Proving Prospective And Retrospective Book Values Agree

Why the Two Book-Value Methods Agree

A bond’s book value at a coupon date can be calculated in either of two ways. The prospective method values the cash flows still due, using the yield at which the bond was originally priced. The retrospective method accumulates the original purchase price to that date and subtracts the coupons already paid, accumulated to the same date. These descriptions look different, but they are two rearrangements of the same original price equation.

The agreement is useful as both a calculation method and a check. It also depends on a precise timing convention: at a coupon date, the formulas below measure the value just after that coupon is paid, with redemption at maturity still due. At the final coupon date, therefore, they give the redemption amount, not zero.

Definition: Consider a bond with $n$ coupon periods, level coupon $C$ paid at the end of each period, redemption amount $R$ paid at the end of period $n$, and purchase price $P$. Let $j$ be the effective yield per coupon period used to price the bond, and let $k$ be a coupon date, where $0\leq k\leq n$. Under the convention used here, the book value at date $k$ is measured after the coupon at that date has been paid, but before the redemption payment at date $n$ is made. The prospective and retrospective values are $$ B_k^{\mathrm{pros}}=C a_{\overline{n-k}|j}+R(1+j)^{-(n-k)} \qquad\text{and}\qquad B_k^{\mathrm{retro}}=P(1+j)^k-C s_{\overline{k}|j}. $$

Here $a_{\overline{m}|j}$ is the present value of $m$ unit payments at the ends of the next $m$ periods, and $s_{\overline{k}|j}$ is the accumulated value at date $k$ of $k$ unit payments made at the ends of periods $1,\ldots,k$. The yield $j$ is per coupon period: if coupons are semiannual, for example, $j$ is the yield for one half-year, not automatically the annual effective yield.

The model assumes level coupons at regular intervals, a fixed redemption amount at maturity, and a constant yield per period. The bond is priced from these same promised cash flows at that yield, so the initial price equation is

P=C a_{\overline{n}|j}+R(1+j)^{-n}.

For a positive yield, the standard accumulated-value expression is $s_{\overline{k}|j}=((1+j)^k-1)/j$. At $j=0$, the sum definition applies directly, so $s_{\overline{k}|0}=k$; the proof below works in either case. The formulas do not describe a value based on a new market yield: changing the valuation yield would change the prospective value and would no longer be the same original-yield book-value calculation.

Derivation From the Original Price Equation

The key is to accumulate every term in the original price equation to date $k$. Coupon payments made by date $k$ become accumulated coupon amounts; the remaining coupons and the redemption payment become their values at date $k$. This separates the accumulated past payments from the remaining bond value.

Start with the original price equation.
$P=C\sum_{t=1}^{n}(1+j)^{-t}+R(1+j)^{-n}$. The coupon sum is the annuity-immediate value.

Accumulate both sides to date $k$.
Multiply each side by $(1+j)^k$. Each coupon at period $t$ is then weighted by $(1+j)^{k-t}$.

Separate coupons already paid from later coupons.
For $t=1,\ldots,k$, the accumulated coupon factors sum to $1+(1+j)+\cdots+(1+j)^{k-1}=s_{\overline{k}|j}$. The terms for $t=k+1,\ldots,n$ are the present-value factors for the remaining coupons at date $k$.

Isolate the remaining bond value.
Subtract the accumulated value of the coupons already paid. What remains is exactly the prospective value of the coupons still due and the redemption amount.

In equations, the accumulated original price is

P(1+j)^k =C\sum_{t=1}^{k}(1+j)^{k-t} +C\sum_{t=k+1}^{n}(1+j)^{k-t} +R(1+j)^{k-n}.

The first sum on the right is $C s_{\overline{k}|j}$. In the second sum, write $u=t-k$, so $u$ runs from 1 to $n-k$. Its factors become $(1+j)^{-u}$, giving $C a_{\overline{n-k}|j}$. The final term is $R(1+j)^{-(n-k)}$. Thus

P(1+j)^k =C s_{\overline{k}|j} +C a_{\overline{n-k}|j} +R(1+j)^{-(n-k)}.

Subtract $C s_{\overline{k}|j}$ from both sides. The result is

P(1+j)^k-C s_{\overline{k}|j} =C a_{\overline{n-k}|j}+R(1+j)^{-(n-k)} =B_k^{\mathrm{retro}}=B_k^{\mathrm{pros}}.

This proves the agreement: the accumulated price, less the accumulated coupons already paid, equals the value at date $k$ of the remaining promised cash flows. The result applies at every coupon date in the stated model, including $k=0$ and $k=n$.

Worked Calculations

Worked Example: Checking Both Methods at an Intermediate Date

A three-period bond pays a coupon of $100 at the end of each period and $1,000 redemption at the end of period 3. Its original yield is 10% per period. Find the book value at date 2, after the second coupon has been paid, using both methods. Verify the original price used in the retrospective calculation.

Method and assumptions. Use $j=0.10$, $n=3$, $C=100$, and $R=1{,}000$, with level end-of-period payments. Since the coupon equals $jR$, the original price is par. Indeed,

P=100a_{\overline{3}|0.10}+1{,}000(1.10)^{-3} =1{,}000\bigl(1-(1.10)^{-3}\bigr)+1{,}000(1.10)^{-3} =1{,}000.

Prospective method. At date 2, the third coupon and redemption are due one period later. Their value at date 2 is

B_2^{\mathrm{pros}} =100a_{\overline{1}|0.10}+1{,}000(1.10)^{-1} =\frac{100}{1.10}+\frac{1{,}000}{1.10} =\frac{1{,}100}{1.10} =1{,}000.

Retrospective method. The original price accumulates for two periods to $1{,}000(1.10)^2=1{,}210$. The first two coupons accumulate to date 2 to $100s_{\overline{2}|0.10}=100(1+1.10)=210$. Therefore,

B_2^{\mathrm{retro}}=1{,}210-210=1{,}000.

Both methods give $1,000. The method avoids the common mistake of subtracting only the coupons’ unaccumulated total, $200, in the retrospective calculation; the first coupon must accumulate one period to date 2.

Worked Example: A Premium Bond With a Nonpar Coupon

A bond has three annual periods, an annual coupon of $120, redemption of $1,000, and an effective yield of 10% per year. Find its book value just after the first coupon using both formulas. Round money to the nearest cent.

Method. The yield is per annual coupon period, so $j=0.10$, $n=3$, $C=120$, $R=1{,}000$, and $k=1$. First calculate the original price from the promised cash flows:

a_{\overline{3}|0.10} =\frac{1-(1.10)^{-3}}{0.10} =2.486852, \qquad (1.10)^{-3}=0.751315,

$$ P=120(2.486852)+1{,}000(0.751315) =298.422240+751.315000 =1{,}049.737240. $$

The small difference between the displayed intermediate price and a calculation using unrounded factors is less than one cent in the final book value; carry unrounded factors through the calculation and round only the final result.

Prospective method. At date 1, two coupons and redemption remain. The two-period factors are $a_{\overline{2}|0.10}=1.735537$ and $(1.10)^{-2}=0.826446$. Thus

B_1^{\mathrm{pros}} =120(1.735537)+1{,}000(0.826446) =208.264440+826.446000 \approx 1{,}034.71.

Retrospective method. Accumulate the original price one period and subtract the first coupon paid at date 1:

B_1^{\mathrm{retro}} =P(1.10)-120 \approx 1{,}049.737240(1.10)-120 =1{,}154.710964-120 \approx 1{,}034.71.

The values agree to the nearest cent. The bond was purchased at a premium because its coupon exceeds the yield times redemption; that does not alter the proof. The relevant check is that the same original yield and cash flows are used in both methods.

Worked Example: Interpreting the Value at Maturity

A bond pays $80 per period for three periods and redeems for $1,100 at the end of period 3. Its yield is 10% per period. Calculate the book value at date 2 and at date 3. At date 3, report both the value under the convention in this tutorial and the value after the redemption payment has also been made. Round money to the nearest cent.

Method. Use $C=80$, $R=1{,}100$, $j=0.10$, and $n=3$. The original price is

P=80a_{\overline{3}|0.10}+1{,}100(1.10)^{-3} =80(2.486851991)+1{,}100(0.751314801) \approx 1{,}025.394440.

At date 2. One coupon and redemption remain, both due at date 3. The prospective value is

B_2^{\mathrm{pros}} =\frac{80+1{,}100}{1.10} =\frac{1{,}180}{1.10} \approx 1{,}072.73.

For the retrospective value, the original price accumulates for two periods. The two coupons already paid accumulate to date 2 to $80s_{\overline{2}|0.10}=80(2.10)=168$. Therefore,

B_2^{\mathrm{retro}} \approx 1{,}025.394440(1.21)-168 \approx 1{,}240.727272-168 \approx 1{,}072.73.

At date 3 under the stated convention. The final coupon has been paid, but redemption remains due at that date. There are no future coupons, and the redemption amount is not discounted. Hence

B_3^{\mathrm{pros}}=80a_{\overline{0}|0.10}+1{,}100(1.10)^0=1{,}100.

The retrospective calculation gives the same result. Since $s_{\overline{3}|0.10}=1+1.10+1.21=3.31$,

B_3^{\mathrm{retro}} \approx 1{,}025.394440(1.331)-80(3.31) \approx 1{,}364.800000-264.80 =1{,}100.

After the redemption payment is also made, the value is zero. In a retrospective calculation measured after both co-timed payments, subtract the accumulated redemption as well; at date 3 this gives $1{,}100-1{,}100=0$. The methods agree when they use the same instant and payment convention.

Common Timing Errors and Why the Result Matters

The most frequent error is to apply the formula at maturity and report zero because the bond is “finished.” That answer describes the instant after redemption has been paid, not the convention in the displayed formulas. Those formulas include redemption at date $n$, so at $k=n$,

B_n^{\mathrm{pros}}=C a_{\overline{0}|j}+R=R, \qquad B_n^{\mathrm{retro}}=P(1+j)^n-C s_{\overline{n}|j}=R.

The second equality follows directly from the original price equation accumulated to maturity: accumulated price equals accumulated coupons plus $R$. Subtracting the accumulated coupons leaves $R$. To measure value after redemption, subtract that payment too; then both methods give zero.

A second error is subtracting the coupons already paid without accumulating them. In the retrospective method, a coupon paid at period $t$ accumulates for $k-t$ periods to date $k$. The full amount to subtract is $C s_{\overline{k}|j}$, not $kC$. Conversely, the prospective method discounts only the remaining cash flows back to date $k$, not all cash flows back to the original purchase date.

This identity matters because it lets you choose the calculation that best fits the information supplied. If the original price and coupons paid are known, the retrospective calculation may be direct; if the remaining cash flows are easier to value, the prospective calculation may be simpler. Either way, agreement is a useful check on timing, yield-period units, and whether redemption has been included consistently. The result is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated value convention: immediately after the coupon at date $k$, with redemption still due at maturity. Round monetary answers to the nearest cent.

A two-period bond pays $60 per period and redeems for $1,000. Its yield is 5% per period. Find its book value at date 1 using the prospective method. (Answer: $1,009.52)
A three-period bond pays $70 per period and redeems for $900. Its yield is 8% per period. Find its book value at date 2. (Answer: $898.15)
A four-period bond pays $45 per period and redeems for $1,000. Its yield is 6% per period. What is its book value at date 4 under the stated convention, and what is its value after redemption is paid? (Answer: $1,000.00 before redemption; $0.00 after redemption)
A three-period bond pays $40 per period and redeems for $950. Its yield is 4% per period. Find its book value at date 1 using the prospective method. (Answer: $953.77)

What You'll Learn