Book Value of a Bond
For accounting purposes, it is necessary to assign a value to any bonds you own. This
value, called the book value of the bond, is the current price for the remaining coupons
and redemption value based on the original yield rate at which the bond was purchased.
If between coupon dates, the book value is the price and doesn’t include any value
accrued toward the next coupon. (see attached comparison of price-plus-accrued versus
the price (i.e., minus accrued) ).
It should be noted that although the bond price is related to the yield rate, as described
above, it is actually the case that the bond yield rate results from the bond price. That
is, after the buyer and seller of the bond reach an agreed upon purchase price P, the
resulting yield rate may be determined by solving for j in P  F  F (r  j )an j .
This requires using your calculator to solve for the unknown interest j (e.g., example 4.3,
page 237).
Amortization of a Bond
Viewing a bond and its periodic coupons similar to a loan with periodic loan payments,
we can construct a schedule where each coupon paid is consider to be a combination of
interest received and principal returned. Consider the purchase price of the bond to be
the loan amount and consider the yield rate of the bond to be the loan’s interest rate. The
outstanding balance on any coupon date is simply the book value of the loan at that time.
That is, the outstanding balance is the present value of the remaining coupons and
redemption value based on the original yield rate.
The size of the payment is the amount of the coupon Fr, except for the last payment of
Fr + F. Since each payment Kt = It + PRt, we may find the interest for each period either
as the difference It = Kt – PRt , or by computing interest on the outstanding balance,
It = j(OBt-1). The principal returned is simply the change in the outstanding balance,
PRt = OBt-1 – OBt , or equivalently PRt = Kt – It. Note that when a bond is purchased at
discount, the price at coupons dates is increasing and so the change in the outstanding
balance is negative and results in a negative value for PRt (e.g., example 4.4, page 241).
Consider a 8%, $100 bond with 6 remaining coupon dates. Given a yield rate of 7%, the
bond was purchased at premium. The amortization of the bond is illustrated below. For
each period, It = 0.035OBt-1 and PRt = Kt – It. Note the final payment Fr + F leaves an
outstanding balance of zero (except for rounding-error).
k Kt
0
1 4
2 4
3 4
4 4
5 4
6 104
It
3.59
3.58
3.56
3.55
3.53
3.52
PRt
OBt
102.66
0.41 102.25
0.42 101.83
0.44 101.39
0.45 100.94
0.47 100.47
100.48 -0.01
In the table above, the principal returned 0.41, 0.42, …, 0.47, 0.48 is the amortization of
the premium.
Callable Bonds
A callable bond is a bond that allows the bond issuer to choose a redemption date, or call
date, that may be prior to the maturity date. This call date must fall within a specified
range of dates and is generally several years after the issue date.
Recall the price and yield of the bond is based on the term of the bond, so the yield can’t
be uniquely determined without knowing the redemption date. The investor should
consider the worst case and determine which potential call date is the most unfavorable to
the investor. That is, determine the date which corresponds to the minimum purchase
price for a given yield rate j. If the investor pays more than this minimum amount and if
the selected call date does correspond to the smaller amount, then the investor has
“overpaid” and the resulting yield is reduced.
Determining the redemption date to be used for computing the price is easy assuming that
the redemption amount is fixed, regardless of the redemption date. For example, when the
bond is bought at premium, r > j and so r – j > 0. Hence, P  F  F (r  j )an j is at a
minimum when an j is small, or equivalently, when n is small as possible. That is,
determine the price based on the earliest potential call date. Conversely, when the bond
is bought at discount, r – j < 0 and so P is minimum when n is large. As a result, the
price should be determined based on the latest potential call date.
That is, we have a rule for determining the price when the redemption amount is fixed:
 when r > j (“bought at premium”), use price based on earliest potential call date.
 when r < j, (“bought at discount”), use price based on latest potential call date.
The situation where the redemption amount varies based on the call date takes more
effort to determine which call date corresponds to the minimum price for a given yield
rate. Example 4.6 (page 246) illustrates one such situation. In the example, there is a
redemption amount fixed for one range of possible call dates, but a different redemption
amount for another range of call dates. Within each range of call dates, the minimum
price is determined based on the rule given above. After computing the minimum price
for each range of call dates, these prices are compared to determine which date
corresponds to the overall minimum purchases price.
The text notes that callable bonds become less common as efforts are made to make the
purchase of bonds more attractive.