Interest rates are a critical concept in finance. In some cases, we assume a particular interest rate and in others, the interest rate remains the unknown quantity to determine. Although the pre-reads have covered the mechanics of time value of money problems, here we first illustrate the underlying economic concepts by explaining the meaning and interpretation of interest rates and then calculate, interpret, and compare different return measures.

OVERVIEW

- An interest rate, r, can have three interpretations: (1) a required rate of return, (2) a discount rate, or (3) an opportunity cost. An interest rate reflects the relationship between differently dated cash flows.

- An interest rate can be viewed as the sum of the real risk-free interest rate and a set of premiums that compensate lenders for bearingdistinct types of risk: an inflation premium, a default risk premium, aliquidity premium, and a maturity premium.

- The nominal risk-free interest rate is approximated as the sum of thereal risk-free interest rate and the inflation premium.

- A financial asset’s total return consists of two components: an incomeyield consisting of cash dividends or interest payments, and a returnreflecting the capital gain or loss resulting from changes in the price ofthe financial asset.

- A holding period return, R, is the return that an investor earns for asingle, specified period of time (e.g., one day, one month, five years).

- Multiperiod returns may be calculated across several holding periodsusing different return measures (e.g., arithmetic mean, geometricmean, harmonic mean, trimmed mean, winsorized mean). Each returncomputation has special applications for evaluating investments.

- The choice of which of the various alternative measurements of mean to use for a given dataset depends on considerations such as thepresence of extreme outliers, outliers that we want to include, whether there is a symmetric distribution, and compounding.■ A money-weighted return reflects the actual return earned on an investment after accounting for the value and timing of cash flows relating to the investment.

- A time-weighted return measures the compound rate of growth of one unit of currency invested in a portfolio during a stated measurement period. Unlike a money-weighted return, a time-weighted return is not sensitive to the timing and amount of cash flows and is the preferred performance measure for evaluating portfolio managers because cashwithdrawals or additions to the portfolio are generally outside of the control of the portfolio manager.

- Interest may be paid or received more frequently than annually. The periodic interest rate and the corresponding number of compoundingperiods (e.g., quarterly, monthly, daily) should be adjusted to computepresent and future values.

- Annualizing periodic returns allows investors to compare different investments across different holding periods to better evaluate andcompare their relative performance. With the number of compounding periods per year approaching infinity, the interest is compound continuously.

- Gross return, return prior to deduction of managerial and administrative expenses (those expenses not directly related to return generation), is an appropriate measure to evaluate the comparative performance of an asset manager.

- Net return, which is equal to the gross return less managerial andadministrative expenses, is a better return measure of what an investoractually earned.

- The after-tax nominal return is computed as the total return minusany allowance for taxes on dividends, interest, and realized gains.

- Real returns are particularly useful in comparing returns across timeperiods because inflation rates may vary over time and are particularlyuseful for comparing investments across time periods and performance between different asset classes with different taxation.

- Leveraging a portfolio, via borrowing or futures, can amplify the portfolio’s gains or losses.

**INTEREST RATES AND TIME VALUE OF MONEY**

The time value of money establishes the equivalence between cash flows occurring on different dates. As cash received today is preferred to cash promised in the future, we must establish a consistent basis for this trade-off to compare financial instruments in cases in which cash is paid or received at different times. An interest rate (or yield), denoted r, is a rate of return that reflects the relationship between differently dated – timed – cash flows. If USD 9,500 today and USD 10,000 in one year are equivalent in value, then USD 10,000 – USD 9,500 = USD 500 is the required compensation for receiving USD 10,000 in one year rather than now. The interest rate (i.e., the required compensation stated as a rate of return) is USD 500/USD 9,500 = 0.0526 or 5.26 percent.

Interest rates can be thought of in three ways:

- First, they can be considered required rates of return—that is, the minimumrate of return an investor must receive to accept an investment.
- Second, interest rates can be considered discount rates. In the previousexample, 5.26 percent is the discount rate at which USD 10,000 in one yearis equivalent to USD 9,500 today. Thus, we use the terms “interest rate” and“discount rate” almost interchangeably.
- Third, interest rates can be considered opportunity costs. An opportunitycost is the value that investors forgo by choosing a course of action. In theexample, if the party who supplied USD 9,500 had instead decided to spendit today, he would have forgone earning 5.26 percent by consuming ratherthan saving. So, we can view 5.26 percent as the opportunity cost of currentconsumption.

**Determinants of Interest Rates**

Economics tells us that interest rates are set by the forces of supply and demand, where investors supply funds and borrowers demand their use. Taking the perspective of investors in analyzing market-determined interest rates, we can view an interest rate r as being composed of a real risk-free interest rate plus a set of premiums that are required returns or compensation for bearing distinct types of risk:

r = Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium.

■ The real risk-free interest rate is the single-period interest rate for a completely risk-free security if no inflation were expected. In economic theory,the real risk-free rate reflects the time preferences of individuals for currentversus future real consumption.

The inflation premium compensates investors for expected inflation andreflects the average inflation rate expected over the maturity of the debt.Inflation reduces the purchasing power of a unit of currency—the amountof goods and services one can buy with it.

The default risk premium compensates investors for the possibility that theborrower will fail to make a promised payment at the contracted time and inthe contracted amount.

The liquidity premium compensates investors for the risk of loss relativeto an investment’s fair value if the investment needs to be converted tocash quickly. US Treasury bills (T-bills), for example, do not bear a liquidity premium because large amounts of them can be bought and sold without affecting their market price. Many bonds of small issuers, by contrast, trade infrequently after they are issued; the interest rate on such bonds includes a liquidity premium reflecting the relatively high costs (including the impact on price) of selling a position.

The maturity premium compensates investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended, in general (holding all else equal). The difference between the interest rate on longer-maturity, liquid Treasury debt and that on short-term Treasury debt typically reflects a positive maturity premium for the longer-term debt (and possibly different inflation premiums as well).

The sum of the real risk-free interest rate and the inflation premium is the nominal risk-free interest rate: The nominal risk-free interest rate reflects the combination of a real risk-free rate plus an inflation premium:

(1 + nominal risk-free rate) = (1 + real risk-free rate)(1 + inflation premium).

In practice, however, the nominal rate is often approximated as the sum of the real risk-free rate plus an inflation premium:

Nominal risk-free rate = Real risk-free rate + inflation premium

Many countries have short-term government debt whose interest rate can be considered to represent the nominal risk-free interest rate over that time horizon in that country. The French government issues BTFs, or negotiable fixed-rate discount Treasury bills, with maturities of up to one year. The Japanese government issues a short-term Treasury bill with maturities of 6 and 12 months. The interest rate on a 90-day US T-bill, for example, represents the nominal risk-free interest rate for the United States over the next three months. Typically, interest rates are quoted in annual terms, so the interest rate on a 90-day government debt security quoted at 3 percent is the annualized rate and not the actual interest rate earned over the 90-day period.

Whether the interest rate we use is a required rate of return, or a discount rate, or an opportunity cost, the rate encompasses the real risk-free rate and a set of risk premia that depend on the characteristics of the cash flows. The foundational set of premia consist of inflation, default risk, liquidity risk, and maturity risk. All these premia vary over time and continuously change, as does the real risk-free rate. Consequently, all interest rates fluctuate, but how much they change depends on various economic fundamentals—and on the expectation of how these various economic fundamentals can change in the future.