The contract price on a forward contract stays fixed for the life of the contract, while a futures contract is rewritten every day. The value of a futures contract is zero at the start of each day. The expected change in the futures price satisfies a formula like the capital asset pricing model. If changes in the futures price are independent of the return on the market, the futures price is the expected spot price. The futures market is not unique in its ability to shift risk, since corporations can do that too. The futures market is unique in the guidance it provides for producers, distributors, and users of commodities. Using assumptions like those used in deriving the original option formula, we find formulas for the values of forward contracts and commodity options in terms of the futures price and other variables.

This paper discusses general principles for choosing bank assets and liabilities, for deciding on when to make a loan and what interest rate to charge, for pricing funds transfer services such as the handling of checks, for establishing compensating balance requirements, and for dealing with government regulation. The discussion assumes markets are efficient and deals first with an unregulated environment and then with policies in the face of regulatory constraints. Most of the policies which would be optimal in an unregulated environment will be optimal in the regulated environment such as in the U.S. today, because it is relatively easy to get around most of the regulations that are applied to banks by the use of non-deposit liabilities, compensating balances and negative checking accounts.

This paper outlines models of capital market equilibrium when there are explicit barriers to international investment in the form of a tax on holdings of assets in one country by residents of another country. There is a corresponding subsidy on short positions in foreign assets. Asset prices deviate from the predictions of the world capital asset pricing model. Investors do not hold a mixture of national market portfolios, but the mix of risky assets is the same for every investor in a country. Optimal portfolios tend to be heavy in domestic assets, and light in foreign assets. Tax free investors, however, tend to hold assets anywhere in the world that are taxed heavily. Estimates of the magnitude of the average tax (or the magnitude of effective barriers to international investment) can be made by comparing the average return on the minimum variance zero β portfolio, z, with the average across countries and time of the short-term interest rate. When barriers are ineffective, the expected return on portfolio z will be the average short-term interest rate, and the world capital asset pricing model will hold.

Stambaugh, and especially to Robert Merton for comments on a previous draft. This paper is part of NBER's research program in Financial Markets and Monetary Economics. Any opinions expressed are those of the author not those

[Most models of the evolution of wealth and consumption assume that wealth volatility and risk premium are constant. But in fact, volatility declines, and risk premium seems to decline, as wealth rises. A model that allows mean reversion in the sense that the risk premium declines as wealth rises can help explain both the "consumption smoothing puzzle" and the "equity premium puzzle." In an example of such a model that gives us an analytic solution, direct and derived risk aversion are both constant, but differ.]

Since people can hold currency at a zero nominal interest rate, the nominal short cannot be negative. The real interest can be and has been negative, since low risk real investment opportunities, like filling in the Mississippi delta, do not guarantee positive returns. The inflation can be and has been negative, most recently (in the U.S.) during the Great Depression. The nominal short is the shadow real interest rate (as defined by the investment opportunity set) plus the inflation rate, or zero, whichever is greater. Thus the nominal short is an option. Longer term interest rates are always positive, since the future short may be positive even when the current short is zero. We can easily build this option element into our interest trees for backward induction or Monte Carlo simulation: just create a distribution that allows negative nominal rates, and then replace each negative with zero.

Since people can hold currency at a zero nominal interest rate, the nominal short rate cannot be negative. The real interest rate can be and has been negative, since low risk real investment opportunities like filling in the Mississippi delta do not guarantee positive returns. The inflation rate can be and has been negative, most recently (in the United States) during the Great Depression. The nominal short rate is the 'shadow real interest rate' (as defined by the investment opportunity set) plus the inflation rate, or zero, whichever is greater. Thus the nominal short rate is an option. Longer term interest rates are always positive, since the future short rate may be positive even when the current short rate is zero. We can easily build this option element into our interest rate trees for backward induction or Monte Carlo simulation: just create a distribution that allows negative nominal rates, and then replace each negative rate with zero.

Since people can hold currency at a zero nominal interest rate, the nominal short rate cannot be negative. The real interest rate can be and has been negative, since low risk real investment opportunities like filling in the Mississippi delta do not guarantee positive returns. The inflation rate can be and has been negative, most recently (in the United States) during the Great Depression. The nominal short rate is the “shadow real interest rate” (as defined by the investment opportunity set) plus the inflation rate, or zero, whichever is greater. Thus the nominal short rate is an option. Longer term interest rates are always positive, since the future short rate may be positive even when the current short rate is zero. We can easily build this option element into our interest rate trees for backward induction or Monte Carlo simulation: just create a distribution that allows negative nominal rates, and then replace each negative rate with zero.

We assume a world like the one that gives the capital asset pricing model, but with many goods and many countries. We assume that investors in a given country have homothetic utility functions with the same weights, and a currency that has a sure end-of-period value using a price index with those weights. Siegel's paradox (derived from Jensen's inequality) makes investors want a positive amount of exchange risk. When average risk tolerance is the same across countries, every investor will hold the same mix of market risk (through the world market portfolio of all assets) and exchange risk (in a diversified basket of foreign currencies). In fact, the ratio of exchange risk to market risk is equal to the average investor's risk tolerance. We can write the ratio of exchange risk to market risk (and the fraction of the market's exchange risk that investors hedge) as depending on an average of world market risk premia, an average of world market volatilities, and an average of exchange rate volatilities. The weights in these averages are the same as the weights of the different countries in the currency basket. Given these averages, the ratio (and the fraction hedged) will not depend directly on exchange rate means or covariances. In equilibrium, we can use the ratio of exchange risk to market risk to measure average risk tolerance: in this model, risk tolerance is observable.