Prospective Book Value From Remaining Bond Cash Flows

Why Value the Cash Flows Still Due?

A bond’s book value on a coupon date can be found by asking what its remaining contractual payments are worth at the bond’s original yield. This is a prospective calculation: list the coupons and redemption still due after the valuation date, then discount each amount back to that date. The coupon just paid is not part of the remaining cash flows.

The essential rate is the yield per payment period. If coupons are semiannual, for example, a nominal annual yield convertible semiannually must be divided by two before it is used to discount semiannual cash flows. Applying the annual rate to each half-year period changes the valuation model.

Definition: Under the fixed-cash-flow bond model, the prospective book value on a coupon date is the present value, at the bond’s original yield per payment period, of all coupons and the redemption payment due after that date. Assume the bond makes its stated payments in full, the yield is used as a constant discount rate, and the valuation date is a coupon date. A coupon already paid on that date is excluded.

The Prospective Valuation Formula

Suppose the bond has level coupon $C$ per payment period, $N$ coupon periods in total, and redemption amount $R$. Let $k$ be the number of coupons already paid, and let $j$ be the original yield per coupon period. Write $B_k$ for the book value immediately after coupon $k$. There are $N-k$ periods remaining: a coupon is due at the end of each of those periods, and $R$ is paid with the final coupon.

B_k=\sum_{t=1}^{N-k} C(1+j)^{-t}+R(1+j)^{-(N-k)}

The coupon sum can be written with the ordinary annuity-immediate notation, using $j$ as the interest rate per period:

B_k=C\,a_{\overline{N-k}|}+R(1+j)^{-(N-k)}

Here $a_{\overline{n}|}$ is the present value of $n$ unit payments, one at the end of each period, at rate $j$. Equivalently, the formula can be used as a direct sum of discounted cash flows; that form is especially helpful when payment amounts vary.

Rate convention: If the original yield is a nominal annual rate $i^{(m)}$, convertible $m$ times per year, and the bond pays coupons $m$ times per year, then the yield per coupon period is $j=i^{(m)}/m$. If instead the yield is an effective annual rate $i$, the equivalent rate per period is $j=(1+i)^{1/m}-1$. Match the period used for discounting to the timing of the cash flows.

Why the Formula Holds

At the valuation date, the first remaining coupon is one period away, so its present value is $C(1+j)^{-1}$. The next coupon is two periods away and has present value $C(1+j)^{-2}$. Continuing this reasoning, the last coupon is discounted for $N-k$ periods. Redemption is paid at that same final date, so its present value is $R(1+j)^{-(N-k)}$. Adding these present values gives the prospective value.

The coupon part is a geometric series. Setting $n=N-k$, its value is

C\sum_{t=1}^{n}(1+j)^{-t} =C\frac{1-(1+j)^{-n}}{j} =C\,a_{\overline{n}|}, \qquad j\ne 0.

The first equality lists each coupon’s discounted value. The second follows from the finite geometric-series formula, and the last is the definition of $a_{\overline{n}|}$. When $j=0$, no discounting applies, so the coupon sum is simply $nC$; the expression with division by $j$ is not used.

Worked Example: A Semiannual Bond After Its Fifth Coupon

A hypothetical six-year bond has face amount $1,000 and pays coupons at a nominal annual coupon rate of 5%, convertible semiannually. Its original yield is 6% nominal, convertible semiannually. Find its book value immediately after the fifth coupon. Assume all promised payments are made.

Method. Count the remaining half-year periods, convert the quoted yield to a per-period rate, and discount the remaining coupons and redemption. This avoids treating a nominal annual yield as though it applied separately to every half-year.

There are $6\times2=12$ coupon periods in the bond’s term. The semiannual coupon is

C=1{,}000\left(\frac{0.05}{2}\right)=\$25.

The yield per half-year is $j=0.06/2=0.03$. After five coupons, $12-5=7$ periods remain. Therefore,

B_5=25a_{\overline{7}|}+1{,}000(1.03)^{-7}.

Calculate the discount factor and annuity factor:

(1.03)^7=1.229873865,\qquad (1.03)^{-7}\approx0.81309148,

a_{\overline{7}|}=\frac{1-0.81309148}{0.03}\approx6.230284.

The coupon value is $25(6.230284)\approx\$155.7571$, and the redemption value is $1{,}000(0.81309148)\approx\$813.0915$. Adding them gives $B_5\approx\$968.8486$, or $968.85 to the nearest cent. As a check, there are seven coupons left, so the coupon component is positive and redemption is discounted for exactly seven periods.

Worked Example: Counting Remaining Semiannual Payments

A hypothetical ten-year bond has face amount $1,000, pays $25 every six months, and has an original yield of 6% nominal, convertible semiannually. Find its book value immediately after the fourth semiannual coupon. Assume the bond pays all coupons and redemption as promised.

Method. Count from the valuation date, not from the bond’s issue date: ten years contain $10\times2=20$ semiannual periods, so $20-4=16$ remain. The yield per period is $j=0.06/2=0.03$. The just-paid fourth coupon is excluded; the next payment is one half-year away.

B_4=25a_{\overline{16}|}+1{,}000(1.03)^{-16}.

The discount factor is

(1.03)^{16}=1.604706438,\qquad (1.03)^{-16}\approx0.62316694.

Thus,

a_{\overline{16}|}=\frac{1-0.62316694}{0.03}\approx12.561102.

The remaining coupons have present value $25(12.561102)\approx\$314.0276$. Redemption has present value $1{,}000(0.62316694)\approx\$623.1669$. Their sum is approximately $\$937.1945$, so the book value is $937.19 to the nearest cent. The final payment includes both a $25 coupon and the $1,000 redemption; both are included in the formula.

Worked Example: Redemption Differs from Face Amount

A hypothetical seven-year bond pays an annual coupon of $80 and redeems for $1,050. Its original yield is 10% effective annually. Find its book value immediately after its fourth annual coupon, assuming all promised payments are made.

Method. Use the stated redemption amount, not the face amount, in the final cash flow. After four of seven coupons, three annual periods remain. Since the yield is already effective annually and coupons are annual, $j=0.10$.

B_4=80a_{\overline{3}|}+1{,}050(1.10)^{-3}.

The discount factor is $(1.10)^{-3}=1/1.331\approx0.751314801$. Hence,

a_{\overline{3}|}=\frac{1-0.751314801}{0.10}\approx2.48685199.

The three remaining coupons are worth $80(2.48685199)\approx\$198.9482$. The redemption payment is worth $1{,}050(0.751314801)\approx\$788.8805$. Therefore, $B_4\approx\$987.8287$, or $987.83 to the nearest cent. The calculation includes three coupons, including the one paid with redemption, and uses $1,050 as the final redemption cash flow.

Book Value Is Not Necessarily the Market Price

The prospective formula uses the bond’s original yield. It does not automatically give the amount for which the bond could be sold on the valuation date. A market price at that date generally reflects the yield investors currently require, which may differ from the original yield. Book value is the value obtained under the stated original-yield convention; the two amounts coincide only when the relevant yields and valuation conventions make them coincide.

This distinction matters in exam problems because a question may give both an original yield and a current market yield. For prospective book value, use the original yield. Do not silently substitute a current market yield unless the problem explicitly asks for a market valuation at that yield.

Common Rate and Timing Traps

Using an annual rate for each subannual period. If a nominal annual yield is convertible semiannually, divide it by two to obtain the half-year rate. Do not use the full annual rate in each half-year discount factor.
Counting from issue rather than from the valuation date. At coupon $k$ of a bond with $N$ total coupons, the number remaining just after that coupon is $N-k$.
Including a coupon already paid. Immediately after a coupon date, that coupon is no longer a future cash flow. The next coupon is one period away.
Using face amount instead of redemption amount. The final payment is the redemption amount plus the final coupon. If redemption differs from face amount, use the stated redemption amount.
Confusing a book value with a sale price. The prospective book value is based on the original yield; a market valuation may use a different yield.

For a numerical illustration of the first trap, suppose a bond pays $30 semiannually, has $1,000 redemption, has six half-year periods remaining, and has an original yield of 8% nominal convertible semiannually. The correct period rate is $0.08/2=0.04$, giving

30a_{\overline{6}|}+1{,}000(1.04)^{-6} \approx30(5.24213685)+1{,}000(0.79031453) \approx\$947.58.

Wrong calculation: using 8% in every half-year period gives $30a_{\overline{6}|}$ at 8% plus $1{,}000(1.08)^{-6}$, which is approximately $768.86. That arithmetic is consistent with an 8% half-year yield, but not with an 8% nominal annual yield convertible semiannually. The correction is to use 4% per half-year, producing the $947.58 value above.

Prospective coupon-date valuation is aligned with the SOA Exam FM (Financial Mathematics) syllabus. In any problem, establish the coupon frequency, the yield convention, the valuation date, the remaining payment count, and the redemption amount before substituting into the formula.

Check Your Understanding

For each problem, assume the bond makes all stated payments. Values are book values immediately after the specified coupon unless stated otherwise.

A bond pays annual coupons of $45, redeems for $1,000, and has an original effective annual yield of 5%. Find its value immediately after the second of five annual coupons. (Answer: $986.38)
A three-year bond has face amount $1,200, pays coupons quarterly at a nominal annual coupon rate of 6% convertible quarterly, and has an original yield of 8% nominal convertible quarterly. Find its value just after the fifth coupon. (Answer: $1,161.17)
A zero-coupon bond will pay $2,000 in six years. Its original effective annual yield is 4%. Find its prospective value at the two-year point, a coupon date for this zero-coupon bond. (Answer: $1,709.61)
A bond pays annual coupons of $80, redeems for $1,050, and has an original effective annual yield of 10%. Find its value just after the fourth coupon of a seven-year term. (Answer: $987.83)
A bond pays $30 semiannually, redeems for $1,000, and has six half-year periods remaining. Its original yield is 8% nominal convertible semiannually. Find its prospective value. (Answer: $947.58)

What You'll Learn