Inferring Yield From Consecutive Book Values

Why Consecutive Book Values Reveal Yield

A bond’s coupon-date book values record how its carrying amount changes as interest accrues and coupons are paid. If the book value at one coupon date, the next book value, and the coupon paid between them are known, the effective-interest recurrence can be rearranged to find the yield used to carry the bond. The key is to use the coupon amount for that period and keep the book-value timing convention consistent.

This method finds the yield implied by the stated book values and coupon. It does not require the bond’s original purchase price or its remaining term. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $B_k$ be the bond’s book value immediately after the coupon payment at coupon date $k$, let $C$ be the coupon paid at each coupon date, and let $i$ be the effective yield per coupon period. The effective-interest recurrence is $B_{k+1}=B_k(1+i)-C$. Given $B_k$, $B_{k+1}$, and $C$, the implied periodic yield is $i=(B_{k+1}+C)/B_k-1$, provided $B_k\ne 0$.

Deriving the Yield Formula

The recurrence follows from the cash-flow sequence within one coupon period. Starting immediately after a coupon, the book value earns interest at the yield for that period. The interest amount is $iB_k$, so the value immediately before the next coupon is $B_k+iB_k=B_k(1+i)$. The coupon $C$ is then paid, leaving the book value immediately after that payment:

B_{k+1}=B_k(1+i)-C.

To solve for $i$, first add the coupon to both sides. Then divide by $B_k$ and subtract 1:

\begin{aligned} B_{k+1}+C &= B_k(1+i),\\ \frac{B_{k+1}+C}{B_k} &= 1+i,\\ i &= \frac{B_{k+1}+C}{B_k}-1. \end{aligned}

An equivalent form is $i=(B_{k+1}-B_k+C)/B_k$. This highlights the interpretation: interest for the period equals the change in book value plus the coupon paid. Dividing that interest by the opening book value gives the effective yield for the period.

Important timing condition: In this tutorial, both book values are measured immediately after their respective coupon payments. The recurrence and rearranged formula must not be used unchanged if one value is measured immediately before a coupon. First express both values at the same point in the coupon-payment cycle.

Worked Examples

Worked Example: Inferring an Annual Coupon-Period Yield

A bond has a book value of $1,040 immediately after a coupon payment. One year later, immediately after the next coupon payment of $52, its book value is $1,014. Find the effective yield per year implied by these values. Check the result by rebuilding the ending book value.

Method. Use the effective-interest recurrence with $B_k=1{,}040$, $C=52$, and $B_{k+1}=1{,}014$. The period is one year, so the yield found is an effective annual yield.

First add the coupon to the ending book value:

B_{k+1}+C=1{,}014+52=1{,}066.

Divide by the opening book value and subtract 1:

i=\frac{1{,}066}{1{,}040}-1 =\frac{26}{1{,}040} =0.025.

The implied effective yield is 2.5% per year. To check, interest for the year is $1{,}040(0.025)=26$. The value before the coupon is $1{,}040+26=1{,}066$, and after paying the $52 coupon it is $1{,}066-52=1{,}014$, as given.

The common mistake this method avoids is treating the decline in book value as a negative amount of interest. Here, interest is positive: the $26 of interest is partly offset by the $52 coupon.

Worked Example: Converting a Semiannual Yield

A bond pays $40 every six months. Its book value is $960 immediately after one coupon and $980 immediately after the next. Find the implied yield per half-year, the nominal annual rate convertible semiannually, and the effective annual yield.

Method. First use the recurrence for one semiannual period. Then convert the periodic yield using the stated coupon frequency of two periods per year.

The implied yield per half-year is:

i_{\text{half-year}} =\frac{980+40}{960}-1 =\frac{1{,}020}{960}-1 =1.0625-1 =0.0625.

Thus, the effective yield per half-year is 6.25%. The nominal annual rate convertible semiannually is twice the periodic rate:

i^{(2)}=2(0.0625)=0.125=12.5\%.

The effective annual yield accounts for compounding over both half-year periods:

i_{\text{annual}}=(1+0.0625)^2-1 =1.12890625-1 =0.12890625.

The effective annual yield is 12.8906% to four decimal places. For a check, one half-year of interest on $960 is $960(0.0625)=60$. Subtracting the $40 coupon gives $960+60-40=980$, matching the ending book value.

The common mistake is to call 6.25% the annual effective yield. It is the yield for one half-year. The nominal annual rate is 12.5%, while the effective annual yield is higher because interest compounds.

Worked Example: Checking a Constant Yield Across Two Periods

A hypothetical bond has face value $1,000 and pays coupons quarterly at an annual coupon rate of 10%. Its book values immediately after three successive quarterly coupons are $1,000, $995, and $989.90. Infer the yield per quarter from each interval, convert it to an effective annual yield, and check that the two intervals agree.

Method. Convert the annual coupon rate to the coupon paid each quarter, then apply the rearranged recurrence separately to each consecutive pair of book values.

The coupon per quarter is:

C=1{,}000\left(\frac{0.10}{4}\right)=25.

For the first interval, use $B_k=1{,}000$ and $B_{k+1}=995$:

i_1=\frac{995+25}{1{,}000}-1 =\frac{1{,}020}{1{,}000}-1 =0.02.

For the second interval, use $B_k=995$ and $B_{k+1}=989.90$:

i_2=\frac{989.90+25}{995}-1 =\frac{1{,}014.90}{995}-1 =1.02-1 =0.02.

Both intervals imply an effective quarterly yield of 2%. There are four quarters in a year, so the effective annual yield is:

$$ (1+0.02)^4-1 =1.08243216-1 =0.08243216. $$

The effective annual yield is 8.2432% to four decimal places. As an interval check, the first ending value is $1{,}000(1.02)-25=995$. The next is $995(1.02)-25=1{,}014.90-25=989.90$. Both match the stated book values.

The common mistake is to use the $1,000 annual coupon amount in a one-quarter recurrence. The recurrence uses the coupon paid during that period, which is $25 here.

Interpreting the Implied Yield

The formula returns a yield for one interval between the two book values. That interval must be identified before converting the result to another rate convention. If the observations are one year apart, the answer is an effective annual yield. If they are six months apart, it is an effective half-year yield. If they are three months apart, it is an effective quarterly yield.

For $m$ equal coupon periods per year and a periodic yield $i$, the corresponding nominal annual rate convertible $m$ times per year is $i^{(m)}=mi$. The effective annual yield is $(1+i)^m-1$. These are different annual rate conventions; state which one is requested rather than reporting them interchangeably.

The implied yield is exact only relative to the given inputs and the recurrence model. It assumes that the book values are measured at the stated coupon dates, the coupon amount is correct, and the same effective yield is applied over the interval. If book values have been rounded to the nearest cent or dollar, the calculated yield is generally an estimate rather than an exact underlying yield.

If several consecutive book values are available, calculate the implied yield for each interval using that interval’s opening book value, ending book value, and coupon. Equal results support a single constant periodic yield under the recurrence. Unequal results may indicate rounding, a changed yield convention, inaccurate inputs, or a mismatch in timing; they should not be forced into one rate without investigating the data.

Common Mistakes to Avoid

Using the wrong sign for the coupon. In the rearranged formula, add the coupon to the ending book value: $i=(B_{k+1}+C)/B_k-1$. The coupon was subtracted in the forward recurrence, so it must be added when solving backward.
Using an annual coupon rate as the period’s cash payment. Convert the stated coupon rate and face value to the actual coupon paid at each date.
Mixing book-value timing conventions. The recurrence here uses values immediately after coupons. A before-coupon value cannot be substituted without adjusting the cash-flow timing.
Reporting a periodic rate as an annual rate. Identify the length of the period first; annualize only with the appropriate rate convention.
Ignoring the model check. Substitute the inferred rate into $B_{k+1}=B_k(1+i)-C$. This catches arithmetic and input-timing errors.

Key takeaway: For consecutive book values measured immediately after coupons, the implied yield per coupon period is the opening book value’s interest divided by that opening value: $i=(B_{k+1}-B_k+C)/B_k$. Use the coupon paid between the observations, then verify the result in the original recurrence.

Check Your Understanding

For each problem, treat the listed book values as immediately after coupon payments. Give rates to four decimal places when needed.

A bond’s book value is $875 after one annual coupon and $868 after the next. The coupon is $42. Find the implied effective annual yield. (Answer: 4.0000%)
A bond pays $27 semiannually. Its book value changes from $780 to $789 across one coupon period. Find the effective yield per half-year and the nominal annual rate convertible semiannually. (Answer: 4.6154% per half-year; 9.2308% nominal annual)
A bond pays quarterly. Its book value changes from $1,500 to $1,512, and the coupon is $48. Find the effective quarterly yield and the effective annual yield. (Answer: 4.0000% per quarter; 16.9859% effective annual)
A book value is $990 immediately after a coupon. The effective yield per period is 3%, and the next coupon is $60. Find the book value immediately after the next coupon. (Answer: $959.70)
Two successive intervals have coupon $50 and book values $1,000, $970, and $940. Compute the implied yield for each interval and decide whether one constant yield fits both. (Answer: 2.0000% for the first interval; 2.0619% for the second; no single constant yield fits both exactly.)

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