Discount Rates Redux

Recall from Chapter 1 the discussion of the role of the discount rate in computing present value. There, we stated that this is the rate at which we are willing to trade present for future consumption. This is the role that the term r in the dividend discount model implicitly plays. Specifically, the one period discount rate is , where r is the relevant interest rate. The discount rate tells us how valuable a dollar one period into the future is to us today. If r = 10 percent, then that dollar feels like about $0.92 today. If r = 5 percent, it feels a little more like $0.95 today. As interest rates fall, the present value of a dollar to be received one year out means more to us. Stated differently, as r falls, we need less tomorrow to compensate us for giving up a dollar today. That is, as r falls to zero, the opportunity cost of consuming a dollar today is zero; there is no foregone payoff by failing to save for the future. In the limit, r equal to zero means we are indifferent between a dollar today and a dollar in the future; the discount rate in this case is one. Therefore, high discount rates follow from lower interest rates. In general, high discount rates are consistent with the idea that the market doesn't value the future as much and this preference shows in the form of low market rates (usually consistent with lots of liquidity, that is, money supply). So, r is a market rate but how is it arrived at? The simple answer is through the interaction of demand and supply in the money markets. But there is a deeper meaning underlying the determination of the discount rate that is meaningful to our development of all our pricing models. It has to do with intertemporal choice. Here is how that works.

Think of the phrase “trading present for future consumption” and consider a two-period model (the present and the future) for a consumer who wants to maximize his utility in both periods. Assume he has a utility function U with diminishing marginal utility (U′ > 0, U′′ < 0). Denote periodic consumption by C_t (present) and C_{t +1} (future). Like all of us, this individual has a budget constraint, that is, his current period income, Y_t, must cover current period consumption C_t, and if there's anything left over, it goes to savings S_t. Thus, the present period constraint is . In the future period, if he has savings, they have grown to . He also has future income, Y_{t +1}, so that the future budget constraint must be . That is, what he gets to consume in the future will depend on future income and what he did not consume in the present. Likewise, we can say that the present value of lifetime consumption must equal his lifetime, or permanent, income:

Now, we can write his general lifetime utility as the sum of present and future utility:

Here, the parameter β is the agent's subjective time preference, that is, his rate of impatience (which is unobserved—more on that later). It represents his preference for present over future consumption; the higher this value, the less he values future consumption. The consumer's objective is to maximize this utility subject to his budget constraint. That is:

The standard way to solve this constrained optimization problem is through the method of Lagrange. Form the Lagrangian L with budget constraint using the Lagrange multiplier λ as follows to maximize the budget-constrained lifetime utility given by:

By taking partial derivatives with respect to present and future consumption (and the constraint λ), we get three first-order conditions:

We can solve the first equation for λ and substitute this into the second equation (thus eliminating λ) to get the Euler equation:

The Euler equation says that, when utility is at an optimum, the value the consumer places on small changes in consumption (either now or in the future) must be the same. Rearranging terms

This result is important to understand. The discount rate is determined by the ratio of future to present marginal utility of consumption. Observed market interest rates are therefore the outcome of the average of agents’ behaviors as they go about the business of optimizing lifetime consumption. For given β, falling market rates (higher discount rates) indicate an increase in the marginal utility of future consumption relative to present consumption and vice versa. During recessions, for example, falling interest rates are reminiscent of households anticipating lower future consumption and, hence, the marginal utility of future consumption rises relative to current and past consumption. In the aftermath of the 2008–2009 credit crisis, household saving rose significantly and current consumption was curtailed in an effort to provide for future consumption. The discount rate therefore signals households’ degree of impatience; higher discount rates (lower interest rates and time value of money) imply more patience and vice versa. High interest rates, on the other hand, push the discount rate down, indicating that the marginal utility of future consumption is low relative to the marginal utility of current consumption. This would suggest that consumers require greater compensation in the form of higher interest rates to induce them to forgo current consumption. John Cochrane (2001) presents a well-written and more rigorous treatment of this problem.