The Term Structure of Interest Rates
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Some loans are for a very short period of time; for example, some are literally just
overnight. The so-called term to maturity on such a loan is 1 day. Other loans are for a longer
period; for example, a 30 year loan to buy a house. The term to maturity on this type of loan is
30 years. How is the interest rate on 1 day loan related to the interest rate on a 30 year, roughly
a 10,950 day, loan? The relationship between interest rates on loans with different terms to
maturity is called the term structure of interest rates and we examine the relationship below.
The term structure of interest rates is summarized by the yield curve. The yield curve
shows the relationship between the term to maturity on a bond and the yield to maturity, or
interest rate, on that bond. The interest rate is measured on vertical axis and the term to maturity
is measured on the horizontal axis. The yield curve for U.S. Treasury securities on May 21,
2004 is shown above.
The yield curve on this date was upward sloping. This means that yields are lower on
securities with shorter terms to maturity. The yield on a security that matured in 1 year was
1.84%, while the yield on an issue that matured in 20 years was 5.5%. This is the typical shape
of a yield curve. It is also typical for yields of differing maturities to move in the same
directions. So, when short term interest rates are rising, long term interest rates are usually rising
also. Note the we have used the weasel words typically and usually. These words mean that the
yield curve is not always upward sloping and that rates on issues of differing maturities don't
always move together. For example, the yield curve was downward sloping in from August of
2000 to December of 2000.1 It turns out the yield curve inversions are relatively rare. The
picture below plots the difference between the yield to maturity on a 5 year Treasury security
and the yield to maturity on a 1 year Treasury security. An inversion occurs when the plot dips
below the zero line. The 5 year Treasury yield exceeds the 1 year yield only about 17% of the
time. There are two key facts, or empirical regularities, with regard to the yield curve : 1) the
1
More specifically, over this period the average monthly yield on a 1 year Treasury security was greater
than the average monthly yield on a 5 year Treasury security.
6
5
4
3
2
1
239
225
211
197
183
169
155
141
127
113
99
85
71
57
43
29
15
0
1
Yield on US Treasury Security
Yield Curve - May 21, 2004
Term to Maturity in Months
yield curve is typically upward sloping, and 2) yields on differing maturities tend to move in the
same direction.
The Pure Expectations Hypothesis
We first assume that short-term
and long-term bonds are perfect
substitutes. In this case an investor saving
Difference Between the Five and One Year Treasury Yields
3
2
over, say, a two year horizon could care
1
less whether she buys a bond that matures
0
in two years, or buys a one year bond this
-1
year and then uses the proceeds to buy
-2
another one year bond next year. The
only thing that matters is which strategy
-3
55
60
65
70
75
80
85
90
95
00
produces the highest yield. To figure out
which strategy produces the greatest
return, we write the yield to maturity at time t on a bond that matures in one year as R t,1 and the
yield to maturity on a one year bond at time t+1 as Rt+1,1. If you follow the strategy of buying a
sequence of two one year bonds, then at the end of the two year period each of your dollars will
have grown to
Strategy 1
future value of $1 two years
hence with a strategy of buying
a sequence of two one year bonds
=
(1+Rt,1)(1+Rt+1,1).
To what value will each dollar grow if we follow the alternative strategy of buying a
bond that matures in two years. We let the annualized yield to maturity at time t on a bond that
matures in two years as Rt,2. If you follow the strategy of buying a bond that matures in two
years and holding it until it matures, then at the end of the two year period each of your dollars
will have grown to
Strategy 2
future value of $1 two years
hence with a strategy of buying
and holding a two year bond
=
(1+Rt,2)(1+Rt,2).
Now suppose that a dollar grows to a greater value by following strategy 1 than from
following strategy 2. In this case
(1+Rt,1)(1+Rt+1,1) > (1+Rt,2)(1+Rt,2),
and people will be eager to sell their long term, two year, bonds in order to buy the more
lucrative short term, one year, bonds. What are the implications of this buying and selling? To
sell the long term bonds, sellers will have to offer these bonds at a lower price, while short term
bonds prices will rise in the face of many eager buyers. The fall in long term prices will increase
Rt,2, while the rise in short term prices will decrease R t,1. This adjustment in prices and yields
will close the gap between the payoffs from strategies one and two. Indeed, this substitution of
short term for long term bonds will continue until the payoffs from the two strategies are the
same. When this equality is reached, any incentive to substitute one type of bond for another
disappears, and we have equilibrium. This equilibrium requires that
(1+Rt,1)(1+Rt+1,1) = (1+Rt,2)(1+Rt,2).
In practice, at time t we do not the value of Rt+1,1 and so we must use its expectation
instead. So, we rewrite the equation above as
(1 + R t,1 )(1 + R et+1,1 ) = (1 + R t,2 )(1 + R t,2 )
where R et+1,1 is the short term rate that people expect will prevail at time t+1. This equation
represents the so-called Pure Expectations Hypothesis (PEH) of the term structure of interest
rates. It helps to interpret this hypothesis if we multiply the expression out. This gives us
1 + R t,1 + R t+1 + R t,1 $ R t+1,1 = 1 + 2 $ R t,2 + R 2t,2.
Now we can cancel the ones on both sides, and since multiplicative terms on each side are small
and offset each other, we have the following approximation
R t,1 + R et+1,1 = 2 $ R t,2
or
(R t,1 + R et+1,1 + $ $ $ +R t+n−1,1 )/n = R t,n
The PEH implies that the current long term rate is the average of the current and expected short
term rates. The argument we used is not specific to just two periods, and so it generalizes. The
relationship between current and expected short term yields and the yield on an n year bond is
give by
(Rt,1 + Ret+1 )/2 = Rt,2
The perfect substitutability of the two bonds is the key assumption behind this result. If
two assets are perfect substitutes, then in equilibrium they must have the same return.
Substitutability also implies that short and long term rates will move in the same direction. For
example, suppose that the current short term interest rate rises. This change increases the payoff
from strategy 1 and so people will buy short term bonds and sell long term bonds. But as we
have seen, this will result in an increase in the long term rate of interest. So, the PEH can
explain or help us understand the second fact of the term structure.
However, the PEH does not help us understand the first fact of the term structure, the
typically upward sloping yield curve. If the PEH is correct, then an upward sloping yield curve
implies that people expect short term rates to rise. To see this result, consider a simple
numerical example. Suppose that the current short rate is 4% and current long rate is 6% so that
the yield curve is upward sloping. The PEH equation implies that
(4% + R et+1,1 )/2 = 6%.
This means that R et+1,1 must equal 8%. People are expecting that the short term rate will rise
from 4% to 8%. You can check for yourself that this result does not depend on the numbers I
selected. If the yield curve is upward sloping, it would then mean that people usually expected
short term rates to rise. This, in turn, implies that short term rates typically rise, as expected, and
so should exhibit a positive trend, or that on average people were wrong. The first possibility is
contradicted by the data. There is no trend in interest rates. The second possibility is
contradicted by economic theory. In particular, people are very unlikely to make systematic, that
is to predictable, errors in matters where their income is concerned. Continual, avoidable errors
will make one poor, and so will be available. In short, the PEH cannot explain why the yield
curve is typically upward sloping.
Liquidity or Risk Premium Hypothesis
The PEH assumes that bonds of differing maturities are perfect substitutes. This
assumption is too strong. While long and short term bonds are substitutes, they differ in an
important characteristic. For a given change in yields, the price of a long term bond changes by
a greater percentage amount than the price of a short term bond.
To illustrate this point consider the following example. Suppose the yield to maturity on
both one year and two year bonds is 5%. To keep things simple we assume that the one year
bond is a promise to pay $100 one year from today. To make the initial prices of the two bonds
the same, we let the two year bond be a promise to pay $51.22 one year from now and then again
two years from now. In this case, given some rounding, we have
P Bt,1 = $100/1.05 = $95.24
and
P Bt,2 =
$51.22
1.05
+
$51.22
(1.05) 2
= $95.24 .
Now suppose that both the long term and short term interest rate increase to 8%. This causes the
price of the short term bond to fall to $92.59, about a 2.8% change. On the other hand, the price
of the two year bond falls to $91.34, over a 4% change. This means that the two year bond is
riskier to hold because it is subject to larger variations in its value.
The PEH ignores this difference and predicts that in equilibrium we will see
(1 + R t,1 )(1 + R et+1,1 ) = (1 + R t,2 )(1 + R t,2 ),
but suppose this equality happens to hold. Would people be indifferent between strategy 1 and
strategy 2? The answer is now if it is riskier to follow strategy 2; that is, to buy the two year
bond. Instead, if the above equality holds, then people would prefer to have the safer short term
bond; and as a result people will be selling long term bonds and buying short term bonds. This,
in turn, will cause yields on long term bonds to rise and yields on short term bonds to fall. In
equilibrium we will have
(1 + R t,1 )(1 + R et+1,1 ) < (1 + R t,2 )(1 + R t,2 ),
or with our approximation we will have
(Rt,1 + Ret+1 )/2 < Rt,2
In general, once we recognize that long term bonds are riskier than short term bonds, the long
term rate will be greater than the average of the current and expected future short rates.
We can rewrite the above inequality as
0 < Rt,2 − (Rt,1 + Ret+1 )/2 = L t,2
The positive difference between the long term rate and the average of short term rates is a risk or
liquidity premium and is labeled Lt,2; and we can write out the risk or Liquidity Premium
Hypothesis (LPH) as
R t,2 = (Rt,1 + R et+1 )/2 + L t,2 .
The LPH takes short and long term bonds to substitutes, but not perfect substitutes.
Nevertheless, because short and long term bonds are substitutes, the LPH predicts that short and
long term yields should usually move in the same direction; so the LPH can explain the first fact.
However, the long and short term bonds are not perfect substitutes because long term bonds are
riskier and so incorporate a liquidity premium into their yield. The liquidity premiums are larger
the longer is the term to maturity. It is this feature of the LPH that explains why the yield curve
is typically upward sloping. The positive slope of the typical yield curve reflects these
increasing liquidity premiums so explains the second fact term structure of interest rates.
A Generalization
There is greater variation in the price of long term bonds than there is in short term
bonds, so, other things the same, shorter term bonds are less risky. In many cases other things
are not the same. For example, suppose I have a payment that I must make in two years.
Perhaps I am planning on buying a house and the payment is the down payment on my new
home, or perhaps I will be helping my child pay for his college education. Suppose this payment
is $10,000. If the current two year rate is 10%, then I can buy $8,264.46 worth of bonds and
know for sure that will have $10,000 in two years. On the other hand, if the one year rate is 10%
and I put $8,264.46 into bonds, then in one year I will have $9,090.01. If the short rate next year
is again 10%, then I can roll my $9,090.01 into another one year bond and meet my goal of
$10,000. However, if the yield on short term bonds fall, my $9,090.01 will not do. I will have to
spend more on bonds to meet my goal. When I have a payment or liability that comes due in two
years, it is safer for me to purchase a bond that also matures in two years. In general, risk is
reduced if you can match the maturity of your liabilities with the maturities of your assets.
Different people or institutions have liabilities that mature at different dates. For
example, a pension plan may have a large amount of its liabilities maturing in 20 or 30 years.
Pension funds may therefore like to hold assets that will mature about the same time. For a
larger retailer, a large amount of their liabilities may arise within a year and so they may prefer
short term bonds. In short, people and institutions may have a preferred habitat and may be will
to pay a premium to hold assets with the preferred maturity. Let this premium be Ht,n for the
premium at date t on an asset that matures in n periods. With this notation we can write a more
general hypothesis down.
Rt,2 = (Rt,1 + Ret+1 )/2 + L t,2 + Ht,2.= (Rt,1 + Ret+1 )/2 + Wt,2
The wedge between the long term yield and the average of the current and expected short
term rates now becomes a bit more complicated. This hypothesis is called the Preferred
Habitat Hypothesis (PHH). This hypothesis does not help us explain the two empirical
regularities of the term structure. In particular, to explain the typically upward sloping yield
curve we must have the increasing risk premiums implied by the LPH. It does help us
understand why the wedge, Wt,2, is complicated and variable.