Yields to Multiple Call Dates from One Price

Why One Bond Can Have Several Yields to Call

A callable bond may be redeemed on more than one permitted date, and the amount paid at redemption may depend on the date. Each possible redemption date therefore defines a different cash-flow scenario. From one observed market price, we can calculate a conditional yield for each scenario: the rate that makes that scenario’s cash flows have present value equal to the price.

These calculations do not predict which call date will occur. They answer separate conditional questions: “What yield would result if the bond were called on this date at this call price?” A later call date generally means more coupon payments, and it may also have a different call price. Both features must be reflected in its equation.

Definition: The yield to a specified call date is the rate that equates the bond’s market price to the present value of all scheduled coupon payments through that date and the call payment on that date. The calculation assumes redemption occurs on that date, regardless of whether it will actually occur.

Unless stated otherwise, assume the bond is valued immediately after a coupon payment, so no accrued interest needs to be included. Let $P$ be its market price, $c$ the coupon paid each period, $n$ the number of periods to a particular call date, $K_n$ the call price at that date, and $r_n$ the yield per coupon period for that scenario. The coupon due on the call date is paid in addition to the call price.

P=\sum_{t=1}^{n-1}\frac{c}{(1+r_n)^t} +\frac{c+K_n}{(1+r_n)^n}

If coupons are not level, replace $c$ in each term with the coupon actually scheduled for that period. For annual coupons, $r_n$ is an annual effective yield. For semiannual coupons, the equation gives the yield per half-year; if the requested quote is nominal annual interest convertible semiannually, multiply the periodic yield by 2.

Why the Present-Value Equation Works

The equation follows from the basic discounting principle. A dollar paid at the end of period $t$ has present value $1/(1+r_n)^t$ dollars when the yield per period is $r_n$. Thus each earlier coupon contributes $c/(1+r_n)^t$. At the call date, the bond pays both its final coupon and its call price, so that period’s cash flow is $c+K_n$.

Adding the present values gives the value of the complete cash-flow scenario. The yield is the rate that makes this sum equal the observed price. Because the cash flows depend on the assumed call date and its call price, a new rate must be calculated for each scenario; the rates are not found by averaging call prices or combining the scenarios.

\text{present value of each payment} = \frac{\text{payment}}{(1+\text{yield per period})^{\text{number of periods}}}

Worked Examples

Worked Example: Three Annual Call Dates from One Price

A hypothetical bond is priced at $P=\$1{,}000$ immediately after an annual coupon payment. It pays an annual coupon of $\$40$. It may be called at the end of year 1 for $\$1{,}000$, year 2 for $\$1{,}020.50$, or year 3 for $\$1{,}063.672$. Calculate the conditional annual effective yield for each call date.

Method. Set up one present-value equation for each date. Use the coupon payments through that date and that date’s call price; the coupon paid on the call date is additional to the call price.

Call at the end of year 1. The only payment is the year-1 coupon plus the call price: $40+1{,}000=\$1{,}040$. The yield $r_1$ satisfies

1{,}000=\frac{1{,}040}{1+r_1}, \qquad 1+r_1=1.04, \qquad r_1=0.04.

The conditional yield to year 1 is 4%. Check: $\$1{,}040/1.04=\$1{,}000$.

Call at the end of year 2. The cash flows are $\$40$ at year 1 and $\$40+\$1{,}020.50=\$1{,}060.50$ at year 2. Test a 5% annual effective yield:

\frac{40}{1.05}+\frac{1{,}060.50}{1.05^2} = 38.095238+961.904762 = 1{,}000.

The conditional yield to year 2 is 5%. The check recovers the same market price.

Call at the end of year 3. The cash flows are $\$40$ at years 1 and 2, then $\$40+\$1{,}063.672=\$1{,}103.672$ at year 3. Test a 6% annual effective yield:

\frac{40}{1.06} +\frac{40}{1.06^2} +\frac{1{,}103.672}{1.06^3} \approx 37.735849+35.599858+926.664293 = 1{,}000.

The conditional yield to year 3 is 6%. Each calculation uses the same price but a different set of payments. The method avoids the mistake of applying the year-1 call price to the later scenarios.

Worked Example: Converting Semiannual Yields to Nominal Annual Quotes

A hypothetical bond has a market price of $\$1{,}000$, pays $\$25$ every six months, and is callable after 1 year for $\$989.90$ or after 2 years for $\$1{,}000$. Find the nominal annual yield convertible semiannually for each call date.

Method. First solve for the yield per half-year, using the number of half-year periods. Then multiply that periodic yield by 2 to obtain the nominal annual quote.

Call after 1 year. There are two coupon periods. The cash flows are $\$25$ after six months and $\$25+\$989.90=\$1{,}014.90$ after one year. The periodic yield $r$ satisfies

1{,}000=\frac{25}{1+r}+\frac{1{,}014.90}{(1+r)^2}.

At $r=0.02$, the first present value is $25/1.02=\$24.509804$. The second is $1{,}014.90/1.0404=\$975.490196$. Their sum is $\$1{,}000$, so the yield per half-year is 2%. The nominal annual yield convertible semiannually is $2(2\%)=$ 4%.

Call after 2 years. There are four coupon periods, with a $\$25$ coupon each period and a $\$1{,}000$ call price at the end. Test $r=0.025$ per half-year. The coupon present values are:

\begin{aligned} \frac{25}{1.025}&=24.390244, & \frac{25}{1.025^2}&=23.795360,\\ \frac{25}{1.025^3}&=23.214985, & \frac{25}{1.025^4}&=22.648766. \end{aligned}

Their sum is $\$94.049355$. The call price present value is $1{,}000/1.025^4=\$905.950645$. The total is $\$1{,}000$, so the yield per half-year is 2.5%. The nominal annual yield convertible semiannually is $2(2.5\%)=$ 5%.

Both checks discount the coupons and call payment to the same market price. The conversion is based on the number of coupon periods per year, not the number of years to the call date.

Worked Example: Include the Coupon Paid on the Call Date

A hypothetical bond is priced at $\$1{,}000$ immediately after an annual coupon. It pays $\$50$ annually and may be called at the end of year 2 for $\$979.60$ or at the end of year 3 for $\$1{,}000$. Calculate both conditional annual effective yields.

Call at year 2. The payments are $\$50$ at year 1 and $\$50+\$979.60=\$1{,}029.60$ at year 2. At a 4% yield:

\frac{50}{1.04}+\frac{1{,}029.60}{1.04^2} = 48.076923+951.923077 = 1{,}000.

The yield to the year-2 call date is 4%.

Call at year 3. The payments are $\$50$ at years 1 and 2, then $\$50+\$1{,}000=\$1{,}050$ at year 3. At a 5% yield:

\frac{50}{1.05}+\frac{50}{1.05^2}+\frac{1{,}050}{1.05^3} = 47.619048+45.351474+907.029478 = 1{,}000.

The yield to the year-3 call date is 5%.

Common mistake. Omitting the coupon paid at year 3 would replace the last $\$1{,}050$ payment with $\$1{,}000$. At 5%, that incorrect calculation gives

\frac{50}{1.05}+\frac{50}{1.05^2}+\frac{1{,}000}{1.05^3} \approx \$956.81,

not the $\$1{,}000$ market price. The correction is to add the year-3 coupon to the year-3 call price. Conversely, do not add the coupon twice if a problem explicitly states that a quoted redemption payment already includes it.

A Reliable Process for Multiple Call Dates

For each date, keep the date-specific cash flows and yield separate. A numerical solver may be used when the equation does not simplify conveniently, but the cash-flow setup still determines whether the answer is correct. For an ordinary bond with positive payments, increasing the yield reduces the present value of those payments; a computed yield should therefore be checked by substituting it back into the price equation.

Count the periods.
Count coupon periods from the valuation date to the particular call date. Do not use the bond’s full term when calculating a yield to an earlier call.

List that scenario’s payments.
Include every coupon due through the call date and the call price at that date. Add the final coupon to the call price unless the terms say it is already included.

Solve using the stated quote convention.
Use a per-period rate in the present-value equation. Convert it to a nominal annual quote only after solving, if the problem requests that convention.

Check against the market price.
Discount every payment using the solved rate. The present values should sum to the given price, subject to rounding.

The resulting figures are conditional yields, not promises or forecasts. A yield for one call date uses the assumption that redemption happens on that date; it does not incorporate a probability of being called. Report each result alongside its call date and rate convention so that two yields cannot be mistaken for one another. This treatment of separate redemption scenarios is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the applicable cash flows before calculating. Assume valuation immediately after a coupon unless stated otherwise.

A bond costs $\$500$, pays $\$30$ annually, and is callable at the end of year 2 for $\$489.75$. Find its annual effective yield to that call date. (Answer: 5%)
A bond costs $\$1{,}000$, pays $\$20$ every six months, and is callable after three half-year periods for $\$1{,}000$. Find the nominal annual yield convertible semiannually. (Answer: 4%)
A bond costs $\$960$, pays $\$45$ annually, and is callable after two years for $\$985.956$. Find its annual effective yield to call. (Answer: 6.000%)
A $\$1{,}000$ bond pays $\$40$ annually and is callable after year 1 for $\$980$ or after year 2 for $\$1{,}020.50$. Find the annual effective yield for each call date. (Answer: 2% to year 1; 5% to year 2)
A bond costs $\$1{,}000$, pays $\$30$ every six months, and is callable after two periods for $\$1{,}000$ or after four periods for $\$979.237421875$. Find each nominal annual yield convertible semiannually. (Answer: 6% to period 2; 5% to period 4)

What You'll Learn