Constructing Cash Flows to Each Callable Bond Redemption Date

Why Callable-Bond Cash Flows Must Be Built by Scenario

A callable bond can be redeemed on one of several dates, but a particular redemption scenario ends the bond’s cash flows at just one date. To form a yield equation, first make a complete timeline for each permitted call date and for maturity. For every timeline, include the coupons paid before redemption and the payment due when redemption occurs.

This is a bookkeeping task with an important consequence: a coupon due on the redemption date is generally paid in addition to the call redemption amount. The redemption amount includes the principal repayment, so the principal must not be added to it a second time.

Definition: A redemption scenario is the assumed outcome in which a bond is redeemed on one specified permitted call date or, if it is not called earlier, at maturity. Its cash-flow stream consists of every scheduled coupon paid up to and including that date, together with the contractual redemption payment on that date. The scenarios are alternatives, not payments that all occur in the same bond stream.

Set the Timing and Payment Conventions First

Suppose a bond has face amount $F$, pays a fixed coupon $C$ at the end of every half-year, and is valued just after a coupon date. Number the future coupon dates $1,2,3,\ldots$. Let $R_k$ denote the contractual redemption payment if the bond is redeemed at coupon date $k$. Here, $R_k$ includes the principal repayment but excludes the coupon due on that date.

For a yield quoted as nominal annual rate $i^{(2)}$, convertible semiannually, the yield per half-year is $q=i^{(2)}/2$. The price $P$ of a given scenario is related to that scenario’s cash flows by discounting each payment for its own number of half-year periods. This equation assumes the stated yield applies to every period and that the listed payments are fixed and certain under the scenario. It does not assign probabilities to the different call outcomes.

Cash-flow rule: If redemption occurs at coupon date $k$, then coupons $C$ are paid at dates $1,\ldots,k-1$, and the final date-$k$ payment is $C+R_k$. The scenario’s yield equation is

P=\sum_{j=1}^{k-1}\frac{C}{(1+q)^j}+\frac{C+R_k}{(1+q)^k}, \qquad q=\frac{i^{(2)}}{2}.

For maturity, use the maturity date as $k$, with the maturity redemption payment in place of $R_k$.

The equation is written for a price measured immediately after a coupon date, so there is no coupon already accrued at the valuation time in this setup. A bond valued between coupon dates requires different timing treatment. Also, if the bond has different coupon amounts on different dates, each actual coupon must be used rather than repeating $C$.

Why the Yield Equation Has This Form

A yield equation is an equation of value: the price at the valuation date equals the present value of the payments in the specified scenario. To see why each term has its exponent, take a payment $X_j$ made $j$ half-year periods after valuation. At yield $q$ per half-year, one period’s discount factor is $1/(1+q)$. Discounting through $j$ periods gives $X_j/(1+q)^j$. Adding the discounted value of each payment gives the scenario equation.

For redemption at date $k$, the coupons at dates $1$ through $k-1$ are each $C$. At date $k$, both the scheduled coupon and redemption payment are due, making the last payment $C+R_k$. Thus the equation is the sum of $k$ discounted payments, with the final payment combining two distinct contractual amounts.

List the permitted endpoints.
Write each eligible call date and the maturity date as a separate scenario.

Count coupon dates through each endpoint.
Include every scheduled coupon before redemption and the coupon due on the redemption date.

Attach the correct redemption payment.
Use the amount specified for that endpoint; it already includes principal repayment.

Discount each dated payment separately.
Use the number of periods from valuation to that payment, and write one equation for the scenario.

Worked Cash-Flow Scenarios

Worked Example: Two Call Dates and a Later Maturity

A hypothetical bond has face amount $1,000 and pays coupons semiannually at a nominal annual coupon rate of 5.4%. It may be called at the end of year 1 for a redemption payment of $1,030, or at the end of year 2 for $1,015. If not called, it matures at the end of year 3 for $1,000. The price is $P$, and the yield is nominal annual $i^{(2)}$. Construct the equation for each scenario.

Method. Convert the coupon rate to a half-year coupon, number the six half-year dates, then combine the final coupon with the redemption payment at each endpoint.

Step 1: Calculate the coupon. There are two coupon periods per year, so each coupon is $1{,}000(0.054)/2=\$27$. The yield per period is $q=i^{(2)}/2$.

Step 2: Call at the end of year 1. This is date 2. The first payment is $27 at date 1. At date 2, the coupon and redemption payment total $\$27+\$1{,}030=\$1{,}057$. The equation is

P=\frac{27}{1+q}+\frac{1{,}057}{(1+q)^2}.

Step 3: Call at the end of year 2. This is date 4. Coupons of $27 are paid at dates 1, 2, and 3. The date-4 payment is $\$27+\$1{,}015=\$1{,}042$. Thus,

P=\frac{27}{1+q}+\frac{27}{(1+q)^2}+\frac{27}{(1+q)^3} +\frac{1{,}042}{(1+q)^4}.

Step 4: Hold to maturity. Maturity is date 6. Coupons of $27 are paid at dates 1 through 5, and the final payment is $\$27+\$1{,}000=\$1{,}027$. The equation is

P=\sum_{j=1}^{5}\frac{27}{(1+q)^j}+\frac{1{,}027}{(1+q)^6}.

Check. The equation for each scenario has exactly one payment at every coupon date up to its endpoint and none after it. The common mistake avoided here is treating $1,030 or $1,015 as the entire final payment and leaving out the coupon due at that date.

Worked Example: Call Dates That Do Not Occur Every Year

A hypothetical $1,000 bond pays a 4.8% nominal annual coupon rate semiannually. It is callable at the end of year 1.5 for $1,040 and at the end of year 2.5 for $1,020. Its maturity is at the end of year 4, with a $1,000 maturity payment. Write all three yield equations using $q=i^{(2)}/2$.

Method. Map each time in years to a half-year coupon-date number. For example, 1.5 years is three half-year periods.

Step 1: Calculate the coupon. Each half-year coupon is $1{,}000(0.048)/2=\$24$. The three endpoints are dates 3, 5, and 8, respectively.

Step 2: First call, date 3. The bond pays $24 at dates 1 and 2. At date 3, it pays the coupon plus the call redemption amount: $\$24+\$1{,}040=\$1{,}064$. Therefore,

P=\frac{24}{1+q}+\frac{24}{(1+q)^2}+\frac{1{,}064}{(1+q)^3}.

Step 3: Second call, date 5. Coupons of $24 are paid at dates 1 through 4. The date-5 payment is $\$24+\$1{,}020=\$1{,}044$, so

P=\sum_{j=1}^{4}\frac{24}{(1+q)^j}+\frac{1{,}044}{(1+q)^5}.

Step 4: Maturity, date 8. Coupons are paid at dates 1 through 7. The final payment is $\$24+\$1{,}000=\$1{,}024$. The maturity equation is

P=\sum_{j=1}^{7}\frac{24}{(1+q)^j}+\frac{1{,}024}{(1+q)^8}.

Check. The first call equation ends at period 3, not period 2: the 1.5-year call date is three half-year periods away. This avoids the time-unit error of treating each year as one coupon period.

Worked Example: Annual Coupons and a Redemption Amount Below Face

A hypothetical bond has face amount $2,000, pays an annual coupon rate of 4.2%, and is valued just after a coupon date. It may be called in two years for a redemption payment of $1,980, or in four years for $2,030. Otherwise, it matures in five years for $2,000. The yield $i$ is an effective annual rate. Construct each equation.

Method. Because coupon dates are annual, use $i$ directly as the one-year yield rate. Count coupons through each endpoint, and do not add face amount again to the stated redemption payment.

Step 1: Calculate the annual coupon. $2{,}000(0.042)=\$84$. The relevant dates are years 2, 4, and 5.

Step 2: Call in year 2. The year-1 coupon is $84. At year 2, the coupon plus redemption payment is $\$84+\$1{,}980=\$2{,}064$. Thus,

P=\frac{84}{1+i}+\frac{2{,}064}{(1+i)^2}.

Step 3: Call in year 4. Coupons of $84 are paid in years 1, 2, and 3. The year-4 payment is $\$84+\$2{,}030=\$2{,}114$. So the equation is

P=\frac{84}{1+i}+\frac{84}{(1+i)^2}+\frac{84}{(1+i)^3} +\frac{2{,}114}{(1+i)^4}.

Step 4: Maturity in year 5. Coupons of $84 are paid in years 1 through 4. The final payment is $\$84+\$2{,}000=\$2{,}084$. Therefore,

P=\sum_{j=1}^{4}\frac{84}{(1+i)^j}+\frac{2{,}084}{(1+i)^5}.

Check. The redemption payments of $1,980, $2,030, and $2,000 each include the principal repayment. For instance, adding another $2,000 to the year-2 call payment would count principal twice. The correct final year-2 cash flow is $2,064, not $4,064.

Common Construction Errors

Before using a yield equation, check both the date count and the cash amount at the endpoint. Errors in either place change the scenario being analyzed.

Omitting the redemption-date coupon. If redemption occurs on a scheduled coupon date, include that coupon as well as the redemption payment, unless the contract explicitly states otherwise.
Adding principal twice. A stated call redemption payment ordinarily already includes repayment of principal. Add the coupon due on that date, not the face amount again.
Discounting by years when periods are shorter. With semiannual coupons, a date two years away is four coupon periods away. The exponent must count half-year periods when the yield is per half-year.
Combining alternative scenarios. A call in year 2 and a call in year 4 are separate possibilities. The year-4 scenario does not include the year-2 redemption payment.
Extending payments after redemption. Once the bond is redeemed, later coupons do not occur in that scenario.

A reliable final check is to read each equation from left to right and match every discounted term to a dated payment on its scenario timeline. There should be one term for each coupon date through redemption, with the final term equal to the coupon plus the applicable redemption payment.

Check Your Understanding

For each problem, construct the stated call-date or maturity yield equations. Use $q=i^{(2)}/2$ for semiannual coupons and $i$ for annual coupons.

A $1,000 bond pays coupons semiannually at a 6% nominal annual coupon rate. It can be called at the end of year 1 for $1,010 or matures at the end of year 2 for $1,000. Write both equations. (Answer: $P=30/(1+q)+1{,}040/(1+q)^2$; $P=\sum_{j=1}^{3}30/(1+q)^j+1{,}030/(1+q)^4$.)
A $1,500 bond pays coupons semiannually at a 4% nominal annual coupon rate. Calls are permitted at periods 3 and 6, with redemption payments $1,480 and $1,515; maturity is period 8 for $1,500. Write all three equations. (Answer: $P=30/(1+q)+30/(1+q)^2+1{,}510/(1+q)^3$; $P=\sum_{j=1}^{5}30/(1+q)^j+1{,}545/(1+q)^6$; $P=\sum_{j=1}^{7}30/(1+q)^j+1{,}530/(1+q)^8$.)
A $2,500 bond pays annual coupons at 5.2%. It can be called in year 3 for $2,470 and matures in year 6 for $2,500. Write both equations. (Answer: $P=130/(1+i)+130/(1+i)^2+2{,}600/(1+i)^3$; $P=\sum_{j=1}^{5}130/(1+i)^j+2{,}630/(1+i)^6$.)
An $800 bond pays coupons semiannually at a 7.5% nominal annual coupon rate. Calls are permitted at periods 4 and 7, with redemption payments $825 and $810; maturity is period 10 for $800. Write all three equations. (Answer: $P=\sum_{j=1}^{3}30/(1+q)^j+855/(1+q)^4$; $P=\sum_{j=1}^{6}30/(1+q)^j+840/(1+q)^7$; $P=\sum_{j=1}^{9}30/(1+q)^j+830/(1+q)^{10}$.)

What You'll Learn