Callable-Bond Yield Problems as Scenario Comparisons
A callable bond does not have one guaranteed redemption date: under the stated terms, the issuer may redeem it on one of several eligible dates or leave it outstanding until maturity. A yield calculation therefore starts by specifying a redemption scenario. The coupons and final payment depend on that scenario, so each eligible redemption date produces its own yield equation.
This workshop combines the call schedule, coupon stream, redemption amount, yield to call, yield to maturity, and yield to worst. Assume annual coupon payments, annual effective yield \(y\), and a valuation date that is a coupon date. Unless stated otherwise, the bond is redeemed immediately after paying the coupon due on its redemption date. The price is the observed price on the valuation date, and taxes, transaction costs, and default are not modeled.
The call schedule tells you which dates can be included in the comparison and the redemption amount on each date. It does not mean that every listed date will actually be used. Yield to worst is a comparison of calculated scenario yields, not a forecast of which action the issuer will take.
Set Up Each Scenario Before Solving
For annual coupons of amount \(C\), a price \(P\), and redemption at the end of year \(k\) for amount \(R_k\), the final cash flow is \(C+R_k\). The yield equation is therefore
Here, \(R_k\) is the redemption amount alone. If the terms say the bond is called for $1,020, the last payment is the call price plus that date's coupon, not just $1,020. If a call price is quoted as a percentage of face amount, first convert that quote to dollars. A redemption premium is already part of the call price; it is not added a second time.
The setup follows directly from discounting each promised payment back to the valuation date at the scenario yield. A coupon paid at time \(t\) has present value \(C/(1+y)^t\); at time \(k\), both the last coupon and redemption payment are due, so their combined present value is \((C+R_k)/(1+y)^k\). Adding the present values and setting their total equal to the observed price gives the equation.
List each eligible call date and call price, and include the maturity date and maturity redemption amount.
Keep coupons through the selected redemption date; put the coupon and redemption amount together at the final date.
Use the same observed price in every equation, but use the cash flows for that particular scenario.
Substitute each yield into its own equation, then select the smallest applicable scenario yield.
Worked Problems
Worked Example: Comparing Two Calls With Maturity
A hypothetical bond has face amount $1,000, annual coupon $40, and observed price $1,000. It matures at the end of year 4 for $1,000. The issuer may call it at the end of year 2 for $1,020.50 or at the end of year 3 for $1,031.53. Find the two yields to call, the yield to maturity, and the yield to worst.
Method. Construct three separate cash-flow streams. In each, include the coupon on the redemption date with that scenario's redemption amount.
For the year-2 call, the payments are $40 at year 1 and $1,060.50 at year 2. Thus
At \(y=5\%\), the present values are \(40/1.05=\$38.0952\) and \(1060.50/1.1025=\$961.9048\). Their sum is $1,000.00, so the year-2 yield to call is 5.00%.
For the year-3 call, the payments are $40, $40, and $1,071.53. At 5%, the present values are \(40/1.05=\$38.0952\), \(40/1.1025=\$36.2812\), and \(1071.53/1.157625=\$925.6279\). The sum is approximately $1,000.0043, which rounds to $1,000.00. The year-3 yield to call is approximately 5.00%.
If held to maturity, the payments are $40 at years 1, 2, and 3, followed by $1,040 at year 4. The yield is exactly 4.00%: at 4%, the coupons and $1,000 redemption amount have a present value of $1,000, since the annual coupon rate equals the yield and the bond is priced at face amount.
The three scenario yields are approximately 5.00%, 5.00%, and 4.00%. The yield to worst is therefore 4.00%, the yield to maturity. The year-3 call price was rounded to cents; the small difference in its present-value check does not change the yield when reported to two decimal places.
Common mistake avoided: The final year-2 call payment is $1,060.50, not $1,020.50. The latter is the call price alone; the final $40 coupon is also due.
Worked Example: A Premium-Priced Bond With a Call and Maturity Scenario
A hypothetical bond is priced at $1,040, pays an annual coupon of $50, and has face amount $1,000. It matures at the end of year 3 for $1,000 and may be called at the end of year 2 for $1,030. Find its yield to call, yield to maturity, and yield to worst.
Method. Solve the year-2 call equation and the year-3 maturity equation separately. The year-2 call's final payment is $1,080, consisting of the $1,030 call price and $50 coupon.
Solving the quadratic gives \(y_{\mathrm{call}}\approx0.0433713\), or 4.337% (4.34% to two decimal places). Substitution gives present values of approximately $47.92 and $992.08, totaling $1,040.00.
For maturity, the cash flows are $50 at year 1, $50 at year 2, and $1,050 at year 3. The equation is
Numerical solution gives \(y_{\mathrm{mat}}\approx0.035703\), or 3.570% (3.57% to two decimal places). Substituting the unrounded solution gives present values of approximately $48.28, $46.61, and $945.11, totaling $1,040.00 to the nearest cent.
The yield to worst is 3.57%, the yield to maturity, because it is lower than the 4.34% yield to call. The fact that the bond is callable does not make its call yield automatically the yield to worst.
Common mistake avoided: Do not use $1,030 as the entire year-2 payment or add the $50 coupon twice. The correct final call payment is $1,030+$50=$1,080.
Worked Example: A Call Produces the Yield to Worst
A hypothetical bond has price $980, annual coupon $30, and face amount $1,000. It matures at the end of year 3 for $1,000 and is callable at the end of year 2 for $990. Calculate both scenario yields and identify the yield to worst.
Method. For the call scenario, the year-2 payment is $1,020. The call equation is
Multiplying by \((1+y_{\mathrm{call}})^2\) gives \(980(1+y_{\mathrm{call}})^2-30(1+y_{\mathrm{call}})-1020=0\). The positive-yield solution is approximately \(y_{\mathrm{call}}=0.035625\), or 3.563% (3.56% to two decimal places). Substitution gives present values of approximately $28.97 and $951.03, totaling $980.00.
For maturity, the payments are $30, $30, and $1,030. Solving
gives approximately \(y_{\mathrm{mat}}=0.03717\), or 3.717%. At this yield, the discounted payments are approximately $28.93, $27.89, and $923.18, totaling $980.00. The yield to worst is therefore 3.563%, the yield to the year-2 call.
Common mistake avoided: A lower call price does not by itself establish which scenario has the lowest yield. Calculate the yield for each scenario from the same observed price, then compare the yields.
Interpreting the Comparison Carefully
The yield-to-worst comparison is meaningful only when the candidate scenarios are formed consistently. Use the same price date and annual effective yield convention in every equation. Do not include coupons after the selected redemption date, and do not omit the coupon due on that date. A bond's maturity redemption amount is usually its face amount, but use the amount specified by the problem rather than assuming it.
A useful check is to substitute a computed yield back into its own cash-flow equation. If the present value differs materially from the stated price, check the final payment, the number of coupon periods, and whether the call price is expressed in dollars or as a percentage of face amount. Round the yield only after solving and checking; intermediate rounding can make two close scenario yields appear equal or reverse their ordering.
Check Your Understanding
For each problem, use annual effective yields and include the coupon due on a redemption date.
- A bond priced at $950 pays annual coupons of $30. It can be called at the end of year 2 for $970. Write its yield-to-call equation and find the yield to four decimal places as a percentage. (Answer: \(950=30/(1+y)+1000/(1+y)^2\); approximately 4.1889%)
- A bond priced at $1,000 pays $60 annually, matures in year 3 for $1,000, and is callable at the end of year 2 for $1,010. Find the yield to maturity and yield to call, then state the yield to worst. (Answer: yield to maturity 6.000%; yield to call approximately 6.484%; yield to worst 6.000%)
- A bond priced at $1,025 pays an annual coupon of $25 and is callable at the end of year 1 for $1,015. Find the yield to call. (Answer: \(1040/1025-1\), approximately 1.4634%)
- A zero-coupon bond costs $900, matures in 5 years for $1,200, and is callable at the end of year 2 for $950. Find the yield to each redemption scenario and identify the yield to worst. (Answer: yield to call approximately 2.740%; yield to maturity approximately 5.922%; yield to worst approximately 2.740%)
- A bond priced at $990 pays $20 annually, matures at the end of year 2 for $1,000, and is callable at the end of year 1 for $995. Find both scenario yields and the yield to worst. (Answer: yield to call approximately 2.525%; yield to maturity approximately 2.519%; yield to worst approximately 2.519%)