Repricing Callable Bonds When Redemption Dates Change

Why a Change in Call Terms Changes the Valuation

A callable bond can be redeemed by its issuer before its stated maturity, subject to the contract’s dates and redemption amounts. A change to either term changes the cash flows an investor receives in that call scenario. The calculation therefore begins by identifying the applicable redemption date—not by valuing the bond to maturity and adjusting the result afterward.

This tutorial values bonds under specified call scenarios. A calculated price or yield is conditional on the assumed redemption date and amount. It does not, by itself, determine whether an issuer will exercise a call or the market price of a bond when future exercise is uncertain. All examples use deterministic contractual cash flows: there is no mortality or other random-life assumption.

Definition: For a valuation immediately after a coupon date, let $F$ be the face amount, $C$ the coupon paid each period, $r$ the yield per coupon period, $n$ the number of periods to the specified call date, and $R_n$ the principal redemption amount paid at that date. If the coupon due at the call date is paid as well as redemption, the scenario cash flows are $C$ at the end of periods $1,\ldots,n$, followed by redemption $R_n$ at the end of period $n$. There are no bond payments after period $n$.

If the coupon rate is a nominal annual rate $c^{(m)}$, convertible $m$ times per year, then $C=F c^{(m)}/m$. If the yield is a nominal annual rate $i^{(m)}$ convertible on the same schedule, then $r=i^{(m)}/m$. For semiannual coupons and yield, for example, there are two coupon periods per year. Use the period yield to discount period cash flows.

Valuing One Specified Call Scenario

The valuation assumes that the bond is redeemed at the specified date and for the specified amount, that each scheduled coupon through that date is paid, and that the yield per period is constant. The valuation date is immediately after a coupon date, so no accrued interest needs to be added. If a contract instead states that the coupon due on redemption is not paid, that coupon must be excluded; the contract determines the cash flows.

Theorem: Under these assumptions, the price at the valuation date for redemption after $n$ periods is $$ P_n=C a_{\overline{n}|r}+R_n(1+r)^{-n}, $$ where $a_{\overline{n}|r}=\dfrac{1-(1+r)^{-n}}{r}$ for $r\ne0$. The coupon at period $n$ is included in the annuity term, and the redemption amount is a separate payment at that same date.

The formula follows by adding the present values of the contractual payments. Coupon $C$ due at period $k$ has value $C(1+r)^{-k}$. Adding these values for $k=1,\ldots,n$ gives $C a_{\overline{n}|r}$. The redemption payment due at period $n$ has value $R_n(1+r)^{-n}$. Therefore

P_n=\sum_{k=1}^{n}C(1+r)^{-k}+R_n(1+r)^{-n} =C a_{\overline{n}|r}+R_n(1+r)^{-n}.

This is a present-value calculation for one specified cash-flow schedule. It does not use the face amount as the final payment unless the contract says redemption is at face value. If $r=0$, each payment is undiscounted, so $P_n=nC+R_n$.

Worked Example: Comparing Two Call Dates at the Same Yield

A hypothetical bond has face amount $1,000 and pays coupons semiannually at a nominal annual coupon rate of 5% convertible semiannually. It is valued immediately after a coupon date. The nominal annual yield is 6% convertible semiannually. Compare the price if the issuer redeems after 3 periods for $1,010 with the price if it redeems after 5 periods for $1,000. In both scenarios, the coupon due on the redemption date is paid.

Method. Calculate the semiannual coupon and yield, then value the cash flows up to each call date. The common mistake to avoid is using the later call date’s cash flows for both scenarios.

The coupon per period is $C=1{,}000(0.05)/2=\$25$, and the yield per period is $r=0.06/2=0.03$. For the 3-period call, the discount factor to period 3 is $(1.03)^{-3}=0.915141659$, and the annuity factor is

a_{\overline{3}|0.03}=\frac{1-0.915141659}{0.03}=2.828611367.

The coupon value is $25(2.828611367)=\$70.715284$. The redemption value is $1{,}010(0.915141659)=\$924.293076$. Thus the price under the 3-period call scenario is $70.715284+924.293076=\$995.008360$, or $995.01 to the nearest cent.

For the 5-period call, $(1.03)^{-5}=0.862608784$, so

a_{\overline{5}|0.03}=\frac{1-0.862608784}{0.03}=4.579707200.

The coupon value is $25(4.579707200)=\$114.492680$, and the redemption value is $1{,}000(0.862608784)=\$862.608784$. Their sum is $\$977.101464$, or $977.10 to the nearest cent.

The earlier call price is higher here because it pays a $1,010 redemption amount rather than $1,000. The later date adds coupons but also discounts every payment for longer; the result depends on the complete cash-flow schedule, not on the number of coupons alone.

Finding the Yield for a Specified Redemption

When a price is given and the call date and amount are specified, the yield is the rate that makes the present value of that scenario’s cash flows equal to the price. In general, this rate must be found by solving an equation. For positive coupons and redemption, the present value decreases as a nonnegative yield increases, so the equation has at most one nonnegative solution. A price alone is not enough to identify a yield unless the relevant cash flows—including the assumed call date and redemption amount—are also specified.

Worked Example: Yield to a Specified Call Date

A hypothetical $1,000 bond pays $30 semiannually. It will be redeemed after 3 periods for $1,030, with the period-3 coupon paid. Immediately after a coupon date, its observed price is $971.45. Find the nominal annual yield convertible semiannually to this specified redemption.

Method. Set the price equal to the present value of three coupons and the redemption amount, and solve for the semiannual yield. Do not discount the redemption amount for three years: there are three semiannual periods, or 1.5 years.

Let $r$ be the semiannual yield. The equation is

971.45=30\frac{1-(1+r)^{-3}}{r}+1{,}030(1+r)^{-3}.

Try $r=0.05$. Then $(1.05)^{-3}=0.863837599$, and

a_{\overline{3}|0.05} =\frac{1-0.863837599}{0.05} =2.723248020.

The coupon present value is $30(2.723248020)=\$81.697441$. The redemption present value is $1{,}030(0.863837599)=\$889.752727$. Together they give $\$971.450168$, which rounds to the observed price of $971.45. Thus the yield is approximately 5% per half-year, equivalent to a nominal annual yield of $2(0.05)=$ 10.00% convertible semiannually.

The rounded observed price is given to cents, so it cannot in general identify an infinitely precise yield. The calculation verifies that 5% per period is consistent with the stated price to the nearest cent.

Keep the Cash-Flow Schedule Tied to the Call Date

A practical way to avoid mixing scenarios is to list the payments by period before calculating. Coupons scheduled before the call remain in the valuation. The coupon due at the call date is included only if the contract pays it. Redemption is entered once, at the call date, for the specified amount. Coupons and principal payments after redemption are excluded.

Fix the valuation date and timing.
Count coupon periods from the valuation date. Convert an annual nominal yield to the yield per coupon period.

Read the call provision.
Record the specified call period, redemption amount, and whether the coupon due on that date is also payable.

Write the scenario’s cash flows.
Include only the coupons through redemption and the one redemption payment. Do not carry forward maturity payments after the call.

Value or solve.
Discount the listed payments at the given yield to find price, or set their present value equal to the observed price and solve for yield.

Worked Example: The Redemption Amount and Date Both Change

A hypothetical bond has face amount $1,000 and pays $20 semiannually. It is valued immediately after a coupon date at a nominal annual yield of 4% convertible semiannually. Under one call provision it is redeemed after 2 periods for $1,040; under another it is redeemed after 4 periods for $980. In either case, the coupon due at redemption is paid. Find both scenario prices.

Method. Use the same yield per period but build a separate cash-flow schedule for each redemption. The trap is to treat $1,000 face value as the redemption amount in both cases, even though the stated amounts differ.

The period yield is $r=0.04/2=0.02$, and the coupon is $C=\$20$. For redemption after 2 periods, $(1.02)^{-2}=0.961168781$ and $a_{\overline{2}|0.02}=1.941560950$. The coupon present value is $20(1.941560950)=\$38.831219$. The redemption present value is $1{,}040(0.961168781)=\$999.615532$. Therefore,

P_2=38.831219+999.615532=1{,}038.446751,

so the price is $1,038.45 to the nearest cent.

For redemption after 4 periods, $(1.02)^{-4}=0.923845426$ and $a_{\overline{4}|0.02}=3.807728700$. The coupon present value is $20(3.807728700)=\$76.154574$. The redemption present value is $980(0.923845426)=\$905.368517$. Thus

$$ P_4=76.154574+905.368517=981.523091, $$

so the price is $981.52 to the nearest cent. The results reflect both the changed payment date and the changed redemption amount. Each calculation has been checked by adding the coupon and redemption present values for that scenario.

Common Mistakes and a Useful Check

Including post-call coupons. If redemption occurs after period 3, coupons at periods 4 and later are not payments in that scenario.
Omitting the call-date coupon. If the contract pays the coupon due on the redemption date, it belongs in the annuity term as well as the redemption amount.
Replacing redemption with face amount. Use the stated call price, even when it is above or below face value.
Using an annual yield for a semiannual period. A nominal annual yield convertible semiannually must be divided by 2 before discounting each half-year cash flow.
Confusing a scenario yield with a guaranteed realized return. A yield calculated to a call date depends on the assumption that redemption occurs on that date and the stated payments are made.

A useful final check is to write the last cash flow explicitly: it is the final coupon, if payable, plus the redemption amount. Then confirm that there are no payments after that date. When calculating a yield, substitute the result back into the price equation; rounding may cause a small difference from the observed price.

Check Your Understanding

For each problem, treat the stated call date as certain, assume the valuation is immediately after a coupon date, and include the call-date coupon when the problem says it is paid. Round dollar prices to the nearest cent.

A $1,000 bond pays $15 semiannually and is called after 2 periods for $1,000. The nominal annual yield is 4% convertible semiannually. Find the price. (Answer: $990.29)
A $1,000 bond pays $25 semiannually and is called after 3 periods for $1,010, including the coupon due then. The nominal annual yield is 4% convertible semiannually. Find the price. (Answer: $1,023.84)
A bond pays $10 per period and is called after 2 periods for $1,000. At a price of $843.80, find the yield per period and, if periods are semiannual, the nominal annual yield convertible semiannually. (Answer: approximately 10% per period; 20% nominal annual)
A $1,000 bond pays $20 semiannually. Compare redemption after 2 periods for $1,020 with redemption after 4 periods for $980, using a nominal annual yield of 4% convertible semiannually. The call-date coupon is paid in both cases. (Answer: approximately $1,019.22 for the 2-period call and $981.52 for the 4-period call)

What You'll Learn