The Relationship Between Interest Rates and Bond
Prices - An Example Using Coupon Bonds
Beginning with the formula Dr. Cogley presented in class, we have
P
c
=
F
2F
1 − (1 + isa )−2m
+ (1 + isa )−2m
isa
which is just a modified version of the equation for the present value of a coupon bond that
we’ve seen before (for proof, see the Appendix). With a little algebra, we get
P
1− =
F
c
2F
isa
− 1 (1 + isa )−2m − 1
(1)
Notice that the far right hand side (RHS) portion is always less than zero
(1 + isa )−2m − 1 < 0
so the sign of the entire RHS will be determined by the sign of
c
2F
isa
− 1 . With this in mind,
we can now talk about what it means if a bond sells at par, a discount, or a premium.
Par
When a coupon bond sells at par, the price is equal to the face value (P = F ). The left
hand side (LHS) of (1) is then equal to zero, implying that
c
= isa
2F
(2)
meaning that the coupon yield is equal to the yield to maturity.
Premium
Selling at a premium means the price is greater than the face value (P > F ). The LHS of
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2006
(1) is then less than zero, meaning the RHS must be less than zero, which in turn implies
c
> isa
2F
(3)
so the coupon yield is greater than the yield to maturity.
Discount
On the other side of things, when a coupon bond sells at at discount, price is lower than
face value (P < F ). The LHS of (1) is then greater than zero, implying that
c
< isa
2F
(4)
and the coupon yield is less than the yield to maturity.
So what?
What does all this really mean? The coupon rate is the value of annual payments as
a percentage of the amount borrowed (or the face value). Since it isn’t a function of the
interest rate, and coupon payments and face value are determined at the time of initial
purchase, the coupon rate won’t change over time (this makes it a good point of comparison
when considering future changes). The yield to maturity is the yield bondholders get if they
hold the bond until maturity (the time of the final payment), and it does change over time.
Whenever the yield to maturity changes, it must be that the price of the bond is changing,
too.
This gives us some insight into how interest rates and bond prices are related. Say you
buy a multi-period bond and try and sell it one period later, but the market interest rate
has gone down. The yield to maturity on your bond will drop, and by equation (3), the price
of your bond will go up. Why?
The face value of the bond hasn’t changed, but the market interest rate is lower. In order
to still reach the face value by maturity under the new lower yield to maturity (which drops
2
to match the market interest rate), we have to have a higher ”starting point”, i.e. price in
the new period. Here’s another way to look at it - consider the equations
P1 = B(1 + i1 )−n =
B
(1 + i1 )n
P2 = B(1 + i2 )−n =
B
(1 + i2 )n
If B and n are the same in both equations, and i1 < i2 then in must be that P1 > P2 . Similar
logic applies in our analysis . . . if the market yield to maturity is higher than the initial yield
to maturity, but all else is the same, the new price must be lower.1 We can follow the logic in
the other direction, too. For example, if the market interest rate is higher than your original
bond yield to maturity, you have to adjust the price (lower it) until the yield to maturity
makes the buyer indifferent between buying your investment and a new one at the current
rate. What’s the moral of the story? Yield to maturity and price respond to interest rate
changes in opposite directions - yield to maturity moves with the interest rate, price moves
against it.
Appendix
Beginning with
P
c
=
F
2F
1 − (1 + isa )−2m
+ (1 + isa )−2m
isa
we multiply both sides by F
c 1 − (1 + isa )−2m
P =
+ F (1 + isa )−2m
2
isa
Pulling i out from inside the brackets
P =
c
2
isa
1 − (1 + isa )−2m + F (1 + isa )−2m
1
Similarly, by equation (4), if the market interest rate (and thus yield to maturity) goes up, the price of
the bond must be going down.
3
where the negative exponent on (1 + isa ) can be written as one over a positive exponent
P =
c
2
isa
1−
1
+ F (1 + isa )−2m
2m
(1 + isa )
(A-1)
Equation (A-1) is the common form we’ve used for the present value of a coupon bond.
The first part is the present value of the mortgage stream resulting from the coupon payments, while the second part is the face value (final payment), with the present value calculated as with a discount bond.
4