Callable Bond Pricing Between Coupon Dates

Why the Valuation Date Matters

A bond’s coupon dates determine when its scheduled payments occur, but an investor may value or purchase the bond on a date between coupons. In that case, the first remaining coupon is not a full coupon period away. Each cash flow must be discounted for the actual number of coupon periods between the valuation date and its payment date, including any fractional period specified by the problem.

For a callable bond, the cash flows also depend on the redemption scenario. If the bond is called on a specified date, payments after that date do not occur; the call-date payment includes both the coupon due on that date and the call price, when the bond terms specify both. If the bond instead remains outstanding to maturity, its coupons continue through maturity and the maturity redemption is paid then. These scenarios produce separate conditional prices, not an average price.

Definition: A conditional price is the present value of the cash flows under one specified redemption scenario, such as redemption on a stated call date or at maturity. Under the fractional-period convention used here, if a cash flow $CF_k$ is due $\tau_k$ coupon periods after the valuation date and the yield per coupon period is $j$, its present value is $CF_k(1+j)^{-\tau_k}$. The full, or dirty, price is the sum of these present values.

This convention is an assumption, not a universal rule for every bond market or every problem. It treats the yield as a constant effective rate per coupon period and extends the corresponding accumulation factor to fractional periods by taking a fractional power. Use it only when the problem states or implies this convention. Actual market quotations can depend on a specified day-count and settlement convention.

Set the Rate and the Fractional Times

Suppose coupons are paid $m$ times per year and the annual quoted yield is the nominal rate $i^{(m)}$, convertible $m$ times per year. The yield per coupon period is $j=i^{(m)}/m$. If a problem instead gives the effective annual rate $i$, the equivalent rate per coupon period is $(1+i)^{1/m}-1$. Do not use an annual rate directly as though it were the rate for each coupon period.

Measure each payment time from the valuation date, not from the previous coupon date. For example, if valuation occurs one-quarter of a coupon period after a coupon date, the next coupon is $0.75$ periods away. A payment two full coupon periods after that earlier coupon date is $1.75$ periods away from valuation.

Fix the convention.
Identify the coupon frequency, the quoted yield basis, the rate per coupon period, and the required fractional-period discounting rule.

Locate the valuation date.
Record what fraction of a coupon period has elapsed since the most recent coupon date.

Build each redemption scenario.
List every coupon up to and including the redemption date, and combine the coupon and redemption amount when they are due together.

Discount and sum.
For each scenario, discount every cash flow using its own time from valuation. Retain unrounded factors through the calculation.

Why Fractional-Period Discounting Works

Under the stated convention, one coupon period of accumulation multiplies a value by $1+j$. The model extends this accumulation rule so that $\tau$ coupon periods multiply a value by $(1+j)^\tau$. If a cash flow of amount $CF$ due in $\tau$ periods has present value $PV$, equivalence of present and accumulated values requires

PV(1+j)^\tau=CF.

Dividing both sides by $(1+j)^\tau$ gives the fractional-period present value. For several payments, present values add because each payment is a separate amount received at a specified time. Thus the conditional price is

P=\sum_k CF_k(1+j)^{-\tau_k}.

This derivation depends on the constant-yield fractional-power convention. It does not establish that every market’s accrued-interest or settlement rules use the same convention.

Worked Example: Call or Maturity After a Quarter-Period Settlement

A hypothetical bond has face value $1,000 and pays $50 every half-year. The annual quoted yield is $i^{(2)}=12\%$, and fractional-period discounting is specified. Valuation occurs $0.25$ coupon periods after a coupon date. The bond may be called at a price of $1,025 on the date two coupon periods after that last coupon date. Otherwise it remains outstanding to a maturity date three coupon periods after that date, when $1,000 is redeemed. Find the conditional full price under each scenario.

Method and rate. The yield per half-year coupon period is $j=0.12/2=0.06$. The next coupon is $1-0.25=0.75$ periods after valuation. The call date is $2-0.25=1.75$ periods after valuation, and maturity is $3-0.25=2.75$ periods after valuation.

Call scenario. The $50 coupon at $0.75$ periods is paid, and the call-date payment is the $50 coupon plus the $1,025 call price, or $1,075. Therefore,

P_{\text{call}} =50(1.06)^{-0.75}+1{,}075(1.06)^{-1.75}.

The discount factors, calculated without rounding in the underlying arithmetic, are $0.957239478$ and $0.903056111$, respectively. The present values are $50(0.957239478)=47.861974$ and $1{,}075(0.903056111)=970.785320$, to six decimal places. Their sum is $1{,}018.647294$, so the call-scenario full price is $1,018.65, rounded to cents.

Maturity scenario. The cash flows are $50 at $0.75$ periods, $50 at $1.75$ periods, and the final coupon plus face value, or $1,050, at $2.75$ periods:

P_{\text{maturity}} =50(1.06)^{-0.75} +50(1.06)^{-1.75} +1{,}050(1.06)^{-2.75}.

The third discount factor is $0.851939728$. The three present values are $47.861974$, $45.152806$, and $894.536714$, to six decimal places. Their sum is $987.551494$, so the maturity-scenario full price is $987.55, rounded to cents.

Check and trap avoided. The coupon at the call date is included with the call price, while the maturity scenario has coupons only through maturity. A common error is to discount all cash flows by integer periods measured from the last coupon date. That would ignore the $0.25$-period elapsed before valuation and would give incorrect discount times.

Comparing Conditional Prices

The call-scenario price and maturity-scenario price answer different conditional questions. They need not have a particular ordering: the call price, coupon schedule, and redemption timing all affect the result. Do not treat the lower of the two calculated prices as an expected value, or assume that the issuer will necessarily choose a scenario based only on these values. The calculation is conditional on the redemption date specified.

A further distinction is between a full price and a clean price. The full price is the present value of the contractual cash flows under the chosen scenario. If the problem asks for a clean price and specifies a linear accrued-interest convention, subtract accrued interest from the full price. For a level coupon $C$, if the elapsed fraction of the coupon period is $f$, this convention gives accrued interest $Cf$. If day-count rules or another accrual convention are stated, use those instead.

Worked Example: Call and Maturity Prices With Accrued Interest

A hypothetical bond has face value $1,000 and pays a $24 coupon every half-year. Its annual nominal yield convertible semiannually is $i^{(2)}=12\%$, so the yield per coupon period is $j=0.06$. Valuation is halfway through a coupon period. The call date is two coupon periods after the most recent coupon date, with a call price of $1,012. If not called, the bond matures three coupon periods after that date for $1,000. Use fractional-period discounting and linear accrued interest. Find the call and maturity full prices and their corresponding clean prices.

Set the times and cash flows. The next coupon is due in $0.5$ periods. The call date is $2-0.5=1.5$ periods after valuation, and the maturity date is $3-0.5=2.5$ periods after valuation. The call-date payment is $1{,}012+24=1{,}036$. At maturity, the final payment is $1{,}000+24=1{,}024$.

Call full price. The discount factors for $0.5$ and $1.5$ periods at $j=0.06$ are $0.971285862$ and $0.916307417$, respectively. Thus,

P_{\text{call}} =24(0.971285862)+1{,}036(0.916307417) =23.310861+949.294484 =972.605345.

Rounded to cents, the call full price is $972.61.

Maturity full price. There are coupons of $24 at $0.5$ and $1.5$ periods, followed by $1,024 at $2.5$ periods. The discount factor for $2.5$ periods is $0.864440959$. Therefore,

P_{\text{maturity}} =24(0.971285862)+24(0.916307417)+1{,}024(0.864440959) =23.310861+21.991378+885.187542 =930.489781.

Rounded to cents, the maturity full price is $930.49. The accrued interest under the stated linear convention is $24(0.5)=$12$. Subtracting $12 gives a call clean price of $960.61 and a maturity clean price of $918.49, each rounded to cents.

Check and trap avoided. Both scenarios have the same accrued-interest deduction because they share the same valuation date and coupon period. Subtracting accrued interest before constructing or discounting the contractual cash flows would mix the clean-price adjustment with the present-value calculation.

Worked Example: A Call Date One Coupon Period Away

A hypothetical bond has face value $1,000 and pays $30 semiannually. Its annual nominal yield convertible semiannually is $i^{(2)}=8\%$. Valuation occurs halfway between coupon dates, and the bond is callable at $1,000 on the next coupon date. Find the call-scenario full price using fractional-period discounting.

Method and timing. The periodic yield is $j=0.08/2=0.04$. Since valuation is halfway through the period, the next coupon and call date are both $0.5$ periods away. The call-date cash flow includes both the $30 coupon and the $1,000 call price, for a total of $1,030. There is no later coupon in this scenario.

P_{\text{call}} =1{,}030(1.04)^{-0.5} =\frac{1{,}030}{\sqrt{1.04}} =1{,}009.998\ldots

The call-scenario full price is $1,010.00, rounded to cents. As a check, the payment is discounted for half a coupon period rather than a full period. Discounting it for one full period would produce $1{,}030/1.04=$990.38$, which does not follow the stated valuation timing.

Common Errors to Avoid

Using the wrong yield unit. With a nominal yield convertible $m$ times per year, divide by $m$ to obtain the periodic rate. Then measure the discount exponent in coupon periods.
Measuring from the last coupon date. The exponent is the time from valuation to payment. If part of the current coupon period has elapsed, subtract that fraction from the coupon-date count.
Omitting a redemption-date coupon. When a coupon is due on the call or maturity date, include it with the redemption amount if the terms say it is paid.
Including post-redemption payments. A call scenario ends at the specified call date. Coupons scheduled after that date are not part of its cash-flow schedule.
Rounding factors too early. Keep full calculator precision for each discount factor and present value, then round the final price to cents. A displayed factor rounded to six decimals may not reproduce the final result if reused as though it were exact.
Confusing full and clean prices. Discount the contractual cash flows to obtain the full price. Apply an accrued-interest deduction only when a clean price is requested and the accrual convention is known.

Key takeaway: A between-coupon callable-bond price is built scenario by scenario. Identify the coupon-period yield, measure each payment time from valuation, include the coupon due at redemption, and discount every cash flow using its own fractional-period exponent. State separately whether the requested result is a full price or a clean price.

Check Your Understanding

For each problem, use the stated fractional-period convention and keep full precision until the final rounding step.

A bond pays $20 semiannually and has annual nominal yield $i^{(2)}=10\%$. Valuation is halfway through a coupon period. It is called on the date two coupon periods after the last coupon date for $980, with the coupon paid on the call date. Find the call-scenario full price to cents. (Answer: $20(1.05)^{-0.5}+1{,}000(1.05)^{-1.5}=$948.95$.)
A bond pays $35 semiannually and has annual nominal yield $i^{(2)}=8\%$. Valuation is $0.25$ periods after a coupon date. Maturity is three coupon periods after that coupon date, with face value $1,000 redeemed at maturity. Write the maturity full price using fractional-period discounting. (Answer: $35(1.04)^{-0.75}+35(1.04)^{-1.75}+1{,}035(1.04)^{-2.75}$.)
A bond pays $15 each half-year. Its annual nominal yield convertible semiannually is $i^{(2)}=6\%$. Valuation is halfway through a coupon period. The call date is two coupon periods after the last coupon date and the call price is $990. Write the call-scenario full price, including all payments due by that date. (Answer: $15(1.03)^{-0.5}+1{,}005(1.03)^{-1.5}$.)
A bond pays $24 semiannually, has $i^{(2)}=12\%$, and is valued halfway through a coupon period. Under linear accrual, find accrued interest and state how to obtain the clean price from a full price of $930.49. (Answer: accrued interest is $24(0.5)=$12$; clean price is $$930.49-$12=$918.49$.)

What You'll Learn