Summary

Returns measure growth rates in asset value over time. They are the observed reward to postponing present for future consumption. Returns may be measured discretely over any frequency such as daily, monthly, or quarterly and higher frequency returns can be converted into lower frequencies—annualizing monthly returns is one such case. The notion of compounding is linked to how often interest is paid; thus, annual returns can represent a single yearly payment, or higher frequency returns can be geometrically linked, or compounded, to form annualized equivalents. In the limit, continuously compounded returns represent the continuous payment of interest. In sum, we can work with returns over any interval and extrapolate those returns to either any longer interval of time or average them to any subinterval of time. An example of extrapolation is annualizing discrete monthly returns, and an example of averaging is finding the geometric average monthly return from an annual return. Caveats relate to the implicit assumption that observed returns will hold into the future.

Discounting links cash flows over time. The discounted present value of a cash flow to be received in the future is the result of finding the amount of cash in present dollars that, when invested at the discount rate, will grow to an amount stipulated by the future cash flow. Discount rates are intimately linked to returns; in the simplest case, the discount rate is the reciprocal of the gross return and the discount rate may be applied discretely or continuously. The role that discount rates play in the trade-off of present over future consumption is a topic that I develop more fully in Chapter 4.

The internal rate of return is an application of discounting in cash management and the yield-to-maturity on a coupon-paying bond is itself an internal rate of return. We can thus link the subject of Chapter 2 on bond pricing to the discount function developed in this chapter. In fact, almost all asset-pricing models will rely on some form of discounting since they all involve the valuation of cash flows that occur over time.

Finally, we recognize the impact that inflation has on the value of cash flows, which requires us to distinguish between inflation adjusted (real) returns and nominal returns and model them accordingly.