoretical developments only insofar as estimates of the discount rate may be obtained from crude application of single period asset pricing theory (but see Brennan 1973; Bogue and Roll 1974; Turnbull 1977; Constantinides 1978). The inadequacy of this approach to capital budgeting is widely acknowledged, although not widely discussed. Its obvious deficiency is its The evaluation of mining and other natural resource projects is made particularly difficult by the high degree of uncertainty attaching to output prices. It is shown that the techniques of continuous time arbitrage and stochastic control theory may be used not only to value such projects but also to determine the optimal policies for developing, managing, and abandoning them. The approach may be adapted to a wide variety of contexts outside the natural resource sector where uncertainty about future project revenues is a paramount concern.

Strategic asset allocation

Abstract This paper analyzes the portfolio problem of an investor who can invest in bonds, stock, and cash when there is time variation in expected returns on the asset classes. The time variation is assumed to be driven by three state variables, the short-term interest rate, the rate on long-term bonds, and the dividend yield on a stock portfolio, which are all assumed to follow a joint Markov process. The process is estimated from empirical data and the investors optimal control problem is solved numerically for the resulting parameter values. The optimal portfolio proportions of an investor with a long horizon are compared with those of an investor with a short horizon such as is typically assumed in ‘tactical asset allocation’ models: they are found to be significantly different. Out of sample simulation results provide encouraging evidence that the predictability of asset returns is sufficient for strategies that take it into account to yield significant improvements in portfolio returns.

A continuous time approach to the pricing of bonds

Abstract This paper develops an arbitrage model of the term structure of interest rates based on the assumptions that the whole term structure at any point in time may be expressed as a function of the yields on the longest and shortest maturity default free instruments and that these two yields follow a Gauss-Wiener process. Arbitrage arguments are used to derive a partial differential equation which must be satisfied by the values of all default free bonds. The joint stochastic process for the two yields is estimated using Canadian data and the model is used to price a sample of Government of Canada bonds.

Analyzing Convertible Bonds

The convertible bond is a hybrid security which, while retaining most of the characteristics of straight debt, offers, in addition, the upside potential associated with theunderlying common stock. As a quid pro quo for the upside potential the convertible bond is typically subordinated to other corporate debt and carries a lower coupon rate than would an otherwise equivalent straight bond.

Electricity prices and power derivatives: Evidence from the Nordic Power Exchange

This paper examines the importance of the regular pattern in the behavior of electricity prices, and its implications for the purposes of derivative pricing. We analyze the Nordic Power Exchanges spot, futures, and forward prices. We conclude that the seasonal systematic pattern throughout the year, in particular, is of crucial importance in explaining the shape of the futures/forward curve. Moreover, in the context of the oneand two factor models analyzed in this paper, a simple sinusoidal functionis adequate in order capture the seasonal pattern of the features and forwardcurve directly implied by the seasonal behavior of spot electricity prices.

The pricing of equity-linked life insurance policies with an asset value guarantee

This paper considers the equilibrium pricing of equity-linked life insurance policies with an asset value guarantee; such policies provide for benefits which depend upon the performance of a reference portfolio subject to a minimum guaranteed benefit. The benefit is decomposed into a sure amount and an immediately exercisable call option on the reference portfolio. A numerical procedure for determining the value of the call option is presented and the risk minimizing investment strategy to be followed by the issuer of the policy is derived.

Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis

Since the seminal article by Black and Scholes on the pricing of corporate liabilities, the importance in finance of contingent claims has become widely recognized. The key to the valuation of such claims has been found to lie in the solution to certain partial differential equations, the best known of which is that derived by Black and Scholes in their original article from the assumption that the value of the asset underlying the contingent claim follows a geometric Brownian motion.

An Equilibrium Model of Bond Pricing and a Test of Market Efficiency

In two previous and related papers ([3], [4]), the authors have reported the results of estimating a particular equilibrium model of bond pricing using quarterly data on Canadian government bonds for the period 1964 to 1979. This paper reports the results of applying a similar model to the pricing of U.S. government bonds for the period 1958–1979 using data from the CRSP Government Bond File. The paper also extends the previous empirical analysis by evaluating the ability of the pricing model to detect underpriced and overpriced bonds: the data reveal a strong relation between price prediction errors and subsequent bond returns.

Rational Pricing of Internet Companies

We apply real-options theory and capital-budgeting techniques to the problem of valuing an Internet company. We formulate the model in continuous time, form a discrete time approximation, estimate the model parameters, solve the model by simulation, and then perform sensitivity analysis. We report that, depending on the parameters chosen, the value of an Internet stock may be rational if growth rates in revenues are high enough. Even with a real chance that a company may go bankrupt, if the initial growth rates are sufficiently high and if this growth rate contains enough volatility over time, then valuations can reach a level that would otherwise appear dramatically high. In addition, the valuation is highly sensitive to initial conditions and exact specification of the parameters, which is consistent with observations that the returns of Internet stocks have been strikingly volatile. Probably no recent investment topic elicits stronger feelings than Internet stocks. The skyrocketing valuations of these companies have made millionaires and billionaires out of many Internet entrepreneurs while the actual companies were generating significant and often growing losses. We developed a simple model to value an Internet company that is fundamentally based on assumptions about the expected growth rate of revenues and on expectations about the cost structure of the company. Because these expectations are likely to change continuously as new information becomes available, the model generates company values and stock prices that are highly volatile, but it provides a systematic way to think about the drivers of value of Internet companies and directs analyst attention to the critical parameters in the valuation. The model basically applies real-options theory and modern capital budgeting to the problem of valuing an Internet stock. We formulate the model in continuous time, form a discrete time approximation, estimate the model parameters, solve the model by simulation, and then perform sensitivity analyses. We found that, depending on the parameters chosen, the value of an Internet stock may be rational if growth rates in revenues are high enough. Even with a real chance that a company will go bankrupt, if the initial growth rates are sufficiently high and if there is enough volatility in this growth over time, valuations can be what would otherwise appear to be unbelievably high. In addition, we found a large sensitivity of the valuation to initial conditions and exact specification of the parameters, which is consistent with the observation that the returns of Internet stocks have been strikingly volatile. To implement the model, we make many assumptions about possible future financing, about future cash distributions to shareholders and bondholders, about the horizon of the estimation, and so on. Alternative assumptions are possible and easily incorporated in the analysis. We expect that potential users of a model such as the one presented will have a deep-enough knowledge of the company and its industry to make more reasonable (perhaps!) assumptions. We illustrate the methodology by applying it to Amazon.com. The basic data used for the valuation included quarterly sales, cost of goods sold, and other expenses for the last 15 quarters. We also used balance sheet data to estimate the loss carry-forward and the amount of cash available at the valuation date. Given the profitability assumed in the valuation (through the cost function), we found that Amazon equity is overpriced. Substantially higher profitability would be needed to obtain model prices that are consistent with those observed in the market. We intend to extend the analysis along a number of important dimensions. One is to make the cost function stochastic to reflect, for example, the uncertainty about future potential competitors, market share, or technological developments. Another is to take into account seasonality. If seasonality is not taken into account when estimating parameters for those industries in which it is characteristic, the volatility of the growth rate in revenues will be overestimated, which can significantly affect company valuation.

Pricing of Options on Commodity Futures with Stochastic Term Structures of Convenience Yields and Interest Rates

We develop a model to value options on commodity futures in the presence of stochastic interest rates as well as stochastic convenience yields. In the development of the model, we distinguish between forward and future convenience yields, a distinction that has not been recognized in the literature. Assuming normality of continuously compounded forward interest rates and convenience yields and log-normality of the spot price of the underlying commodity, we obtain closed-form solutions generalizing the Black-Scholes/Mertons formulas. We provide numerical examples with realistic parameter values showing that both the effect of introducing stochastic convenience yields into the model and the effect of having a short time lag between the maturity of a European call option and the underlying futures contract have significant impact on the option prices.