BOND VALUATION
The financial value of any asset, be it a security, real estate, business, etc., is the
present value of all future cash flows. The easiest thing to value (conceptually) is a bond since
the promised cash flows are known with certainty.
Consider a bond that pays a 10% coupon (or stated) rate of interest, has a par (or
stated) face value of $1,000 and matures in 5 years. Suppose also that the market rate of
interest for such a bond (i.e., your required rate of return, k) is 8%. Thus,
Par = $1,000
Coupon Rate = 10%
Maturity = 5 years
K = 8%
The cash flows that are promised by the company include interest payments of $100 per
year (although most corporate bonds pay interest semi-annually, we will assume annual
payments—we have already seen how to adjust for semi-annual cash flows) for five years and
the payment of the face value (stated, or par, value) of $1,000 at the end of five years.
0
1
2
3
4
100
100
100
100
5
100
1,000
1,100
PVIFA 8%,4 = 3.3121
331.21
PVIF 8%,5 = .6806
748.66
$1,079.87
The value of the bond is $1,079.87 which is selling at a premium relative to the par value
of $1,000. (A bond selling at less than par is said to be selling at a discount.)
What does the premium represent? As we saw when we looked at present values, it
represents the present value of the additional interest of $20 per year (because it pays $100 in
interest when we only require $80 for a $1,000 investment ($20 * 3.9927 = $79.85 with two
cents rounding error). Any time the market rate of interest is less than the coupon rate of
interest, the bond will sell at a premium. Similarly, when market rates of interest are greater
than the coupon rate, the bond will sell at a discount. Recall from economics that, when interest
rates go up, bond prices go down, and when interest rates go down, bond prices go up. This is
a consequence of the mathematics of present value calculations.
Suppose we purchase the bond for $1,079.87. After one year, we collect $100 in
interest. The $100 represents a 9.26% return on our investment of $1,079.87, not an 8% rate of
return. What are we ignoring?
The 9.26% is referred to as the current yield (as in accounting, where “current” refers to
within one year). What is being ignored is the fact that we paid a premium for the bond which,
at maturity will be worth only $1,000. Thus, over the five years to maturity, the value of the bond
will decrease. Let’s look at what the bond will be worth one year from now. In one year, there
will only be four years left to maturity:
0
1
2
3
4
100
100
100
100
1,000
PVIFA 8%,4 = 3.3121
331.21
PVIF 8%,4 = .7350
735.00
$1,066.21
Note that this time, the interest payment in the last year was included as a part of the
present value of an annuity calculation while the par value was discounted as a lump sum of
$1,000. As indicated, the value of the bond when only four years to maturity remain is only
$1,066.21. This is a decrease in value of $13.66. When expressed as a percentage of the
original value of $1079.87, this represents a loss of 1.26%. The total return of 8% that we built
into our valuation when the bond had five years left to maturity is comprised of two components:
Total Yield = Current Yield + Capital Gain Yield
Current Yield = One Year’s Interest/Current Price
Total Yield = 9.26% + <1.26%>
= 8.00%
Note that the premium for the four-year bond is smaller than the premium for the fiveyear bond since we are only paying for four years’ worth of additional interest payments.
Bond Maturities & Premiums/Discounts
If a five-year bond sells at a premium of $1,079.87, what do you think the premium for a
ten-year bond will be? (Recall that the premium is the present value of the additional amount of
interest being paid.)
A ten-year 10%, $1,000 par value bond should sell at a larger premium since we are
paying for ten years’ worth of an extra $20 per year of interest. For example,
Par = $1,000
Coupon Rate = 10%
Maturity = 10 years
K = 8%
0
1
100
2
3
4
5
6
7
8
9
100
100
100
100
100
100
100
100
PVIFA 8%,10 = 6.7101
10
100
1,000
671.01
PVIF 8%,10 = .4632
$ 463.20
$1,134.21
As was expected, the additional five years’ worth of an extra $20 per year in interest
payments results in a larger premium for a ten-year bond relative to a five-year bond.
Sensitivity to Changes in Interest Rates
As we determined previously, as interest rates fall, bond prices rise. Which type of bond
rises more, short-term or long-term bonds? (Hint: Do we really care what interest rates do
today for a bond that matures tomorrow?)
Suppose that interest rates fall from 8% to 6%. Let’s see what happens to the values of
our five-year and ten-year bond prices.
0
1
2
3
4
5
100
100
100
100
100
1,000
PVIFA 6%,5 = 4.2124
421.24
PVIF 6%,5 = .7473
747.30
$1,168.54
The value of the five-year bond has increased from $1,079.87 to $1,168.54 or $88.67
due to the fall in market rates of interest from 8% to 6%. The $88.67 increase in price
represents an 8.2% appreciation relative to its original value.
The ten-year bond’s increase in price is calculated in the following manner:
0
1
100
2
3
4
5
6
7
8
9
100
100
100
100
100
100
100
100
PVIFA 6%,10 = 7.3601
10
100
1,000
736.01
PVIF 6%,10 = .5584
$ 558.40
$1,294.41
The increase in price for the ten year bond amounts to $160.20 or 14.1%. Why do we
calculate the change in price as a percent of its original value?
The reason the change in price is much larger for a long-term bond is due to the fact that
the longer period of time for compounding has a more pronounced effect on the ten-year bond
than it does on a five-year bond since, on average, the five-year bond is generating cash flows
much sooner than the ten-year bond. If long-term bonds are more sensitive to changes in
interest rates than short-term bonds, can you guess whether a high coupon bond or a low
coupon bond is more sensitive to changes in interest rates? (See Handout #2.)
The equation for the value of a bond can be written as follows:
N Interest
Par
Bond Value = 
+
t=1 (1+k)t
(1+k)N
= Interest (PVIFA) + Par (PVIF)
= Present Value of the Interest Payments + Present Value of the Par
Perpetuities
There is a type of bond that never matures called a perpetuity, or a consol. (The term
“consol” comes from the fact that the first perpetuities were issued by the British government
following the Napoleonic Wars to “consolidate” their war debts.) Canada issued some
perpetuities in the late 1970s. If long-term bonds are more sensitive to changes in interest rates
than short-term bonds, what type of bond is the most sensitive to interest rate changes? (A
consol, of course.)
When N is infinity, the value of a perpetual bond reduces to
Interest
Value of a perpetuity =
K
While there are not a lot of perpetuities that trade in the marketplace, there is a financial
security which is, essentially, a perpetuity. Do you know what security pays a constant dollar
amount each year and never matures?
Preferred Stock
The classic version of preferred stock is a share that pays a fixed dollar amount of
dividend and never matures. It is, therefore, a perpetuity. The formula for the value of a share
of preferred stock is
Dividend
Value of Preferred Stock =
Kp
So if you expect that interest rates are going to decrease in the future, what type of bond
would you want to buy?
If you expect that interest rates are going to rise in the future, what type of bond do you
want to buy?