Step-Down Call Prices in Callable Bond Yield Equations

Why Step-Down Call Prices Need Separate Yield Equations

A callable bond may give the issuer the right to redeem the bond on several dates, with the permitted call price decreasing over time. A lower later call price does not by itself tell you whether the yield for that later redemption scenario is higher or lower. The yield calculation depends on the bond’s price, every coupon paid along the way, the redemption date, and the call price assigned to that date.

The safe method is to treat each eligible call date as its own cash-flow scenario. For each scenario, include all coupons through that date and redeem the bond at that date’s stated call price. Then solve a separate yield equation. These are conditional yields: each one assumes that redemption occurs on a specified date. They do not predict which date the issuer will choose.

Definition: A conditional yield to call is the yield per coupon period that equates a bond’s price at the valuation date to the present value of the coupons and call-price payment, assuming the bond is redeemed on one specified eligible call date. A step-down call schedule has call prices that decrease at successive eligible dates.

Assume for now that the valuation date is a coupon date, coupons are level, and the bond is redeemed together with the coupon due on the assumed call date. Let $P$ be the bond’s price, $C$ the coupon paid each period, $R_t$ the call price at the end of period $t$, and $y_t$ the conditional yield per coupon period for redemption at that date. If the coupon periods are equal, the equation is

P=\sum_{k=1}^{t-1}\frac{C}{(1+y_t)^k}+\frac{C+R_t}{(1+y_t)^t}.

The call price $R_t$ is not a substitute for the coupon due at time $t$: under the stated assumption, the final cash flow is $C+R_t$. If the bond’s terms say the coupon is not paid on redemption, use the contractual cash flows instead of adding that coupon.

Why the Conditional-Yield Equation Holds

The equation follows from the present-value principle used for bond pricing. For a particular redemption date, discount each payment in that scenario at the same yield per period. The coupons before redemption are $C$ at the ends of periods $1$ through $t-1$; the final payment is the coupon plus the call price at period $t$. Adding their present values gives the scenario’s price.

\underbrace{\frac{C}{1+y_t}+\frac{C}{(1+y_t)^2}+\cdots+\frac{C}{(1+y_t)^{t-1}}}_{\text{coupons before the call}} +\underbrace{\frac{C+R_t}{(1+y_t)^t}}_{\text{coupon and call price at the call}} =P.

This derivation also explains why the equation changes when the assumed call date changes. A later date adds coupon payments and discounts the redemption payment for more periods; it may also have a different call price. Keeping the cash-flow timeline explicit prevents the call schedule from being treated as if it were a single redemption payment.

If a bond pays coupons $m$ times per year and the equation gives a yield $y$ per coupon period, the corresponding nominal annual yield convertible $m$-thly is $m y$. The per-period yield and nominal annual yield are different measures; label which one is being reported.

Set Up Each Redemption Scenario

A useful way to organize the work is to make a timeline for every eligible date before solving. The call price must come from the row for that date, not from the first call date, the maturity date, or an average of the scheduled prices.

List the eligible dates and prices.
Record each call date and its own contractual call price $R_t$.

Write the cash flows through that date.
Include each coupon paid before redemption and the contractual payment at redemption.

Set price equal to present value.
Discount the scenario’s payments using one yield per coupon period.

Solve and label the yield.
State the assumed redemption date and, if needed, convert the periodic yield to a nominal annual yield.

Worked Examples

Worked Example: Two Annual Call Dates With a Step-Down Price

A hypothetical bond is priced at $P=\$1{,}000$ immediately after a coupon date. It pays an annual coupon of $\$30$. The issuer may call it after one year for $\$1{,}080$, or after two years for $\$1{,}041$. The coupon due on a call date is paid with the call price. Find the conditional yield for each date.

Method. Write one equation for each assumed redemption date, using annual periods and the price of $\$1{,}000$.

Call after one year. There is one payment: the $\$30$ coupon plus the $\$1{,}080$ call price. Thus

1000=\frac{30+1080}{1+y_1} =\frac{1110}{1+y_1}.

Therefore $1+y_1=1.11$, so $y_1=0.11$, or 11% per year.

Call after two years. The bond pays $\$30$ after year one. At the end of year two it pays the $\$30$ coupon and the $\$1{,}041$ call price, for a final cash flow of $\$1{,}071$. The equation is

1000=\frac{30}{1+y_2}+\frac{1071}{(1+y_2)^2}.

At $y_2=0.05$, the first payment has present value $30/1.05=\$28.5714286$. The second has present value $1071/(1.05)^2=1071/1.1025=\$971.4285714$. Their sum is $\$1{,}000.00$, so the conditional yield is 5% per year.

Check and trap avoided. The call price decreases from $\$1{,}080$ to $\$1{,}041$, and the conditional yield also decreases from 11% to 5% in this example. The second equation uses the second date’s price and includes both coupons. Using $\$1{,}080$ at year two, or omitting the year-one coupon, would describe a different cash-flow scenario.

Worked Example: Semiannual Coupons and Two Different Periodic Yields

A hypothetical bond costs $\$1{,}000$ and pays $\$25$ every half-year. It can be called after one half-year for $\$1{,}100$, or after one year for $\$1{,}051.25$. A coupon is paid at redemption. Find each conditional yield per half-year and its nominal annual equivalent convertible semiannually.

Method. First solve using half-year periods. Convert to a nominal annual yield by multiplying the half-year yield by two.

Call after one half-year. The only payment is $25+1100=\$1{,}125$. Therefore

1000=\frac{1125}{1+y_1}, \qquad 1+y_1=\frac{1125}{1000}=1.125, \qquad y_1=0.125.

The conditional yield is 0.125, or 12.5% per half-year. Its nominal annual equivalent convertible semiannually is $2(0.125)=0.25$, or 25.00% per year nominal.

Call after one year. There are two half-year periods. The first cash flow is $\$25$; the second is $25+1051.25=\$1{,}076.25$. At a yield of $0.05$ per half-year, the first payment has present value $25/1.05=\$23.8095238$. The second has present value $1076.25/(1.05)^2=1076.25/1.1025=\$976.1904762$. Hence

\frac{25}{1.05}+\frac{1076.25}{(1.05)^2} =23.8095238+976.1904762 =1000.

So the second conditional yield is 0.05, or 5% per half-year. Its nominal annual equivalent convertible semiannually is $2(0.05)=0.10$, or 10.00% per year nominal.

Check and trap avoided. Both discounted cash-flow totals equal the bond price. In the first scenario, $1125/1000-1=0.125$, not 12.5 as a decimal rate. Also, 12.5% per half-year converts to 25.00% nominal annually, not 12.5% nominally.

Worked Example: Three Call Dates and Three Separate Equations

A hypothetical bond has a price of $\$1{,}000$ just after an annual coupon date and pays $\$40$ annually. It is callable after one year for $\$1{,}100$, after two years for $\$1{,}041.20$, or after three years for $\$1{,}031.525$. The coupon is paid on each call date. Find the conditional yield for each date.

Method. Build the complete cash-flow stream through each assumed call date. The final payment includes both the annual coupon and that date’s call price.

One-year call. The payment is $40+1100=\$1{,}140$, so

1000=\frac{1140}{1+y_1} \quad\Longrightarrow\quad y_1=0.14.

The conditional yield is 14% annually.

Two-year call. The payments are $\$40$ at year one and $40+1041.20=\$1{,}081.20$ at year two. At $y_2=0.06$, their present values are $40/1.06=\$37.7358491$ and $1081.20/(1.06)^2=1081.20/1.1236=\$962.2641509$. Their sum is $\$1{,}000.00$, so the conditional yield is 6% annually.

Three-year call. The cash flows are $\$40$, $\$40$, and $40+1031.525=\$1{,}071.525$ at years one, two, and three. At $y_3=0.05$, the present values are

\frac{40}{1.05}=38.0952381,\qquad \frac{40}{(1.05)^2}=36.2811791,\qquad \frac{1071.525}{(1.05)^3}=925.6235828.

Adding them gives $38.0952381+36.2811791+925.6235828=\$1{,}000.00$. Thus the conditional yield is 5% annually.

Check and trap avoided. The call prices step down from $\$1{,}100$ to $\$1{,}041.20$ to $\$1{,}031.525$, while the yields in these scenarios are 14%, 6%, and 5%. Each result checks by discounting that date’s entire cash-flow stream to $\$1{,}000$. A declining call price does not eliminate the need to count intervening coupons or determine a new yield.

Comparing Conditional Yields Correctly

A comparison is meaningful only when the yield conventions match. If coupons are semiannual, compare yields per half-year or convert all of them consistently to nominal annual yields convertible semiannually. Do not compare one scenario’s periodic yield with another scenario’s annualized yield.

Keep the distinction between call price and total cash flow at redemption clear. The call price is the amount paid to redeem the bond; when a coupon is also due, the cash flow at that date is their sum. The timing matters as well: a coupon paid before a later call remains in the later scenario’s stream.

The conditional yields rely on the stated cash flows and the assumption that redemption occurs on the selected date. They are not probabilities of a call, and a lower conditional yield does not guarantee that the issuer will call the bond on that date. The call schedule determines the possible scenarios; the yield equation values each scenario separately.

Common Mistakes

Using one call price for every date. Read the scheduled price for the specific assumed redemption date.
Leaving out coupons before a later call. A later redemption scenario includes every coupon paid up to that date.
Omitting the coupon due with redemption. Include it when the bond’s terms provide for that payment.
Calling the call price the final total payment. If a coupon is due at redemption, the final cash flow is coupon plus call price.
Mixing rate units. Solve per coupon period, then convert to the requested annual convention.
Assuming yields must follow the call-price schedule. The yield also depends on the purchase price, coupon stream, and time to redemption.

Key takeaway: For each eligible call date, pair that date’s call price with all coupons through redemption, equate the present value to the bond price, and report the yield using a clearly stated period and annual convention.

Check Your Understanding

For each problem, use the stated coupon and redemption assumptions. Treat the valuation date as immediately after a coupon date.

A bond costs $\$980$, pays an annual coupon of $\$35$, and can be called after one year for $\$1{,}000$. Find its conditional annual yield. (Answer: $1035/980-1=0.0561224$, or approximately 5.6122%.)
A bond costs $\$1{,}000$, pays $\$40$ annually, and can be called after two years for $\$1{,}020.50$. A coupon is paid on the call date. Verify the conditional yield of 5% by evaluating the two discounted cash flows. (Answer: $40/1.05+1060.50/(1.05)^2=38.0952381+961.9047619=\$1{,}000$, so 5% annually.)
A bond costs $\$1{,}200$, pays $\$24$ every half-year, and can be called after one half-year for $\$1{,}212$, including the redemption-date coupon. Find the periodic yield and nominal annual yield convertible semiannually. (Answer: $1236/1200-1=0.03$, or 3% per half-year and 6% nominal annually convertible semiannually.)
A bond costs $\$990$, pays $\$25$ annually, and is callable after one year for $\$1{,}100$ or after two years for $\$1{,}040.225$. Coupons are paid through redemption. Find both conditional annual yields. (Answer: First call, $1125/990-1=0.1363636$, or approximately 13.6364%; second call, $25/1.05+1065.225/(1.05)^2=23.8095238+966.1904762=\$990$, so 5%.)
A two-year call scenario has annual coupons of $\$30$, a call price of $\$1{,}041$, and a bond price of $\$1{,}000$. Write the conditional-yield equation, identifying the final cash flow. (Answer: $1000=30/(1+y)+1071/(1+y)^2$; the year-two payment is the $\$30$ coupon plus the $\$1{,}041$ call price.)

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