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 P= F + F( r - j ) a resulting yield rate may be determined by solving for j in n 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 It PRt OBt 0 102.66 1 4 3.59 0.41 102.25 2 4 3.58 0.42 101.83 3 4 3.56 0.44 101.39 4 4 3.55 0.45 100.94 5 4 3.53 0.47 100.47 6 104 3.52 100.48 -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 P= F + F( r - j ) a bond is bought at premium, r > j and so r – j > 0. Hence, n 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.