We have seen that under certainty, 1 plus the yield to maturity on a zero-coupon bond is simply the geometric average of 1 plus the future short rates that will prevail over the life of the bond. This is the meaning of equation 15.3, which we give in general form here:

When future rates are uncertain, we modify equation 15.3 by replacing future short rates with forward rates:

1 + y„ = [(1 + r1)(1 + f2)(1 + f3) . . . (1 + fn)]1/n (15.6)

Thus there is a direct relationship between yields on various maturity bonds and forward interest rates. This relationship is the source of the information that can be gleaned from an analysis of the yield curve.

First, we ask what factors can account for a rising yield curve. Mathematically, if the yield curve is rising, fn+1 must exceed yn. In words, the yield curve is upward sloping at any maturity date, n, for which the forward rate for the coming period is greater than the yield at that maturity. This rule follows from the notion of the yield to maturity as an average (albeit a geometric average) of forward rates.

If the yield curve is to rise as one moves to longer maturities, it must be the case that extension to a longer maturity results in the inclusion of a "new" forward rate that is higher than the average of the previously observed rates. This is analogous to the observation that if a new student's test score is to increase the class average, that student's score must exceed the class's average without her score. To raise the yield to maturity, an above-average forward rate must be added to the other rates in the averaging computation.

For example, if the yield to maturity on three-year zero-coupon bonds is 9%, then the yield on four-year bonds will satisfy the following equation:

Iff4 = .09, then y4 also will equal .09. (Confirm this!) Iff4 is greater than 9%, y4 will exceed 9%, and the yield curve will slope upward.

CONCEPT CHECK ^ QUESTION 6

Look back at Tables 15.1 and 15.2. Show that y4 would exceed y3 if and only if the interest rate for period 4 had been greater than 9.66%, which was the yield to maturity on the three-year bond, y3.

Given that an upward-sloping yield curve is always associated with a forward rate higher than the spot, or current, yield, we ask next what can account for that higher forward rate. Unfortunately, there always are two possible answers to this question. Recall that the forward rate can be related to the expected future short rate according to this equation:

fn = E(rn) + Liquidity premium

where the liquidity premium might be necessary to induce investors to hold bonds of maturities that do not correspond to their preferred investment horizons.

By the way, the liquidity premium need not be positive, although that is the position generally taken by advocates of the liquidity premium hypothesis. We showed previously that if most investors have long-term horizons, the liquidity premium could be negative.

In any case, the equation shows that there are two reasons that the forward rate could be high. Either investors expect rising interest rates, meaning that E(rn) is high, or they require a large premium for holding longer-term bonds. Although it is tempting to infer from a rising yield curve that investors believe that interest rates will eventually increase, this is not a valid inference. Indeed, Figure 15.5A, provides a simple counterexample to this line of reasoning. There, the spot rate is expected to stay at 10% forever. Yet there is a constant 1% liquidity premium so that all forward rates are 11%. The result is that the yield curve continually rises, starting at a level of 10% for one-year bonds, but eventually approaching 11% for long-term bonds as more and more forward rates at 11% are averaged into the yields to maturity.

Therefore, although it is true that expectations of increases in future interest rates can result in a rising yield curve, the converse is not true: A rising yield curve does not in and of itself imply expectations of higher future interest rates. This is the heart of the difficulty in drawing conclusions from the yield curve. The effects of possible liquidity premiums confound any simple attempt to extract expectations from the term structure. But estimating the market's expectations is a crucial task, because only by comparing your own expectations to those reflected in market prices can you determine whether you are relatively bullish or bearish on interest rates.

One very rough approach to deriving expected future spot rates is to assume that liquidity premiums are constant. An estimate of that premium can be subtracted from the forward rate to obtain the market's expected interest rate. For example, again making use of the example plotted in Figure 15.5A, the researcher would estimate from historical data that a typical liquidity premium in this economy is 1%. After calculating the forward rate from the yield curve to be 11%, the expectation of the future spot rate would be determined to be 10%.

This approach has little to recommend it for two reasons. First, it is next to impossible to obtain precise estimates of a liquidity premium. The general approach to doing so would be to compare forward rates and eventually realized future short rates and to calculate the average difference between the two. However, the deviations between the two values can be quite large and unpredictable because of unanticipated economic events that affect the realized short rate. The data do not contain enough information to calculate a reliable estimate of the expected premium. Second, there is no reason to believe that the liquidity premium should be constant. Figure 15.6 shows the rate of return variability of prices of long-term Treasury bonds since 1971. Interest rate risk fluctuated dramatically during the period. So might we expect risk premiums on various maturity bonds to fluctuate, and empirical evidence suggests that term premiums do in fact fluctuate over time.

Still, very steep yield curves are interpreted by many market professionals as warning signs of impending rate increases. In fact, the yield curve is a good predictor of the business cycle as a whole, since long-term rates tend to rise in anticipation of an expansion in the economy. When the curve is steep, there is a far lower probability of a recession in the next year than when it is inverted or falling. For this reason, the yield curve is a component of the index of leading economic indicators.

The usually observed upward slope of the yield curve, especially for short maturities, is the empirical basis for the liquidity premium doctrine that long-term bonds offer a positive liquidity premium. In the face of this empirical regularity, perhaps it is valid to interpret a

466 PART IV Fixed-Income Securities

Figure 15.6 Price volatility of long-term Treasury bonds.

Figure 15.6 Price volatility of long-term Treasury bonds.

downward-sloping yield curve as evidence that interest rates are expected to decline. If term premiums, the spread between yields on long- and short-term bonds, generally are positive, then anticipated declines in rates could account for a downward-sloping yield curve.

Figure 15.7 presents a history of yields on 90-day Treasury bills and long-term Treasury bonds. Yields on the longer-term bonds generally (roughly two-thirds of the time) exceed those on the bills, meaning that the yield curve generally slopes upward. Moreover, the exceptions to this rule seem to precede episodes of falling short rates, which if anticipated, would induce a downward-sloping yield curve. For example, 1980-82 were years in which 90-day yields exceeded long-term yields. These years preceded a drastic drop in the general level of rates.

Why might interest rates fall? There are two factors to consider: the real rate and the inflation premium. Recall that the nominal interest rate is composed of the real rate plus a factor to compensate for the effect of inflation:

1 + Nominal rate = (1 + Real rate)(1 + Inflation rate)

or approximately,

Nominal rate ~ Real rate + Inflation rate

Therefore, an expected change in interest rates can be due to changes in either expected real rates or expected inflation rates. Usually, it is important to distinguish between these two possibilities because the economic environments associated with them may vary substantially. High real rates may indicate a rapidly expanding economy, high government budget deficits, and tight monetary policy. Although high inflation rates can arise out of a rapidly

CHAPTER 15 The Term Structure of Interest Rates

Figure 15.7 Yields on long-term versus 90-day Treasury securities: term spread.

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- Long-term T-bonds 90-day T-bills ■ Difference

- Long-term T-bonds 90-day T-bills ■ Difference

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expanding economy, inflation also may be caused by rapid expansion of the money supply or supply-side shocks to the economy such as interruptions in oil supplies. These factors have very different implications for investments. Even if we conclude from an analysis of the yield curve that rates will fall, we need to analyze the macroeconomic factors that might cause such a decline.

15.5

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