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Exam FM · Premium, Discount, and Book Value · Tutorial 378 of 998

Semiannual Book Value With Periodic Yields

Calculate semiannual coupon-date book values with periodic yields, and translate consistently between half-year calculations and nominal annual rate notation.

Intermediate 10 min read

What You'll Learn

  • Identify the coupon amount and yield per half-year from annual bond terms.
  • Calculate coupon-date book value from the remaining bond cash flows.
  • Use the premium-or-discount form to check a prospective valuation.
  • Advance book value across coupon dates using the periodic-yield recurrence.
  • Reconcile two half-year steps with annual nominal-rate notation.
  • Recognize why applying an annual quoted yield to each half-year is incorrect.

Why the Half-Year Period Matters

A bond with semiannual coupons makes two coupon payments in each year. Its book value at a coupon date must therefore be calculated using the yield earned over a half-year and the amount of a half-year coupon. Using an annual yield for each coupon period, or using a full annual coupon at each date, changes the cash-flow valuation.

This tutorial considers a fixed-yield book value: the bond is valued at its original yield, and the coupon-date book value is measured immediately after that date's coupon has been paid. The calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Suppose a bond has redemption value \(R\), an annual coupon rate \(c\), and semiannual coupons. The coupon paid every six months is \(C=Rc/2\). If the yield is quoted as a nominal annual rate \(i^{(2)}\) convertible semiannually, the yield per half-year is \(j=i^{(2)}/2\). If \(n\) coupons remain immediately after a coupon date, the book value \(B\) is the present value at rate \(j\) of those \(n\) coupons and the redemption payment.

The nominal annual rate \(i^{(2)}\) is not itself the rate applied over one half-year. Each coupon period lasts half a year, so the periodic rate in this convention is \(j=i^{(2)}/2\). The formula \(C=Rc/2\) likewise converts the annual coupon amount into the amount actually paid at each semiannual date.

In this topic, “annual-rate notation” means writing the quoted yield as \(i^{(2)}\) and explicitly converting it to \(j\) for the half-year calculations. The quoted annual coupon rate \(c\) is also converted to the semiannual payment \(C\). These conversions keep the time unit of the rate consistent with the spacing of the cash flows.

Prospective Book Value at a Coupon Date

Let \(n\) be the number of coupon payments remaining immediately after the current coupon has been paid. Under the fixed-yield assumption, each remaining coupon \(C\) is discounted by one, two, and up to \(n\) half-year periods, while redemption \(R\) is discounted by \(n\) periods. With \(v_j=1/(1+j)\), the prospective value is:

$$ B=C a_{\overline{n}|j}+R v_j^n =C\frac{1-v_j^n}{j}+R v_j^n. $$

Here \(a_{\overline{n}|j}\) is the present-value annuity factor for \(n\) payments, with \(j\) as the rate per payment period. Writing the periodic rate explicitly helps distinguish this half-year factor from an annuity factor calculated at an annual rate.

Useful equivalent form: Since \(1-v_j^n=j a_{\overline{n}|j}\), the prospective value can also be written as \(B=R+(C-Rj)a_{\overline{n}|j}\). The quantity \(C-Rj\) compares the coupon per half-year with the interest that would accrue on redemption value over a half-year.

If \(C>Rj\), the bond is priced above redemption value and its book value is above \(R\); if \(C<Rj\), the book value is below \(R\). This comparison is a useful check on a calculation, not a substitute for using the correct coupon-period rate.

Worked Example: A Premium Bond with Four Coupons Remaining

A hypothetical bond has redemption value \(R=\$1{,}000\), an annual coupon rate of \(10\%\), and four semiannual coupons remaining after the coupon just paid. Its yield is a nominal annual rate of \(8\%\) convertible semiannually. Find its book value immediately after that coupon. Round the final answer to the nearest cent.

Method: prospective valuation. The semiannual coupon and half-year yield are:

$$ C=\frac{\$1{,}000(0.10)}{2}=\$50, \qquad j=\frac{0.08}{2}=0.04. $$

There are four periods to redemption, so first calculate the discount factor:

$$ v_j^4=(1.04)^{-4}=0.8548041907. $$

The annuity factor is:

$$ a_{\overline{4}|j} =\frac{1-0.8548041907}{0.04} =3.6298952325. $$

The value of the remaining coupons is \(50(3.6298952325)=\$181.4947616\). The present value of redemption is \(1{,}000(0.8548041907)=\$854.8041907\). Therefore:

$$ B=\$181.4947616+\$854.8041907 =\$1{,}036.2989523 \approx \boxed{\$1{,}036.30}. $$

Check. The interest on redemption value for one half-year is \(1{,}000(0.04)=\$40\), less than the \(\$50\) coupon. The premium form gives \(B=1{,}000+(50-40)(3.6298952325)=\$1{,}036.2989523\), confirming the result. A common mistake would be to use \(8\%\) rather than \(4\%\) to discount each half-year cash flow.

Book Value Recurrence Across Coupon Dates

A book value can also be advanced from one coupon date to the next. Let \(B_{k-1}\) be the value immediately after one coupon, and \(B_k\) the value immediately after the next coupon. Both are measured at the same fixed yield, and one half-year separates them.

1
Express the earlier value in terms of the next one.
At the earlier date, the next coupon \(C\) and the book value \(B_k\) immediately after that coupon are one half-year away. Thus \(B_{k-1}=v_j(C+B_k)\).
2
Rearrange for the later book value.
Multiplying by \(1+j\) gives \(B_k=(1+j)B_{k-1}-C\).

This recurrence has a direct interpretation: accrue interest on the earlier book value for one half-year, then subtract the coupon just paid. It uses the same \(j\) and \(C\) as the prospective valuation. Applying one recurrence step for each coupon period is therefore a way to check or update the prospective book value.

Result: Under a fixed nominal annual yield \(i^{(2)}\) convertible semiannually, the coupon-date recurrence is \(B_k=(1+i^{(2)}/2)B_{k-1}-C\). One step represents one half-year, not one full year.

Worked Example: Advancing Book Value Across One Year

Immediately after a coupon, a hypothetical bond has a book value of \(\$1{,}050\). It pays \(\$35\) every six months, and its fixed yield is a nominal annual rate of \(8\%\) convertible semiannually. Find the book value immediately after the second subsequent coupon. Assume both coupons occur before maturity. Round the final answer to the nearest cent.

Method: apply the coupon-date recurrence twice. The per-half-year yield is \(j=0.08/2=0.04\). After the first half-year and coupon:

$$ B_1=\$1{,}050(1.04)-\$35 =\$1{,}092-\$35 =\$1{,}057. $$

After the second half-year, accrue at \(4\%\) again and subtract the second coupon:

$$ B_2=\$1{,}057(1.04)-\$35 =\$1{,}099.28-\$35 =\boxed{\$1{,}064.28}. $$

Annual-rate reconciliation. Two half-year accruals multiply the starting book value by \((1+0.08/2)^2\), not by \(1+0.08\). Because the first coupon is paid one half-year before the second, it accrues for one more half-year in this two-step expression:

$$ B_2 =\$1{,}050(1.04)^2-\$35(1.04)-\$35 =\$1{,}135.68-\$36.40-\$35 =\$1{,}064.28. $$

The two methods agree. The annual quote \(8\%\) is still the input; dividing by two gives the rate for each of the two coupon periods. Using \(8\%\) at each half-year would incorrectly produce \(B_1=\$1{,}050(1.08)-\$35=\$1{,}099\), rather than \(\$1{,}057\).

Worked Example: A Discount Bond with Three Coupons Remaining

Worked Example: A Discount Bond with Three Coupons Remaining

A hypothetical bond has redemption value \(\$2{,}000\), an annual coupon rate of \(6\%\), and three semiannual coupons remaining immediately after a coupon date. The yield is a nominal annual rate of \(8\%\) convertible semiannually. Calculate the book value at that date, rounded to the nearest cent.

Method: prospective valuation with a premium-or-discount check. The coupon and periodic yield are:

$$ C=\frac{\$2{,}000(0.06)}{2}=\$60, \qquad j=\frac{0.08}{2}=0.04. $$

The three-period discount factor and annuity factor are:

$$ v_j^3=(1.04)^{-3}=0.8889963587, \qquad a_{\overline{3}|j}=\frac{1-0.8889963587}{0.04}=2.7750910325. $$

The coupons have present value \(\$60(2.7750910325)=\$166.50546195\), and redemption has present value \(\$2{,}000(0.8889963587)=\$1{,}777.9927174\). Therefore:

$$ B=\$166.50546195+\$1{,}777.9927174 =\$1{,}944.49817935 \approx \boxed{\$1{,}944.50}. $$

Check. Half-year interest on redemption value is \(\$2{,}000(0.04)=\$80\), which exceeds the \(\$60\) coupon. The discount form gives \(B=2{,}000+(60-80)(2.7750910325)=\$1{,}944.49817935\), consistent with the prospective calculation. The common error to avoid is treating the annual coupon of \(\$120\) as though it were paid at each semiannual date.

Checks and Common Exam Traps

Before finalizing a semiannual book value, check that the cash-flow count, coupon amount, and interest rate all use the same period. At a date immediately after a coupon, count only the coupons still due; do not include the coupon just paid. The redemption payment occurs with the final coupon and is discounted for the same number of remaining half-year periods.

  • Using the annual quote as the period rate. If the yield is \(i^{(2)}\) convertible semiannually, use \(i^{(2)}/2\) for each half-year.
  • Using a full-year coupon at each date. Convert the annual coupon amount \(Rc\) to \(Rc/2\) per semiannual payment.
  • Miscounting remaining periods. If \(n\) coupons remain after the current payment, redemption is also \(n\) half-year periods away.
  • Mixing methods with different assumptions. The prospective value and recurrence agree when both use the same fixed yield and coupon schedule.

A final sign check is useful: compare the coupon \(C\) with the half-year interest \(Rj\) on redemption value. A coupon above that interest indicates a premium; a coupon below it indicates a discount. This check can reveal a rate or coupon conversion error before rounding obscures it.

Check Your Understanding

For each hypothetical bond, use the fixed yield and measure book value immediately after the stated coupon date. Round dollar answers to the nearest cent.

  1. A bond with redemption value \(\$5{,}000\) has an annual coupon rate of \(5.6\%\), six semiannual coupons remaining, and a nominal annual yield of \(6.4\%\) convertible semiannually. Find the book value. (Answer: \(\$4{,}892.20\))
  2. A bond with redemption value \(\$1{,}200\) pays semiannual coupons at an annual coupon rate of \(7\%\). There are five coupons remaining, and the nominal annual yield convertible semiannually is \(6\%\). Find its book value. (Answer: \(\$1{,}227.48\))
  3. A book value immediately after a coupon is \(\$980\). The next two semiannual coupons are each \(\$48\), and the nominal annual yield convertible semiannually is \(9.6\%\). Find the book value immediately after the second coupon. (Answer: \(\$978.03\))
  4. A bond with redemption value \(\$1{,}000\) pays coupons at an annual rate of \(5\%\). It has 12 semiannual coupons remaining, and its nominal annual yield convertible semiannually is \(6\%\). Find its book value. (Answer: \(\$950.23\))

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