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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 semiannually, we will assume annual paymentswe have already seen how to adjust for semiannual 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 5
100 100 100 100 100
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. Lets 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 Years Interest/Current Price
Total Yield = 9.26% + <1.26%>
= 8.00%
Note that the premium for the fouryear 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 fiveyear bond sells at a premium of $1,079.87, what do you think the premium for a tenyear bond will be? (Recall that the premium is the present value of the additional amount of interest being paid.)
A tenyear 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 2 3 4 5 6 7 8 9 10
100 100 100 100 100 100 100 100 100 100
PVIFA 8%,10 = 6.7101 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 tenyear bond relative to a fiveyear bond.
Sensitivity to Changes in Interest Rates
As we determined previously, as interest rates fall, bond prices rise. Which type of bond rises more, shortterm or longterm 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%. Lets see what happens to the values of our fiveyear and tenyear 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 fiveyear 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 tenyear bonds increase in price is calculated in the following manner:
0 1 2 3 4 5 6 7 8 9 10
100 100 100 100 100 100 100 100 100 100
PVIFA 6%,10 = 7.3601 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 longterm bond is due to the fact that the longer period of time for compounding has a more pronounced effect on the tenyear bond than it does on a fiveyear bond since, on average, the fiveyear bond is generating cash flows much sooner than the tenyear bond. If longterm bonds are more sensitive to changes in interest rates than shortterm 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 longterm bonds are more sensitive to changes in interest rates than shortterm 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?
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