Callable Zero-Coupon Bonds Under Alternative Redemption Dates

Why Alternative Redemption Dates Need Separate Calculations

A callable zero-coupon bond makes no periodic coupon payments, but it may have more than one possible payment date. The issuer might redeem it on an eligible call date; if that does not happen, the bond may continue to a later call date or maturity. Each possibility specifies a different single payment at a different time. To value or calculate a yield for one possibility, use the redemption amount and date for that scenario together.

This is a conditional calculation: it answers what the price or yield is if redemption occurs under the specified scenario. The calculation does not, by itself, determine which date the issuer will choose or the bond’s market price after accounting for call behavior. For that, additional assumptions about the issuer’s action and the valuation model would be needed.

Definition: A redemption scenario is one specified outcome for a callable bond, identified by a redemption date $t$ and a payment $R_t$. The conditional price is the present value of $R_t$ under that scenario and a stated yield. The conditional yield is the annual effective rate that accumulates the observed price to $R_t$ over $t$ years.

Assume the valuation date is time 0, each redemption date is an integer number of years from that date, and a single annual effective yield applies over the scenario’s term. The bond has no interim payments, and the specified redemption payment is received with certainty within that scenario. These calculations do not include taxes, default risk, or the probability of different call outcomes.

Pricing Each Redemption Scenario

For a zero-coupon bond, there is just one payment in a specified scenario. If the payment $R_t$ is due at the end of year $t$, and the annual effective yield is $y$, its conditional price $P_t$ is:

P_t=\frac{R_t}{(1+y)^t}

The redemption amount and time must come from the same scenario. For example, a call payment at year 2 is discounted for 2 years, even if the bond has a later maturity date. If there are several eligible call dates, calculate a separate conditional price for each specified call or maturity outcome.

Worked Example: Pricing a Call Scenario and a Maturity Scenario

A hypothetical callable zero-coupon bond has a call payment of $1,100 at the end of year 2 and a maturity payment of $1,250 at the end of year 5. Calculate the conditional price for each scenario at a 5% annual effective yield. State which common mistake the method avoids.

Method. Discount each single payment using its own date. The 5% annual effective accumulation factors are:

At year 2, $1.05^2=1.1025$. At year 5, $1.05^5=1.2762815625$.

Call scenario. The $1,100 payment is due in 2 years:

$P_2=1{,}100/1.1025=997.732426\ldots$, so the conditional price is $997.73.

Maturity scenario. The $1,250 payment is due in 5 years:

$P_5=1{,}250/1.2762815625=979.407708\ldots$, so the conditional price is $979.41.

Check. Using the unrounded prices, $997.732426\ldots(1.05)^2=1{,}100$ and $979.407708\ldots(1.05)^5=1{,}250$. Rounding to cents only at the end gives the stated prices.

The method avoids discounting the call payment for 5 years just because year 5 is the bond’s maturity, or discounting the maturity payment for only 2 years. Each price is conditional on its own specified payment date.

These scenario prices are not necessarily interchangeable. Here, the call scenario has a higher payment but is received sooner; the maturity scenario has a still larger payment but is discounted for longer. Which price is greater depends on the amounts, dates, and yield together. A comparison of redemption amounts alone does not establish a comparison of present values.

Calculating a Conditional Yield

If the price $P$, redemption amount $R_t$, and redemption date $t$ are known, solve the pricing equation for the annual effective yield. The result is a yield for that specific scenario:

P=\frac{R_t}{(1+y_t)^t} \qquad\Longrightarrow\qquad y_t=\left(\frac{R_t}{P}\right)^{1/t}-1

The exponent $1/t$ is essential: it converts the total accumulation factor $R_t/P$ over $t$ years into an annual effective rate. When a bond has several possible redemption dates, calculate a separate conditional yield for each date. Do not combine alternative payments as though they were all received.

Justification: Under an annual effective yield $y_t$, one dollar at time 0 accumulates to $(1+y_t)^t$ dollars at year $t$. Therefore a price $P$ accumulates to $P(1+y_t)^t$. In the specified scenario this accumulated amount must equal the single redemption payment, so $P(1+y_t)^t=R_t$. Dividing by $P$, taking the $t$-th root, and subtracting 1 gives $y_t=(R_t/P)^{1/t}-1$. This argument relies on one payment at the specified date and the stated annual effective yield convention.

Worked Example: Finding a Conditional Yield to a Call Date

A hypothetical callable zero-coupon bond costs $880. If called at the end of year 4, it pays $1,065.60. Find the conditional annual effective yield to that call date. Identify a useful calculation check.

Method. Use the price and redemption amount for the year-4 call scenario:

$y_4=(1{,}065.60/880)^{1/4}-1$.

First calculate the ratio:

$1{,}065.60/880=1.2109090909\ldots$.

Taking the fourth root gives approximately $1.0490058$. Thus:

$y_4\approx1.0490058-1=0.0490058$, or 4.9006% to four decimal places as a percentage.

Check. Substituting the unrounded rate into the accumulation equation gives $880(1.0490058)^4\approx1{,}065.60$, as required. The unrounded rate, rather than the displayed rounded percentage, is used for this check.

This method avoids treating the four-year total accumulation factor, about 1.2109091, as though it were an annual rate. The fourth root is needed because the accumulation covers four years.

Calculating Yields for Several Alternative Dates

A single observed price can be used to calculate a conditional yield for each specified redemption scenario. Each yield answers a different question: what annual effective rate equates that price to the payment under that particular redemption outcome? The results need not be equal because both the payment and the time to payment may differ.

Worked Example: Comparing Conditional Yields to Call and Maturity

A hypothetical callable zero-coupon bond is priced at $900. It may be redeemed for $1,012.24 at the end of year 3 or for $1,262.30 at the end of year 5. Calculate both conditional yields and check each against its own payment date.

Method. Apply $y_t=(R_t/P)^{1/t}-1$ separately to the two alternatives.

Year-3 call scenario. The ratio is $1{,}012.24/900=1.124711111\ldots$. Taking the cube root and subtracting 1 gives:

$y_3=(1.124711111\ldots)^{1/3}-1\approx0.039953$, or 3.9953%.

Year-5 maturity scenario. The ratio is $1{,}262.30/900=1.402555556\ldots$. Taking the fifth root and subtracting 1 gives:

$y_5=(1.402555556\ldots)^{1/5}-1\approx0.070001$, or 7.0001%.

Check. With the unrounded conditional yields, $900(1+y_3)^3\approx1{,}012.24$ and $900(1+y_5)^5\approx1{,}262.30$. Each price is accumulated for the number of years attached to its own scenario.

The method avoids using the year-3 call date with the year-5 maturity amount. For example, $(1{,}262.30/900)^{1/3}-1$ would calculate a rate under a different, incorrect assumption: that the maturity amount were paid after 3 years.

Reading the Results Correctly

A conditional yield is tied to its stated redemption outcome. It does not say that the issuer will call the bond, and it is not a forecast of the bond’s realized return unless that payment and date actually occur under the assumed conditions. Likewise, a conditional price at an assumed yield is a scenario value, not automatically the market price of a callable security.

When a problem gives alternative call and maturity terms, organize the data before calculating:

List each outcome.
Record the payment and date for each eligible call or maturity scenario.

Choose the direction of calculation.
For a price, use the stated yield and discount each scenario. For a conditional yield, use the price and solve separately for each scenario.

Match payment to time.
Use the exponent belonging to the date of that payment, not another call date or the final maturity by default.

Check the equation.
For a price, accumulate it to the specified payment. For a yield, confirm that the price accumulated over the scenario term gives its redemption amount.

A frequent exam trap is to average conditional yields or combine call and maturity payments in one equation. Those payments are alternatives: under the terms described, the bond is redeemed under one applicable scenario, not all of them. Without probabilities and a suitable model for call behavior, a probability-weighted price cannot be inferred from the scenario yields alone.

Key takeaway: For a callable zero-coupon bond, a conditional price or yield belongs to one redemption amount at one redemption date. Pair the correct amount and date, state the annual effective yield convention, and verify the result by discounting or accumulating the single payment.

Check Your Understanding

For each problem, treat the stated redemption date as the date the payment is made. Round prices to cents and yields to four decimal places as percentages where needed.

A hypothetical bond is redeemable for $1,102.50 at the end of year 2 or for $1,276.2815625 at the end of year 5. At a 5% annual effective yield, calculate the conditional price for each scenario. (Answer: $1,000.00 for each scenario.)
A bond costs $800 and is redeemable for $968 at the end of year 2 under the specified call scenario. Calculate the conditional annual effective yield and verify the two-year accumulation. (Answer: 10.0000%; $800(1.10)^2=\$968$.)
A callable zero-coupon bond costs $900. It pays $1,089 if called at the end of year 2, or $1,317.69 if held to maturity at the end of year 4. Find the conditional yield for each scenario. (Answer: 10.0000% to year 2 and 10.0000% to year 4.)
At a 4% annual effective yield, a bond has a $1,081.60 call payment at the end of year 2 and a $1,169.85856 maturity payment at the end of year 4. Calculate both conditional prices, then state whether their equality alone establishes a market price for the callable bond. (Answer: Both conditional prices are $1,000.00; no, equality under this stated yield does not by itself establish a market price accounting for call behavior.)

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