Callable Bond Duration Under Fixed Redemption Assumptions

Why a Callable Bond Needs a Cash-Flow Assumption

A bond’s duration is calculated from its expected payment times and present values. For a callable bond, the payment schedule may depend on whether and when the issuer exercises its right to redeem the bond early. A duration calculation therefore needs an explicit cash-flow scenario. If the problem specifies a call date and redemption amount, treat them as fixed assumptions and calculate duration from the resulting payments.

This calculation does not determine whether the issuer will call the bond. It also does not assign probabilities to different call dates or value the embedded call option. It answers a narrower question: if the stated call scenario occurs, what is the duration of those assumed cash flows at the stated yield?

Definition: Under a fixed redemption assumption, a callable bond is modeled as paying its scheduled coupons through the specified call date, plus the stated redemption amount on that date. Unless the problem says otherwise, the redemption amount is separate from the coupon due on the call date. Macaulay duration is the present-value-weighted average time of these assumed payments.

For an annual-coupon bond with annual effective yield $i$, face amount $F$, annual coupon amount $C$, call date $k$, and redemption amount $K$ (excluding the coupon), the assumed cash flows are $C$ at times $1,\ldots,k-1$ and $C+K$ at time $k$. Its price $P$ and Macaulay duration $D$ under this scenario are

P=\sum_{t=1}^{k-1}\frac{C}{(1+i)^t}+\frac{C+K}{(1+i)^k}, \qquad D=\frac{\displaystyle\sum_{t=1}^{k-1}t\frac{C}{(1+i)^t} +k\frac{C+K}{(1+i)^k}}{P}.

The face amount and coupon rate determine $C$, but the final assumed payment depends on the redemption amount. For annual effective yield and fixed annual cash flows, modified duration is $D_{\mathrm{mod}}=D/(1+i)$. If the problem uses another payment frequency or yield convention, the time units and modified-duration conversion must match that convention.

Why the Present-Value-Weighted Formula Works

Each payment contributes to the bond’s price according to its present value. A payment received later receives a larger time weight; a payment with a larger present value receives a larger weight. Dividing the sum of time multiplied by present value by the total price normalizes those weights to sum to one.

w_t=\frac{CF_t(1+i)^{-t}}{P}, \qquad \sum_t w_t=1, \qquad D=\sum_t t\,w_t.

Here $CF_t$ is the payment at time $t$, and $w_t$ is its share of the bond’s price. Thus duration is the weighted average payment time for the cash-flow scenario being analyzed. The fixed-cash-flow assumption also explains the modified-duration relation: differentiating $P(i)=\sum_t CF_t(1+i)^{-t}$ gives $-P'(i)/P=D/(1+i)$. This relation uses cash flows that do not change as yield changes; it is not, by itself, a model of how an issuer’s call decision may respond to yields.

Calculate Duration Under the Stated Call Scenario

Worked Example: A Call at the End of Year 3

A hypothetical bond has face amount $1,000 and pays annual coupons at a 6% coupon rate. Assume it is called at the end of year 3 for $1,020, with the year-3 coupon paid in addition to redemption. The annual effective yield is 5%. Find its price, Macaulay duration, and modified duration under this assumption.

Method. First write the cash flows implied by the specified call scenario. The annual coupon is $0.06(1{,}000)=\$60$. The payments are $60 at time 1, $60 at time 2, and $60+1{,}020=\$1{,}080$ at time 3.

At 5%, the discount factors are $1/1.05=0.9523809524$, $1/1.05^2=1/1.1025=0.9070294785$, and $1/1.05^3=1/1.157625=0.8638375985$. The present values are

PV_1=60(0.9523809524)=57.142857143,\quad PV_2=60(0.9070294785)=54.421768707,

PV_3=1{,}080(0.8638375985)=932.944606412.

Therefore, the price is $57.142857143+54.421768707+932.944606412=1{,}044.509232262$, or $1,044.51 to the nearest cent.

For Macaulay duration, multiply each present value by its payment time. The weighted total is $1(57.142857143)+2(54.421768707)+3(932.944606412)=2{,}964.820213794$. Hence

D=\frac{2{,}964.820213794}{1{,}044.509232262} =2.83848\text{ years}\approx 2.8385\text{ years}.

Modified duration is $2.83848/1.05=2.703314$, or 2.7033 years. As a check, the three present-value weights sum to 1, and the duration is between the first and last payment times, 1 and 3 years. The common mistake avoided here is omitting the year-3 coupon when adding the redemption amount.

Redemption Amount and Coupon Timing Matter

The call-date payment is generally not just the redemption amount. If the coupon due on that date is paid, the final cash flow is the coupon plus redemption. Conversely, if a problem explicitly says the bond is redeemed between coupon dates or that the final coupon is not paid, use the cash flows it specifies rather than automatically adding a full coupon.

Worked Example: A Premium Redemption at the End of Year 2

A hypothetical bond has face amount $2,000, an annual coupon rate of 4%, and an annual effective yield of 5%. It is assumed called at the end of year 2 for $2,030, with the year-2 coupon paid separately. Find the price and Macaulay duration.

Method. The annual coupon is $0.04(2{,}000)=\$80$. Thus the assumed payments are $80 at time 1 and $80+2{,}030=\$2{,}110$ at time 2. Discounting at 5% gives

PV_1=\frac{80}{1.05}=76.190476190, \qquad PV_2=\frac{2{,}110}{1.05^2} =\frac{2{,}110}{1.1025}=1{,}913.832199546.

The price is $76.190476190+1{,}913.832199546=1{,}990.022675736$, or $1,990.02 to the nearest cent. The time-weighted present-value total is $1(76.190476190)+2(1{,}913.832199546)=3{,}903.854875282$. Therefore,

D=\frac{3{,}903.854875282}{1{,}990.022675736} =1.961714\text{ years}\approx 1.9617\text{ years}.

The duration is greater than 1 year and less than 2 years, as required for a positive-weighted average of payments at those times. The common mistake avoided is using $2,030 as the entire year-2 cash flow: it is the redemption amount, so the separately paid $80 coupon must also be included.

Changing the Assumed Call Date Changes Duration

A call date is not just a label attached to the same payment stream. Moving it changes which coupons are paid and when the redemption amount is received. Even if the bond, yield, and redemption amount stay the same, duration can change because the present-value weights and payment times change.

Worked Example: Comparing Two Fixed Call-Date Assumptions

A hypothetical bond has face amount $1,000, pays annual coupons at an 8% coupon rate, and has an annual effective yield of 8%. Compare Macaulay duration if it is assumed redeemed at par, with the coupon paid, at the end of year 2 versus the end of year 3.

Method. The coupon is $80. For either scenario, the redemption payment is $1,000, in addition to the coupon due on the call date. The discount factors for years 1, 2, and 3 are $1/1.08=0.925925926$, $1/1.08^2=0.857338820$, and $1/1.08^3=0.793832241$.

Call at year 2. The assumed payments are $80 and $1,080. Their present values are $80(0.925925926)=74.074074$ and $1{,}080(0.857338820)=925.925926$. The price is $1,000, and the time-weighted total is $74.074074+2(925.925926)=1{,}925.925926$. Thus the duration is $1{,}925.925926/1{,}000=\mathbf{1.9259}$ years.

Call at year 3. The assumed payments are $80, $80, and $1,080. Their present values are $74.074074$, $68.587106$, and $857.338820$. The price is again $1,000. The weighted total is $74.074074+2(68.587106)+3(857.338820)=2{,}783.264746$, so the duration is $2{,}783.264746/1{,}000=\mathbf{2.7833}$ years.

Both prices equal par because the coupon rate equals the yield and redemption is at par. The durations nevertheless differ: the year-3 scenario places more present value at a later payment time. This comparison checks the central point: duration belongs to the assumed cash-flow schedule, not simply to the bond’s name or stated maturity. The common mistake avoided is assuming that equal prices imply equal durations.

Interpretation, Limits, and Exam Traps

Under a specified call date and redemption amount, use the fixed cash flows exactly as stated. The resulting Macaulay duration is a scenario-based measure. It can support a modified-duration price estimate when the cash flows are treated as unchanged for a small yield movement, but that estimate does not account for the possibility that a different yield could change the issuer’s call decision.

If a problem gives several possible call dates without specifying which one to assume, a unique fixed-scenario duration cannot be calculated until a cash-flow scenario is selected. Do not silently choose the earliest call date, the stated maturity date, or the date that seems most favorable to the issuer. State what is assumed and calculate from that schedule. A fuller treatment of the call option would require modeling how exercise and cash flows depend on market conditions; that is a different question from fixed-redemption duration.

Do not omit the call-date coupon. Add it to redemption when the problem says it is paid.
Do not add a coupon twice. If the stated redemption payment already includes the final coupon, use the stated total as the final cash flow.
Keep time units consistent. Annual coupon dates and an annual effective yield give duration in years. For another frequency, use matching periods and yield conventions.
Do not call a scenario result universal. A different assumed call date or redemption amount generally produces different present values and duration.

Key takeaway: For fixed-call duration, specify the assumed call date, redemption amount, and coupon treatment; construct the resulting cash flows; discount them at the stated yield; and take their present-value-weighted average time. The answer is conditional on that cash-flow scenario.

Check Your Understanding

For each problem, identify the assumed cash flows before calculating. Round dollar prices to the nearest cent and durations to four decimal places where needed.

A hypothetical $1,000 bond pays annual coupons at 5% and is assumed called at the end of year 2 for $1,010, with the year-2 coupon paid. The annual effective yield is 4%. Find the price and Macaulay duration. (Answer: price $1,028.11; duration 1.9532 years)
A hypothetical $1,000 bond pays annual coupons at 10% and is assumed redeemed at par at the end of year 2, with the year-2 coupon paid. Its annual effective yield is 10%. Find the price, Macaulay duration, and modified duration. (Answer: price $1,000.00; Macaulay duration 1.9091 years; modified duration 1.7355 years)
A hypothetical $1,000 bond pays annual coupons at 8% and is assumed redeemed at par at the end of year 3, with the final coupon paid. Its annual effective yield is 8%. Find its price and Macaulay duration. (Answer: price $1,000.00; duration 2.7833 years)
A hypothetical $1,000 bond pays annual coupons at 5%, has a 5% annual effective yield, and is redeemable at par with the coupon paid. Compare the duration under an assumed call at the end of year 1 with the duration under an assumed call at the end of year 3. Which scenario has greater duration, and by how much? (Answer: year-1 duration 1.0000 years; year-3 duration 2.8594 years; the year-3 scenario is greater by 1.8594 years)
A problem states that a hypothetical bond may be called at the end of year 2 or year 4, but gives no assumed call date or exercise rule. Can one fixed-scenario Macaulay duration be calculated from that information alone? State what additional assumption is needed. (Answer: No; specify the assumed call date and corresponding redemption cash flow, or provide an exercise rule that determines the cash-flow scenario.)

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