Supplemental Notes on Interest Rates, Yield Curve, and Bond Valuation
Techniques
Introduction
This brief tutorial will cover the expectation theory of interest rates; yield curves and
what they mean, and the method to value bonds in a market setting.
Interest Rates
One of the most powerful tools in understanding interest rates is the expectations theory
of interest rates. This theory can be summarized in the following equation:
k  k real  IP  DRP  MRP  LP (Equation 1)
Where:
k is the rate of interest
kreal is the real interest rate
IP is the inflation premium
DRP is the default risk premium
MRP is the maturity risk premium
LP is the liquidity premium
This means that the interest rate expected by investors on any given investment is
composed of the sum of five different expectations by investors. The real rate of interest
(kreal) is the interest an investor would expect to earn on a risk free investment after
inflation effects are taken out of the equation. The inflation premium (IP) is the average
inflation rate expected by investors over the term of the financial instrument being
considered. Default Risk Premium (DRP) is the premium investors demand to
compensate them for the risk that the firm will not pay back their principal or interest
promised on the loan. Maturity Risk Premium (MRP) represents a small premium
investors demand for investing in long term investments because of the risk of tying up
their money for a longer period of time. Finally, Liquidity Premium (LP) is the amount
investors demand due to the risk that they might not be able to sell their bond when they
wish to sell it.
So for a government bond, we can see that DRP = LP = 0 because the government can’t
default on its obligations (they would just print more money) and because the market for
government bonds is so large as to make any individual investor too small to matter in
terms of selling and buying any reasonable amount of financial instruments.
Expectations theory can yield valuable insights into investor thinking. For example, if
the interest rates on a 30-day government note increase by 5 basis points (100 basis
points is equal to 1% yield change), what does this mean about investor expectations?
One thing this could mean is that investors believe events will yield an inflation rate that
is 5/100% higher than it is today. If this is a longer-term trend, then the 30-year bond
will move much more than the 30-day note. This kind of thinking makes analysis of the
effect of interest rate moves across the maturity spectrum an important component of
financial analysis.
Yield Curves
A yield curve is the graph formed by plotting interest rate verses time to maturity of a
series of the same investment. Usually, government bonds are used, since they exist in a
wide range of maturities and are freely traded on the secondary markets. A positively
sloping yield curve is said to be normal (see Figure 1) because the shorter-term rates are
lower in general than the longer-term rates due to the higher risk of inflation and the
MRP inherent in longer maturities.
Figure 1. Normal Yield Curve
7
6
Yield (percent)
5
4
3
2
1
0
0
5
10
15
20
Maturity (years)
25
30
35
While rarely a smooth line, the general direction of the yield curve can tell us a lot about
what investors are thinking and what is happening in the economy. For example, if the
yield curve is normal, investors demand a higher return for longer-term debt, owing
mostly to the expectation of increased inflation (and therefore lower money value due to
the time value of money concept). When inflation expectations are high among bond
investors, it signals that bond investors believe the economy will grow rapidly in the near
to intermediate term. This is normally a favorable sign for stocks.
Figure 2 shows an inverted yield curve, one in which inflation is expected to be lower in
the future than the present. Lower expectations of inflation among bond investors is
usually a signal that the bond market thinks the economy will soon cool down, and grow
at a lower or negative rate.
Figure 2. Inverted Yield Curve
8
Yield (percent)
7
6
5
4
3
2
1
0
0
5
10
15
20
25
30
35
Maturity (years)
The inverted yield curve is a condition often seen when the Federal Reserve abruptly
raises short term interest rates to cool down the economy with the goal of lowering
inflation. The yield curve was most recently inverted at the peak of the 1999/2000 stock
market bubble. To see how the yield curve reacts over time, visit www.bigcharts.com on
the Internet.
Bond Valuation
The value of a bond depends on the relative interest rate between that in the market and
that of the bond. The interest rate carried by the bond is called the coupon rate. This is
the rate of interest that the bond issuer promises to pay the bondholder in return for the
use of his capital. The market rate is also called the yield to maturity (YTM). The yield
to maturity of a bond is the rate of interest that the bond would yield if it were sold today
by the same bond issuer. The coupon rate is fixed, but the YTM of a bond changes in
response to market conditions.
The value of a bond moves inversely to interest rates. To value a bond, use the following
formula:
Vb  COUPON ( PVIFA(n, YTM ))  PAR( PVIF (n, YTM )) (Equation 2)
Where:
Vb is the bond value (dollars)
COUPON is the coupon payment (equal to the coupon rate times the PAR value
of the bond)
PVIFA is the present value interest factor for an annuity
n is the number of years between the present and the maturity date of the bond
YTM is the market rate or Yield to Maturity
PAR is the face or par value of the bond
PVIF is the present value interest factor
To illustrate, suppose you bought a $1000 bond from your broker. The market rate of
interest for the bond is 6%, but this bond was issued 10 years ago and carries an 8%
coupon. The bond was a 30-year bond when issued. What is the fair market price you
should pay your broker?
The first step is to figure the coupon, which is the par value times the coupon rate. This
bond is an 8% coupon bond with a $1000 face value. This means the coupon is 1000 x
0.08 or $80.00.
Now look up the factors from the charts for PVIF and PVIFA
PVIF(20,6%)=0.3118
PVIFA(20,6%)=11.4699
Now you can value the bond:
Vb=80 x 11.4699 + 1000 x 0.3118
Vb=917.59+311.80
Vb=$1,229.39
You can see that a 2% decrease in the going rate for this bond yielded a 22.9% increase
in the value of the bond.
Discussion Questions
1. Given the inflationary expectation shown below, compute and graph a yield curve
for a series of government bonds if the real interest rate in the economy is 1.5%
and the MRP demanded by investors is 5 basis points per year of maturity.
Year
1
2
3
4
5
6 to 10
10 to 30
Inflationary
Expectation
5%
6%
10%
15%
8%
7% each
year
6.5% each
year
2. If the real rate of interest is 1.5% and the inflationary expectation changes from
3% to 10% for a ten-year period, what is the effect on the present value of a
$2,000,000 payment to be received 10 years from the present time?
3. What is the value of a $1000 par value bond with a 5% coupon if the going rate
for the bond in the market is now 7%?