Calculating Yield to Maturity When a Bond Is Callable

What Yield to Maturity Assumes for a Callable Bond

A callable bond gives its issuer the right to redeem the bond before its stated maturity, according to the terms of the bond. That possibility matters when describing the bond’s possible cash-flow scenarios. In a yield-to-maturity calculation, however, the scenario is specified: assume every scheduled coupon is paid and the maturity redemption amount is paid on the maturity date. The yield is the rate that discounts those assumed payments to the bond’s observed price.

This is a conditional calculation, not a prediction that the bond will remain outstanding until maturity. If the issuer calls the bond earlier, the investor receives a different sequence of payments, and the yield calculated on the maturity assumption will not describe that realized cash flow.

Definition: A callable bond’s yield to maturity (YTM) is the yield rate that makes the present value of all scheduled coupons plus the maturity redemption amount equal to the bond’s price, assuming the bond is held to maturity and those payments occur as scheduled. The call feature does not change the cash flows in this specified calculation.

The model assumes the price is measured at a coupon date, the coupon dates and amounts are known, and payments are discounted at one constant yield per coupon period. It also assumes the maturity redemption amount is known and that there is no default or missed payment. The calculation does not model the issuer’s decision to call the bond.

Build the Maturity Cash-Flow Equation

Suppose a bond has $n$ remaining coupon periods, pays coupon $C$ at the end of each period, and pays redemption amount $R$ with the final coupon. Let $P$ be its price and $j$ the yield per coupon period. Under the maturity assumption, the first $n-1$ payments are $C$, and the final payment is $C+R$.

P=\frac{C}{(1+j)}+\frac{C}{(1+j)^2}+\cdots+\frac{C+R}{(1+j)^n}

The periodic rate $j$ is the rate used to discount one coupon period. With $m$ coupon periods per year, a nominal annual yield convertible $m$ times is $i^{(m)}=mj$. Its equivalent effective annual rate is $(1+j)^m-1$. These are different annual rate conventions, so a reported yield should state which one is being used. For annual coupons, $m=1$, and the yield per period is the effective annual rate $i$.

When coupon amounts are level, the equation can also be written using the annuity-immediate notation $a_{\overline{n}|}$, evaluated at the periodic yield $j$:

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

The annuity term represents the $n$ scheduled coupons, including the last coupon. The separate redemption term represents only $R$, so it is added to the final coupon rather than substituted for it.

Why Discounting Gives the Yield Equation

The equation follows from the definition of present value. At yield $j$, one dollar due in period $k$ has present value $(1+j)^{-k}$. Therefore, a payment $X_k$ at that date has present value $X_k(1+j)^{-k}$. Adding the present values of all the assumed payments gives the price that corresponds to the yield.

P=\sum_{k=1}^{n}X_k(1+j)^{-k}, \qquad X_k= \begin{cases} C, & 1\le k<n,\\ C+R, & k=n. \end{cases}

This is why the maturity redemption amount appears only at period $n$, and why the final coupon remains part of the period-$n$ payment. A call price belongs in this equation only if the specified cash-flow scenario is redemption on that call date; it is not included in a YTM equation that assumes maturity redemption.

Identify the maturity scenario.
Count the remaining coupon periods and record the maturity date and redemption amount. Do not use an earlier call date for this calculation.

List each payment.
Enter each coupon as a separate payment. Add the redemption amount to the final coupon.

Set present value equal to price.
Discount each payment by the number of coupon periods until it is due, using the unknown periodic yield.

Report the yield convention.
Convert the periodic yield to the requested annual quote, and check that the resulting present value matches the price.

Worked Calculations

Worked Example: Semiannual Coupons and a Premium Call Price

A hypothetical bond is priced at $990.11. It pays $70 every six months, has three coupon periods remaining, and pays $1,020 at maturity. The issuer may call it after the first coupon for $1,040. Calculate the YTM, quoted as a nominal annual rate convertible semiannually, assuming the bond is held to maturity.

Method. Use the three-period maturity cash flows. The first two payments are $70 each, and the final payment is $70+$1,020=$1,090. The $1,040 call price is not a cash flow in this specified maturity scenario.

990.11=\frac{70}{1+j}+\frac{70}{(1+j)^2}+\frac{1{,}090}{(1+j)^3}

Test a periodic yield of $j=0.08$. The discount factors are $1/1.08=0.9259259$, $1/(1.08)^2=0.8573388$, and $1/(1.08)^3=0.7938322$. The corresponding present values, calculated before rounding, are:

\frac{70}{1.08}=64.8148148,\qquad \frac{70}{1.08^2}=60.0137174,\qquad \frac{1{,}090}{1.08^3}=865.2771427

Adding the unrounded values gives $990.1056749, which rounds to $990.11. Thus the periodic yield is approximately 8.000%. Since there are two periods per year, the nominal annual YTM convertible semiannually is $i^{(2)}=2(0.08)=0.16$, or 16.00% nominal, convertible semiannually.

Check and trap avoided. Substitution reproduces the stated price to the cent. Using $1,040 as the final redemption would incorrectly mix a call-date scenario into a maturity calculation; the maturity redemption is $1,020.

Worked Example: Annual Coupons with a Call Before Maturity

A hypothetical callable bond has a price of $948.46, pays an annual coupon of $60, and will pay $1,000 at the end of its third year if held to maturity. It is callable at the end of year two for $1,035. Find the YTM under the maturity assumption.

Method. There are three annual periods, so let $i$ be the effective annual yield. The cash flows are $60 at the end of years one and two, then $1,060 at the end of year three.

948.46=\frac{60}{1+i}+\frac{60}{(1+i)^2}+\frac{1{,}060}{(1+i)^3}

Test $i=0.08$. The discount factors for years one, two, and three are $0.9259259$, $0.8573388$, and $0.7938322$, respectively. Therefore:

60(0.9259259)=55.5556,\qquad 60(0.8573388)=51.4403,\qquad 1{,}060(0.7938322)=841.4622

The sum using unrounded present values is $948.4581, which rounds to $948.46. The YTM is therefore approximately 8.00% effective annually.

Check and trap avoided. The discounted maturity cash flows reproduce the price. Replacing the final $1,060 payment with the year-two call price would omit the third coupon and calculate a different redemption scenario, not YTM.

Worked Example: Converting a Semiannual Yield to an Annual Quote

A hypothetical bond costs $980.69. It pays $25 every six months, has eight semiannual periods remaining, and pays $1,020 at maturity. Although it is callable after four periods, calculate its YTM on the maturity assumption and report both the nominal annual rate convertible semiannually and the equivalent effective annual rate.

Method. The final payment is $25+$1,020=$1,045. Test a periodic yield of $j=0.03$, then convert the periodic rate using each requested convention.

P=25a_{\overline{8}|}+1{,}020(1.03)^{-8}

First, $(1.03)^4=1.12550881$, so $(1.03)^8=1.2667701$ and $(1.03)^{-8}\approx0.789409$. Thus:

a_{\overline{8}|}=\frac{1-0.789409}{0.03}\approx7.0197

The present value of the coupons is $25(7.0197)\approx$175.4925$, and the present value of the maturity redemption is $1{,}020(0.789409)\approx$805.1972$. Using unrounded factors, their sum is approximately $980.69, matching the price.

The periodic yield is 3.00% per half-year. The nominal annual quote is $i^{(2)}=2(0.03)=0.06$, or 6.00% convertible semiannually. The equivalent effective annual rate is $(1.03)^2-1=0.0609$, or 6.09%.

Check and trap avoided. The maturity cash flows reproduce the quoted price. Reporting 3.00% as an annual yield without stating that it is a half-year rate confuses the periodic yield with an annual rate.

Interpretation and Common Errors

YTM summarizes the discount rate that equates the current price with one specified set of future payments. For a callable bond, that set is the coupons through maturity and the maturity redemption amount. The call feature does not make this computed rate a guaranteed return: an issuer can exercise its call right, changing the timing and amount of payments. This calculation answers only the maturity-scenario question.

This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus. In an exam problem, write down the assumed redemption scenario before entering numbers into the price equation. That small step helps keep a call price, maturity value, and final coupon from being confused.

Using the call price in a YTM equation. The call price is used for a call-date cash-flow scenario. For YTM, use the stated maturity redemption amount.
Omitting the final coupon. If redemption is $R$ and the final coupon is $C$, the final payment is $R+C$, not $R$ alone.
Counting years instead of coupon periods. For semiannual coupons over three years, there are six periods, not three. Each exponent counts periods.
Misreporting the rate. A periodic yield of $j$ with $m$ coupon periods per year corresponds to nominal annual rate $i^{(m)}=mj$, not automatically to effective annual rate $mj$.
Rounding too early. Keep discount factors and present values unrounded during the calculation; round the final price comparison and reported rate only at the end.

Key takeaway: For a callable bond’s YTM, discount every scheduled coupon and the maturity redemption amount at one periodic yield until their present values sum to the price. State the periodic and annual rate conventions, and treat early calling as outside this particular cash-flow assumption.

Check Your Understanding

For each problem, use the stated maturity assumption and report the yield using the requested convention.

A callable bond costs $972.77, pays annual coupons of $40, has three years remaining, and redeems for $1,000 at maturity. It may be called after year two for $1,025. Find its YTM as an effective annual rate. (Answer: approximately 5.00%)
A bond costs $947.58 and pays $30 every six months for six periods, with a maturity redemption of $1,000. It is callable after two periods. Find the nominal annual YTM convertible semiannually. (Answer: approximately 8.00%)
A callable bond costs $1,036.13, pays annual coupons of $75 for four years, and redeems for $980 at maturity. Find the YTM as an effective annual rate. (Answer: approximately 6.00%)
A bond costs $870.12, pays $20 semiannually for six periods, and redeems for $1,030 at maturity. If its call price after the second period is $1,050, find the nominal annual YTM convertible semiannually under the maturity assumption. (Answer: approximately 10.00%)

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