The Term Structure of Interest Rates Dr. David A. Stangeland

MBA 8415: Managerial Finance

Warsaw School of Economics

The Term Structure of Interest Rates Dr. David A. Stangeland

The simplest way to explain interest rates is that the current interest rate is that rate that equates the supply of capital to the demand for capital. Unfortunately, while the supply/demand analysis is good for explaining short term interest rates, it does not give obvious explanations for why interest rates differ for loans with different terms-to-maturity. The effects of the maturity preferences of borrowers and lenders combined with expectations and uncertainties about inflation must be taken into account when explaining the term structure of interest rates.

The term structure of interest rates is the relationship between current spot rates of interest for loans of differing terms-to-maturity (holding default risk constant). E.g., let ri be the observed rate of interest for a loan starting now that is to be paid back in i time periods. If we observe r1 = 8%, r2 = 9%, r3 = 9.5%, r4 = 9.75% and r5 = 9.875% then the current term structure of interest rates is represented by plotting these spot rates against their terms-to-maturity.

The n period current spot rate of interest denoted rn is the current interest rate (fixed today) for a loan (where the cash is borrowed now) to be repaid in n periods. Note: all spot rates are expressed in the form of an effective interest rate per year. In the example above, r1, r2, r3, r4, and r5 are all current spot rates of interest.

The one period forward rate of interest denoted fn is the interest rate (fixed today) for a one period loan to be repaid at some future time period, n. I.e., the money is borrowed in period n-1 and repaid in period n. E.g., given the data plotted above, we find f2 = 10.0092593% is the rate agreed upon today for a loan where the money is borrowed in one year and repaid the following year. Note: forward rates are also expressed as effective interest rates per year.

The future one period spot rate of interest, denoted n-1rn is the future interest rate (fixed in the future) for a loan to be repaid at some future time period, n. I.e., the money is borrowed in period n-1 and repaid in period n and the interest rate for the loan is fixed in period n-1. E.g., 2r3 = ? is the one period future spot rate for a loan to be made two time periods from now that is to be repaid three time periods from now. Since the rate is not fixed until the future time period (in this example, period 2), the future spot rate is not known today. Expectations of future spot rates are a major concern for anyone who needs to borrow, refinance, invest or reinvest in the future. For instance, suppose people expect the one period spot rate 1r2 to be 10.3%, then we write E[1r2] = 10.3%. The actual future spot rate of interest will likely differ from the expected future spot rate depending on the accuracy of the forecasting technique and unanticipated changes to the economy.

Note: r1 = f1 = Or1

and 1 + fn = (1+rn)n / (1+rn-1)n-1

Theories of the term structure of interest rates:

The pure expectations theory of the term structure states that expected future one-year spot rates are equal to the one-year forward rates that can be determined today. I.e., the one year forward rates of interest are the best unbiased estimates of future one year spot rates. If this theory is true, then an implication is that we can use forward rates (that we can easily calculate) to determine expected future spot rates. E.g., E[1r2] is simply f2.

The liquidity preference theory of the term structures states that investors value the greater liquidity of shorter-term investments and will therefore insist on a higher expected return from holding longer-term investments before they will switch from short-term to long-term investments. E.g. Given r1, then r2 = r1 + premium2,

r3 = r1 + premium3,.....etc. (premiumn > .... > premium3 > premium2)

The augmented expectations theory of the term structure combines the pure expectations and liquidity preference theories (and is sometimes also referred to as the Liquidity-Preference Theory). It states that longer term loans have a liquidity premium built into their interest rates and thus calculated forward rates will incorporate the liquidity premium and will overstate the expected future one-period spot rates. Restated, the one-period forward rate is higher than the expected future one-period spot rate because the forward rate is calculated from a longer-term rate that has a built in liquidity premium. I.e., f2 > E[1r2] because f2 is calculated from rates that include a liquidity premium. Similarly, f3 > E[2r3], f4 > E[3r4], etc. Given increasing liquidity premiums for longer term loans, the difference between the forward rates and the expected future spot rates should be larger for later time periods.

The segmented-markets / preferred-habitat theory of the term structure states that different groups of investors will prefer different term-to-maturity investments, but each group will be willing to invest in other “non-preferred” term-to-maturity investments if the return on these other investments contains a large enough premium to compensate for the fact that the term-to-maturity is not as desired.

Note: Each of the above theories are just that — theories! None is proven beyond a reasonable doubt. Current empirical evidence tends to support the augmented expectations theory (i.e., a combination of the pure expectations theory and the liquidity preference theory) as forward rates are usually positively biased estimates of future spot rates.

Notes on types of averages:

Arithmetic average rate of return:

If an investment earns 100% the first year, 50% the second year, and -66% the third year, then the arithmetic average rate of return is calculated as

(1.00 + 0.50 - 0. 666666667) ÷ 3 = 0.2777778 = 27. 77778% per year.

Geometric average rate of return (or average compound rate of return): If an investment earns 100% the first year, 50% the second year, and -66% the third year, then the geometric average rate of return is calculated as

[(1+1.00)(1+0.50)(1-0.666666667)]1/3 - 1 = 0.00 = 0% per year.

(Note: the n period current spot rate of interest, rn, is simply the geometric average of r1, f2, f3, . . . fn.)

Consider an original investment of $1000. How much is the investment worth after the three years given the above returns per year? Which average gives you the best indication of the investment’s performance?