BAC 1644
Ch 6
3.
The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this
problem assumes an annual coupon. The price of the bond will be:
P = $70({1 – [1/(1 + .10)]
8
} / .10) + $1,000[1 / (1 + .10)
8
]
P = $839.95
We would like to introduce shorthand notation here. Rather than write (or type, as the case may be)
the entire equation for the PV of a lump sum, or the PVA equation, it is common to abbreviate the
equations as:
PVIF
R,t
= 1 / (1 +
R
)
t
which stands for P
resent V
alue I
nterest F
actor
PVIFA
R,t
= ({1 – [1/(1 +
R)
]
t
} /
R
)
which stands for P
resent V
alue I
nterest F
actor of an A
nnuity
These abbreviations are shorthand notation for the equations in which the interest rate and the
number of periods are substituted into the equation and solved. We will use this shorthand notation
in the remainder of the solutions key. The bond price equation for this problem would be:
P = $70(PVIFA
10%,8
) + $1,000(PVIF
10%,8
)
P = $839.95
4.
Here, we need to find the YTM of a bond. The equation for the bond price is:
P = $1,145.70 = $100(PVIFA
R%
,9
) + $1,000(PVIF
R
%,9
)
Notice the equation cannot be solved directly for
R
. Using a spreadsheet, a financial calculator, or
trial and error, we find:
R
= YTM = 7.70%
If you are using trial and error to find the YTM of the bond, you might be wondering how to pick an
interest rate to start the process. First, we know the YTM has to be lower than the coupon rate since
the bond is a premium bond. That still leaves a lot of interest rates to check. One way to get a
starting point is to use the following equation, which will give you an approximation of the YTM:
Approximate YTM = [Annual interest payment + (Par value – Price) / Years to maturity] /
[(Price + Par value) / 2]
Solving for this problem, we get:
Approximate YTM = [$100 + (–$145.70 / 9)] / [($1,145.70 + 1,000) / 2]
Approximate YTM = .0781 or 7.81%
This is not the exact YTM, but it is close, and it will give you a place to start.