Yield To Call From A Callable Bond’s Cash Flows

Introduction

A callable bond may be redeemed before its stated maturity. If a particular call date is assumed, the yield-to-call is the rate that equates the bond’s purchase price to the present value of the cash flows received up to and including that date. The central task is to build the cash-flow sequence correctly: coupons paid before the call date are included, and the call price is received on the call date.

This tutorial focuses on one specified call date. It uses the present-value approach from bond yield calculations and is aligned with the SOA Exam FM (Financial Mathematics) syllabus. The yield is conditional on the bond being called on that date; it is not a promise that the issuer will call it.

Definition: The yield-to-call for a specified call date is the interest rate that makes the present value of the bond’s coupons through that date, together with the call price paid on that date, equal to the bond’s price at the valuation date. If there are $n$ coupon periods to the call date and the nominal annual yield convertible $m$-thly is $i^{(m)}$, the yield per coupon period is $i^{(m)}/m$, when coupons are paid $m$ times per year.

Set the Call-Date Cash Flows

Assume the bond is valued immediately after a coupon date, so its price $P$ is the full value at that date and there is no accrued coupon to add. Suppose it pays a level coupon $C$ at the end of every coupon period, and the call occurs after $n$ periods for a call price $K$. If a coupon is scheduled on the call date, the investor receives both that coupon and $K$.

Let $r$ denote the yield per coupon period. When all coupons through the call date are due, the yield equation is

P=C\left(\frac{1-(1+r)^{-n}}{r}\right)+K(1+r)^{-n},\qquad r=\frac{i^{(m)}}{m}.

The annuity term represents $n$ coupons, including the coupon at time $n$. The call price is a separate payment at time $n$, so it is discounted for the same $n$ periods. This equation assumes regular coupon periods and a call date that coincides with a coupon date. For irregularly dated payments or a valuation between coupon dates, the cash flows require period-specific discounting and an appropriate price at the valuation date.

Count coupon periods.
Measure the number $n$ of coupon intervals from the valuation date to the specified call date. Do not count years as periods unless coupons are annual.

List payments by date.
Record each coupon due before the call date. At the call date, record the coupon if due and the call price $K$.

Write the present-value equation.
Discount each payment using the yield per coupon period. For level coupons, the coupon stream can be written as an annuity.

Solve and check.
Convert the periodic yield to the requested nominal annual form, then substitute it back into the equation to confirm that it reproduces the price.

Why the Equation Has This Form

The formula follows directly from discounting each dated payment. A coupon paid at time $t$ has present value $C(1+r)^{-t}$. Thus the present value of all $n$ coupons is the finite geometric sum below; the call price is another payment at time $n$.

\begin{aligned} C\sum_{t=1}^{n}(1+r)^{-t} &=C\left((1+r)^{-1}+(1+r)^{-2}+\cdots+(1+r)^{-n}\right)\\ &=C\frac{1-(1+r)^{-n}}{r},\\ PV(K)&=K(1+r)^{-n}. \end{aligned}

Adding these two present values gives the yield-to-call equation. In particular, the final coupon is already included in the coupon sum. The call price does not replace that coupon; it is an additional payment at the same date.

Worked Example: Infer Yield From a Price and a Single Call Date

A hypothetical bond is purchased for $1,000.83 immediately after a coupon date. It pays $40 every six months and is callable after three half-year periods for $970. Find its nominal annual yield convertible semiannually. Assume the coupon due on the call date is paid.

Method. Use the present value of three semiannual coupons plus the call price at time 3. The coupon period is half a year, so the periodic yield $r$ is half the nominal annual yield.

The equation is

1000.83=40\left(\frac{1-(1+r)^{-3}}{r}\right)+970(1+r)^{-3}.

Test $r=0.0300$, corresponding to a nominal annual yield of $2(0.0300)=0.0600$, or 6.000% convertible semiannually. The discount factor for three periods is

(1.03)^{-3}=\frac{1}{1.092727}=0.9151417.

The present value of the coupons is

40\left(\frac{1-0.9151417}{0.03}\right) =40(2.828610) =\$113.1444.

The present value of the call price is

970(0.9151417)=\$887.6874.

Adding the two components gives $113.1444+887.6874=\$1,000.8318$, which rounds to the stated purchase price of $1,000.83. Therefore, the periodic yield is approximately 3.000%, and the nominal annual yield convertible semiannually is 6.000%, to three decimal places. The check confirms that the three coupons and the call price are all represented.

Worked Example: Include the Coupon Due on the Call Date

A hypothetical bond costs $1,031.67. It pays a coupon of $28 every six months and is called after seven half-year periods for $1,015. A coupon is due on the call date. Find the nominal annual yield convertible semiannually, and identify the payments at the final date.

Method. Include all seven coupons in the annuity and discount the call price for seven periods. Try a periodic yield of 2.500%, equivalent to a nominal annual rate of $2(0.025)=0.050$, or 5.000% convertible semiannually.

P=28\left(\frac{1-(1.025)^{-7}}{0.025}\right)+1015(1.025)^{-7}.

The seven-period discount factor is

(1.025)^{-7}=0.8412652.

The coupon present value is $28[(1-0.8412652)/0.025]=28(6.349392)=\$177.7830$. The call-price present value is $1015(0.8412652)=\$853.8842$. Their sum is $177.7830+853.8842=\$1,031.6672$, which rounds to $1,031.67.

Thus the yield is 5.000% nominal annual, convertible semiannually, to three decimal places. At the final date the holder receives both the $28 coupon and the $1,015 call price, for a total payment of $1,043. The separate present-value terms in the equation are a useful check that neither payment has been omitted.

Worked Example: Solve the Yield and Reprice the Bond

A hypothetical bond is priced at $1,074.68 immediately after a coupon date. It pays $32 semiannually and will be called after five half-year periods for $1,020. The final coupon is paid on the call date. Estimate the nominal annual yield convertible semiannually to three decimal places.

Method. Set the present value of the five coupons and the call price equal to $1,074.68, then solve for the periodic yield. A financial calculator or numerical yield method can be used; the result should be verified by repricing.

1074.68=32\left(\frac{1-(1+r)^{-5}}{r}\right)+1020(1+r)^{-5}.

Solving gives a periodic yield of approximately $r=0.02000$. Therefore the nominal annual yield convertible semiannually is $2(0.02000)=0.04000$, or 4.000% to three decimal places.

Check by substitution. Since $1.02^5=1.1040808$, the five-period discount factor is approximately $0.9057308$. The coupon present value is $32[(1-0.9057308)/0.02]=32(4.713460)=\$150.8307$. The call-price present value is $1020(0.9057308)=\$923.8454$. The sum is $\$1,074.6761$, which rounds to the stated price of $1,074.68. The final coupon is included as the fifth coupon in the annuity.

Common Mistake: Omitting the Final Coupon

A frequent error is to discount the call price correctly but stop the coupon annuity one period early. That calculation treats the call date as though no coupon were due, even when the bond terms specify a coupon on that date.

\begin{aligned} \text{Incorrect at }r=0.025:\quad P&=28\left(\frac{1-(1.025)^{-6}}{0.025}\right)+1015(1.025)^{-7},\\ \text{Correct at }r=0.025:\quad P&=28\left(\frac{1-(1.025)^{-7}}{0.025}\right)+1015(1.025)^{-7}. \end{aligned}

The incorrect expression leaves out the seventh coupon, whose present value is $28(1.025)^{-7}=\$23.5554$, approximately. It therefore understates the present value at the proposed yield by that amount. If the price is given and yield is being solved, omitting the coupon instead leads to a different, incorrect yield. Always write the payment at each date before compressing the coupons into an annuity formula.

A second timing error is counting the call date as an extra period after the stated number of coupon periods. If the call occurs after $n$ half-year periods, both the final coupon (if due) and the call price are discounted by $(1+r)^{-n}$, not by $(1+r)^{-(n+1)}$.

Interpreting and Checking the Result

For positive coupons and a positive call price, the present value of the specified cash flows decreases as the yield increases. Consequently, when a numerical method is used, a candidate yield can be checked by evaluating the present value at that rate and comparing it with the bond price. A correct yield reproduces the price, apart from rounding.

The nominal annual rate is not the same number as the rate per coupon period. For semiannual coupons, multiply the periodic yield by two to report a nominal annual yield convertible semiannually. For annual coupons, the periodic yield and nominal annual yield coincide. State the conversion convention whenever reporting the result.

Yield-to-call is conditional on the assumed call date and payments. If the bond is not called then, the investor’s actual cash flows differ and the calculated yield is not the realized return. The yield equation is also not directly applicable without adjustment when coupon dates are irregular, the valuation date falls between coupon dates, or the call terms specify a different payment timing.

Key takeaway: For one specified call date, discount every coupon paid through that date and the call price paid on that date. If the coupon is due at the call date, include it as well as the call price, then verify the yield by repricing.

Check Your Understanding

For each problem, assume the bond is valued immediately after a coupon date. Round dollar answers to the nearest cent and yields to three decimal places.

A hypothetical bond pays $40 annually and is callable after three years for $950. At a 6.000% annual yield, find its price, including the coupon due at the call date. (Answer: $904.56)
A hypothetical bond pays $24 every six months and is callable after five half-year periods for $1,000. Find its price at a nominal annual yield of 6.000% convertible semiannually. Include the final coupon. (Answer: $972.52)
A hypothetical bond costs $1,055.74, pays $35 semiannually, and is callable after four half-year periods for $1,020, with a coupon due at the call date. Estimate the nominal annual yield convertible semiannually. (Answer: 5.000%)
A hypothetical bond pays $18 every six months and is callable after six half-year periods for $990, with the final coupon paid on that date. Find its price at a nominal annual yield of 4.000% convertible semiannually. (Answer: $979.92)

What You'll Learn