Yield to Worst Across Callable Redemption Scenarios

Why Yield to Worst Requires Several Calculations

A callable bond may be redeemed before maturity, but the investor does not know in advance which eligible redemption will occur. Each possible redemption date and price creates a different set of cash flows and therefore a different yield. Yield to worst summarizes these specified scenarios by taking the lowest yield among them, under the stated convention.

The comparison must include every eligible call scenario being considered and the maturity scenario. A low call price may produce a low yield, but the price alone does not determine the answer: the purchase price, coupons, and time to redemption all matter. The coupons paid before redemption matter too, including a coupon paid on the redemption date when the bond’s terms provide for it.

Definition: For a stated set of eligible call and maturity scenarios, the yield to worst is the minimum of the yields calculated separately for those scenarios using the specified yield convention. It is a scenario-based measure, not a probability-weighted or expected yield.

The examples below use annual coupon payments, annual effective yields, a purchase on a coupon date, and the stated price as the amount paid. Each call scenario assumes the bond is redeemed on that date at its specified call price, after that date’s coupon is paid. The maturity scenario includes the final coupon and maturity value. These assumptions make the cash-flow timing explicit; a different coupon frequency or price convention must be handled consistently with the bond’s terms.

Set Up One Yield Equation per Scenario

Let $P$ be the purchase price, $C$ the annual coupon, $K$ a specified call price, and $F$ the maturity value. If a bond is called at the end of year $k$, its cash flows are the coupons at the ends of years $1$ through $k$, followed by the call price at year $k$. The yield $y_k$ for that scenario solves

P=\sum_{j=1}^{k}\frac{C}{(1+y_k)^j}+\frac{K}{(1+y_k)^k}.

The final term is the call price, not the call price plus a second redemption amount. The coupon for year $k$ is already included in the sum. For maturity at the end of year $n$, the corresponding equation is

P=\sum_{j=1}^{n}\frac{C}{(1+y_{\mathrm{mat}})^j}+\frac{F}{(1+y_{\mathrm{mat}})^n}.

Here $y_{\mathrm{mat}}$ denotes yield to maturity for this particular cash-flow scenario. Once the yields have been determined, the yield-to-worst calculation is

\mathrm{YTW}=\min\{y_1,y_2,\ldots,y_r,y_{\mathrm{mat}}\},

where $y_1,\ldots,y_r$ are the yields for the eligible call dates included in the comparison. If two or more scenarios have the same lowest yield, that yield is the yield to worst, and the scenarios are tied.

Why the Scenario Equations Work

For a fixed redemption scenario, the yield is the annual effective rate that discounts that scenario’s future payments back to the purchase date to equal the price. To see why, a dollar due at the end of year $j$ has present value $1/(1+y)^j$ dollars under annual effective yield $y$. Thus a coupon $C$ due at year $j$ contributes $C/(1+y)^j$ to the present value, and a redemption payment contributes its discounted amount at its own date. Adding the present values gives the scenario equation.

The minimum is then a direct consequence of the stated convention: among the specified redemption outcomes, the least scenario yield is the worst one for the investor when yields are compared on the same basis. It does not assert that the lowest-yield scenario is the one most likely to occur. The calculation also does not include a call date or price that is not part of the scenarios being analyzed.

Worked Yield-to-Worst Calculations

Worked Example: A Tie Between a Call Scenario and Maturity

A bond costs $1,000, pays a $40 coupon at each year-end, and has a $1,000 maturity value at the end of year 3. It may be called at the end of year 1 for $1,010 or at the end of year 2 for $1,000. Find the yield to worst, using annual effective yields.

Method. Calculate a yield for each of the two call scenarios and for maturity. For a call at the end of year 1, the investor receives the $40 coupon and the $1,010 call price at that date:

1000=\frac{40+1010}{1+y_1}, \qquad y_1=\frac{1050}{1000}-1=0.05=5\%.

For the year-2 call, the cash flows are $40 at year 1 and $40 plus $1,000 at year 2:

1000=\frac{40}{1+y_2}+\frac{1040}{(1+y_2)^2}.

At $y_2=4\%$, the two present values are $40/1.04=38.4615$ and $1040/1.0816=961.5385$. Their sum is $1,000, so the year-2 call yield is 4%.

For maturity, the cash flows are $40 at years 1 and 2, then $1,040 at year 3:

\frac{40}{1.04}+\frac{40}{1.0816}+\frac{1040}{1.124864} =38.4615+36.9822+924.5563 =1000.0000.

Therefore, the maturity yield is also 4%. Comparing all three results gives $\min\{5\%,4\%,4\%\}=4\%$. The yield to worst is 4%, tied between the year-2 call and maturity. This comparison avoids assuming that the earliest call must be the worst scenario.

Worked Example: Maturity Has the Lowest Yield

A bond costs $1,040, pays a $40 annual coupon, and matures in two years for $1,000. It can be called at the end of year 1 for $1,030. Find the yield to worst.

Method. First calculate the one-year call yield. The year-1 cash flow is the coupon plus the call price, or $1,070:

y_{\mathrm{call}}=\frac{1070}{1040}-1 =0.02884615\ldots =2.8846\%.

Next solve the maturity equation, which includes the year-1 coupon and the year-2 coupon plus maturity value:

1040=\frac{40}{1+y_{\mathrm{mat}}}+ \frac{1040}{(1+y_{\mathrm{mat}})^2}.

Set $x=1+y_{\mathrm{mat}}$, multiply through by $x^2$, and rearrange:

1040x^2-40x-1040=0, \qquad x=\frac{1+\sqrt{2705}}{52} =1.0194157\ldots.

Thus $y_{\mathrm{mat}}=x-1=0.0194157\ldots$, or 1.9416% to four decimal places as a percentage. Substitution checks the result: at this $x$, $40/x+1040/x^2=1040$ to the shown precision. Comparing 1.9416% with the call yield of 2.8846% gives yield to worst of 1.9416%, at maturity. The mistake avoided here is selecting the call yield merely because the bond is callable; maturity is an eligible scenario too.

Worked Example: A Later Call Has the Lowest Yield

A bond costs $1,020, pays $30 annually, and has a maturity value of $1,000 at the end of year 3. The stated call scenarios are redemption for $1,010 at the end of year 1 and for $995 at the end of year 2. Find the yield to worst using annual effective yields.

Method. For the year-1 call, total proceeds at redemption are $30+$1,010=$1,040. Therefore,

y_1=\frac{1040}{1020}-1 =0.0196078\ldots =1.9608\%.

For the year-2 call, the year-1 payment is $30 and the year-2 payment is $30+$995=$1,025. With $x=1+y_2$, solve:

1020=\frac{30}{x}+\frac{1025}{x^2}, \qquad 1020x^2-30x-1025=0,

x=\frac{30+\sqrt{4{,}182{,}900}}{2040} =1.0172617\ldots, \qquad y_2=1.7262\%.

A substitution check gives $30/1.0172617+1025/(1.0172617)^2\approx29.4909+990.5091=1,020.0000$. For maturity, solve

1020=\frac{30}{1+y_{\mathrm{mat}}} +\frac{30}{(1+y_{\mathrm{mat}})^2} +\frac{1030}{(1+y_{\mathrm{mat}})^3}.

Numerical solution gives $y_{\mathrm{mat}}\approx2.3024\%$; substituting this yield into the equation gives a present value of approximately $1,020. Comparing 1.9608%, 1.7262%, and 2.3024% shows that yield to worst is 1.7262%, for the year-2 call. The first eligible call is not necessarily the lowest-yield scenario.

Common Mistakes and Practical Checks

A reliable calculation begins by listing the cash flows for each outcome before solving for its yield. In particular, write down the coupon and redemption payment separately when both occur on the same date. Then compare all the resulting yields, not just the call yields.

Omitting the redemption-date coupon. In the first example, using $1010/1000-1=1\%$ for the year-1 call is wrong because it leaves out the $40 coupon. The correct calculation is $(1010+40)/1000-1=5\%$.
Comparing redemption amounts instead of yields. A lower call price does not by itself prove that its yield is lower. The price paid and the timing and amount of every payment also enter the equation.
Leaving maturity out of the comparison. Yield to worst is the minimum over the stated call scenarios and maturity when maturity is included in the convention being used.
Counting later coupons after a call. A call scenario stops at its redemption date. Payments after that date belong to a different scenario and must not be added to the called-bond cash flows.
Treating yield to worst as a forecast. It is the lowest yield among the specified scenarios, not the expected yield and not a statement about which redemption is most likely.

For a final arithmetic check, substitute each solved yield into its own present-value equation and confirm that the discounted cash flows reproduce the purchase price, up to rounding. Keep the yield convention and time units consistent across scenarios. This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, calculate the yield for every stated redemption scenario and select the lowest yield.

A bond costs $950, pays $30 annually, matures in two years for $1,000, and can be called at the end of year 1 for $930. Compare the one-year call yield with the maturity yield. (Answer: 1.0526%, year-1 call)
A bond costs $1,000, pays $50 annually, matures in two years for $1,000, and can be called at the end of year 1 for $980. Find the yield to worst. (Answer: 3.0000%, year-1 call)
A bond costs $990, pays $30 annually, matures in two years for $1,000, and can be called at the end of year 1 for $950. Include the year-1 coupon when calculating the call yield. (Answer: -1.0101%, year-1 call)
A bond costs $1,000, pays $40 annually, matures in three years for $1,000, and is callable at the end of year 1 for $1,020 and at the end of year 2 for $990. Calculate the year-2 call yield and compare it with the other scenarios. (Answer: approximately 3.5086%, year-2 call)
A bond costs $1,020, pays $20 annually, matures in two years for $1,000, and can be called at the end of year 1 for $1,000. Calculate the call yield and determine whether the positive maturity yield changes the result. (Answer: 0.0000%, year-1 call)

What You'll Learn