9 Bond Prices and Yields 323
Percent Percent Percent Percent
6.50 6.50 8.60 9.00
4.50 7.40 7.00
3 61 10 3 61 23 5 3010 3 61 2345710 30 3 61 2345710 30
Months Year Year Months Maturities
Falling yield curve
The Expectations Theory
Suppose everyone in the market believes firmly that while the current one-year interest rate is
8%, the interest rate on one-year bonds next year will rise to 10%. What would this belief im-
ply about the proper yield to maturity on two-year bonds issued today?
It is easy to see that an investor who buys the one-year bond and rolls the proceeds into an-
other one-year bond in the following year will earn, on average, about 9% per year. This value
Term Structure of Interest Rates
The bond section of the Smart Money website has a section called the Living Yield Cur ve.
It has a graph that allows you to compare the shape of the yield curve at different points
in time. Go to Smart Money’s website at http://www.smartmoney.com/onebond/
index.cfm?story yieldcur ve. Then, use the site to answer the following questions:
1. What is consider ed a normal yield cur ve?
2. Compare the yield cur ve for December 2001 with the average yield cur ve.
According to their explanations what would the market be expecting with a steep
upward sloping yield cur ve?
3. What type of yield curve was present in March 1980? How does that cur ve
compar e with the typical yield curve?
324 Part THREE Debt Securities
is just the average of the 8% earned this year and the 10% expected for next year. More pre-
cisely, the investment will grow by a factor of 1.08 in the first year and 1.10 in the second
year, for a total two-year growth factor of 1.08 1.10 1.188. This corresponds to an annual
growth rate of 8.995% (because 1.089952 1.188).
For investments in two-year bonds to be competitive with the strategy of rolling over one-
year bonds, these two-year bonds also must offer an average annual return of 8.995% over the
two-year holding period. This is illustrated in Figure 9.12. The current short-term rate of 8%
and the expected value of next year’s short-term rate are depicted above the time line. The
two-year rate that provides the same expected two-year total return is below the time line. In
this example, therefore, the yield curve will be upward sloping; while one-year bonds offer an
8% yield to maturity, two-year bonds offer an 8.995% yield.
This notion is the essence of the expectations hypothesis of the yield curve, which asserts
that the slope of the yield curve is attributable to expectations of changes in short-term rates.
Relatively high yields on long-term bonds are attributed to expectations of future increases in
The theory that yields rates, while relatively low yields on long-term bonds (a downward-sloping or inverted yield
to maturity are curve) are attributed to expectations of falling short-term rates.
determined solely by One of the implications of the expectations hypothesis is that expected holding-period re-
expectations of turns on bonds of all maturities ought to be about equal. Even if the yield curve is upward
future short-term sloping (so that two-year bonds offer higher yields to maturity than one-year bonds), this does
interest rates. not necessarily mean investors expect higher rates of return on the two-year bonds. As we’ve
seen, the higher initial yield to maturity on the two-year bond is necessary to compensate in-
vestors for the fact that interest rates the next year will be even higher. Over the two-year pe-
riod, and indeed over any holding period, this theory predicts that holding-period returns will
be equalized across bonds of all maturities.
Suppose we buy the one-year zero-coupon bond with a current yield to maturity of 8%. If its
face value is $1,000, its price will be $925.93, providing an 8% rate of return over the com-
9.10 EXAMPLE ing year. Suppose instead that we buy the two-year zero-coupon bond at its yield of 8.995%.
Holding-Period Its price today is $1,000/(1.08995)2 $841.76. After a year passes, the zero will have a re-
Returns maining maturity of only one year; based on the forecast that the one-year yield next year will
be 10%, it then will sell for $1,000/1.10 $909.09. The expected rate of return over the
year is thus ($909.09 $841.76)/$841.76 .08, or 8%, precisely the same return provided
by the one-year bond. This makes sense: If risk considerations are ignored when pricing the
two bonds, they ought to provide equal expected rates of return.
In fact, advocates of the expectations hypothesis commonly invert this analysis to infer the
market’s expectation of future short-term rates. They note that we do not directly observe the
expectation of next year’s rate, but we can observe yields on bonds of different maturities.
FIGURE 9.12 2-year cumulative
Returns to two 2-year expected returns
1.08 1.10 1.188
r1 8% E(r2 ) 10
y2 8.995% 1.089952 1.188
9 Bond Prices and Yields 325
Suppose, as in this example, we see that one-year bonds offer yields of 8% and two-year
bonds offer yields of 8.995%. Each dollar invested in the two-year zero would grow after two
years to $1 1.089952 $1.188. A dollar invested in the one-year zero would grow by a fac-
tor of 1.08 in the first year and, then, if reinvested or “rolled over” into another one-year zero
in the second year, would grow by an additional factor of 1 r2. Final proceeds would be
$1 1.08 (1 r2).
The final proceeds of the rollover strategy depend on the interest rate that actually tran-
spires in year 2. However, we can solve for the second-year interest rate that makes the ex-
pected payoff of these two strategies equal. This “breakeven” value is called the forward rate for ward rate
for the second year, f2, and is derived as follows:
The inferred short-
term rate of interest
1.089952 1.08 (1 f2 )
for a future period
which implies that f2 .10, or 10%. Notice that the forward rate equals the market’s expecta- that makes the
tion of the year 2 short rate. Hence, we conclude that when the expected total return of a long-
expected total return
term bond equals that of a rolling over a short-term bond, the forward rate equals the expectedof a long-term bond
equal to that of
short-term interest rate. This is why the theory is called the expectations hypothesis.
rolling over short-
More generally, we obtain the forward rate by equating the return on an n-period zero- term bonds.
coupon bond with that of an (n 1)-period zero-coupon bond rolled over into a one-year
bond in year n:
(1 yn)n (1 yn 1)n 1 (1 fn)
The actual total returns on the two n-year strategies will be equal if the short-term interest rate
in year n turns out to equal fn.
Suppose that two -year maturity bonds offer yields to maturity of 6%, and three-year bonds
have yields of 7%. What is the forward rate for the thir d year? We could compare these two
strategies as follows:
1. Buy a three-year bond. Total proceeds per dollar invested will be
$1 (1.07)3 $1.2250
2. Buy a two-year bond. Reinvest all proceeds in a one-year bond in the third year, which will
provide a return in that year of r3. Total proceeds per dollar invested will be the result of
two years’ growth of invested funds at 6% plus the final year’s growth at rate r3:
$1 (1.06)2 (1 r3) $1.1236 (1 r3)
The forward rate is the rate in year 3 that makes the total return on these strategies equal:
1.2250 1.1236 (1 f3 )
We conclude that the forward rate for the third year satisfies (1 f3) 1.0902, so that f3 is
The Liquidity Preference Theory
The expectations hypothesis starts from the assertion that bonds are priced so that “buy and
hold” investments in long-term bonds provide the same returns as rolling over a series of
short-term bonds. However, the risks of long- and short-term bonds are not equivalent.
We have seen that longer term bonds are subject to greater interest rate risk than short-term
bonds. As a result, investors in long-term bonds might require a risk premium to compensate
them for this risk. In this case, the yield curve will be upward sloping even in the absence of
any expectations of future increases in rates. The source of the upward slope in the yield curve
is investor demand for higher expected returns on assets that are perceived as riskier.
326 Part THREE Debt Securities
This viewpoint is called the liquidity preference theory of the term structure. Its name de-
rives from the fact that shorter term bonds have more “liquidity” than longer term bonds, in
the sense that they offer greater price certainty and trade in more active markets with lower
The theory that bid–ask spreads. The preference of investors for greater liquidity makes them willing to hold
investors demand these shorter term bonds even if they do not offer expected returns as high as those of longer
a risk premium on term bonds.
long-term bonds. We can think of a liquidity premium as resulting from the extra compensation investors
liquidity premium demand for holding longer term bonds with lower liquidity. We measure it as the spread be-
tween the forward rate of interest and the expected short rate:
The extra expected
return demanded fn E(rn) Liquidity premium
by investors as
In the absence of a liquidity premium, the forward rate would equal the expectation of the fu-
ture short rate. But generally, we expect the forward rate to exceed that expectation to com-
the greater risk of
pensate investors for the lower liquidity of longer term bonds.
longer term bonds.
Advocates of the liquidity preference theory also note that issuers of bonds seem to prefer
to issue long-term bonds. This allows them to lock in an interest rate on their borrowing for
long periods. If issuers do prefer to issue long-term bonds, they will be willing to pay higher
yields on these issues as a way of eliminating interest rate risk. In sum, borrowers demand
higher rates on longer term bonds, and issuers are willing to pay higher rates on longer term
bonds. The conjunction of these two preferences means longer term bonds typically should
offer higher expected rates of return to investors than shorter term bonds. These expectations
will show up in an upward-sloping yield curve.
If the liquidity preference theory is valid, the forward rate of interest is not a good estimate
of market expectations of future interest rates. Even if rates are expected to remain unchanged,
for example, the yield curve will slope upward because of the liquidity premium. That upward
slope would be mistakenly attributed to expectations of rising rates if one were to use the pure
expectations hypothesis to interpret the yield curve.
Suppose that the short-term rate of interest is currently 8% and that investors expect it to re-
main at 8% next year. In the absence of a liquidity premium, with no expectation of a change
9.12 EXAMPLE in yields, the yield to maturity on two-year bonds also would be 8%, the yield cur ve would be
Liquidity flat, and the for ward rate would be 8%. But what if investors demand a risk premium to in-
Premia and vest in two-year rather than one-year bonds? If the liquidity premium is 1%, then the for ward
the Yield Curve rate would be 8% 1% 9%, and the yield to maturity on the two -year bond would be de-
(1 y2)2 1.08 1.09 1.1772
implying that y2 .085 8.5%. Here, the yield cur ve is upward sloping due solely to the
liquidity premium embedded in the price of the longer term bond.
9. Suppose that the expected value of the interest rate for year 3 remains at 8% but
CHECK > that the liquidity premium for that year is also 1%. What would be the yield to ma-
turity on three-year zeros? What would this imply about the slope of the yield
9 Bond Prices and Yields 327
A maturity Illustrative
Yield curve yield curves
Of course, we do not need to make an either/or choice between expectations and risk premi-
ums. Both of these factors influence the yield curve, and both should be considered in inter-
preting the curve.
Figure 9.13 shows two possible yield curves. In Figure 9.13A, rates are expected to rise
over time. This fact, together with a liquidity premium, makes the yield curve steeply upward
sloping. In Figure 9.13B, rates are expected to fall, which tends to make the yield curve slope
downward, even though the liquidity premium lends something of an upward slope. The net
effect of these two opposing factors is a “hump-shaped” curve.
These two examples make it clear that the combination of varying expectations and li-
quidity premiums can result in a wide array of yield-curve profiles. For example, an upward-
sloping curve does not in and of itself imply expectations of higher future interest rates,
because the slope can result either from expectations or from risk premiums. A curve that is
more steeply sloped than usual might signal expectations of higher rates, but even this infer-
ence is perilous.
Figure 9.14 presents yield spreads between 90-day T-bills and 10-year T-bonds since 1970.
The figure shows that the yield curve is generally upward sloping in that the longer-term
bonds usually offer higher yields to maturity, despite the fact that rates could not have been
expected to increase throughout the entire period. This tendency is the empirical basis for the
liquidity premium doctrine that at least part of the upward slope in the yield curve must be be-
cause of a risk premium.
328 Part THREE Debt Securities
FIGURE 9.14 20
Term spread. Yields
to maturity on 90-day
T-bills and 10-year
90 day T-bills
1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002
• Debt securities are distinguished by their promise to pay a fixed or specified stream of
income to their holders. The coupon bond is a typical debt security.
• Treasury notes and bonds have original maturities greater than one year. They are issued at
or near par value, with their prices quoted net of accrued interest. T-bonds may be callable
during their last five years of life.
• Callable bonds should offer higher promised yields to maturity to compensate investors
for the fact that they will not realize full capital gains should the interest rate fall and the
bonds be called away from them at the stipulated call price. Bonds often are issued with a
period of call protection. In addition, discount bonds selling significantly below their call
price offer implicit call protection.
• Put bonds give the bondholder rather than the issuer the option to terminate or extend the
life of the bond.
• Convertible bonds may be exchanged, at the bondholder’s discretion, for a specified
number of shares of stock. Convertible bondholders “pay” for this option by accepting a
lower coupon rate on the security.
• Floating-rate bonds pay a fixed premium over a referenced short-term interest rate. Risk is
limited because the rate paid is tied to current market conditions.
• The yield to maturity is the single interest rate that equates the present value of a security’s
cash flows to its price. Bond prices and yields are inversely related. For premium bonds,
the coupon rate is greater than the current yield, which is greater than the yield to maturity.
The order of these inequalities is reversed for discount bonds.
• The yield to maturity often is interpreted as an estimate of the average rate of return to an
investor who purchases a bond and holds it until maturity. This interpretation is subject to
error, however. Related measures are yield to call, realized compound yield, and expected
(versus promised) yield to maturity.
• Treasury bills are U.S. government-issued zero-coupon bonds with original maturities of
up to one year. Prices of zero-coupon bonds rise exponentially over time, providing a rate
of appreciation equal to the interest rate. The IRS treats this price appreciation as imputed
taxable interest income to the investor.