Comparing Yield to Call and Yield to Maturity

Why the Assumed Redemption Date Matters

A callable bond may be redeemed before its stated maturity date, subject to the bond’s terms. Its yield therefore depends on which redemption scenario is being evaluated. Yield to call is calculated by assuming redemption on a specified call date; yield to maturity is calculated by assuming the bond remains outstanding until maturity. These are different cash-flow assumptions, not two names for the same calculation.

For either calculation, compare the purchase price with the present value of the cash flows under the selected scenario. The scenario determines both when the final payment occurs and how much that payment is. A call-date equation uses the call redemption amount, while a maturity-date equation uses the maturity redemption amount. In each case, include the coupon scheduled for that date if the bond terms provide for it.

Definition: For a bond purchased at price $P$, yield to call is the annual effective rate $i$ that equates $P$ to the present value of the coupons through a specified call date and the call redemption amount at that date. Yield to maturity is the annual effective rate $i$ that equates $P$ to the present value of the coupons through maturity and the maturity redemption amount at maturity.

The examples below assume annual coupon payments, purchase on a coupon date, and a positive purchase price. The quoted yield is an annual effective rate, so a cash flow at time $k$ years is discounted by $v^k$, where $v=1/(1+i)$. If coupons occur more frequently, the yield rate and discount periods must instead be consistent with that payment schedule. A yield calculation describes a stated cash-flow scenario; it does not establish that the issuer will call the bond.

Build One Equation for Each Scenario

Suppose annual coupons of amount $C$ are paid at the ends of years $1,2,\ldots,n$, and the bond’s purchase price is $P$. If the bond is assumed to be redeemed at the end of year $n$ for amount $R$, its final cash flow is $C+R$: the coupon plus redemption. Thus the yield equation is

P=\sum_{k=1}^{n-1} C v^k+(C+R)v^n.

For yield to maturity, $n$ is the number of years to maturity and $R$ is the maturity redemption amount. For yield to call, replace $n$ with the number of years to the specified call date and replace $R$ with the call redemption amount. Coupons after that assumed redemption date are not included.

This equation follows directly from the present-value principle: each payment is discounted from its payment date to the purchase date, and the sum of those discounted payments must equal the price. The final coupon and redemption are both paid at time $n$, so their combined present value is $(C+R)v^n$. This also explains why changing either the assumed date or redemption amount changes the yield equation.

Worked Comparisons

Worked Example: A Call Premium Produces a Higher Yield to Call

A hypothetical bond costs $1,000, pays an annual coupon of $40, is callable at the end of year 2 for $1,040, and matures at the end of year 4 with a $1,000 redemption amount. Calculate its yield to call and yield to maturity, then compare them.

Method and scenario setup. For yield to call, include the coupons at years 1 and 2 and the $1,040 call payment at year 2. The year-2 total is $40+$1,040=$1,080. For yield to maturity, include coupons at years 1 through 4 and the $1,000 maturity payment at year 4.

Yield to call. Let $i_c$ denote the annual effective yield under the call-date assumption and set $x=1+i_c$. Then

1000=\frac{40}{x}+\frac{1080}{x^2}, \qquad 1000x^2-40x-1080=0, \qquad 25x^2-x-27=0.

The quadratic formula gives $x=(1+\sqrt{2701})/50$. Since $\sqrt{2701}\approx51.9711$, $x\approx1.0594229$, so $i_c\approx0.0594229$, or 5.942% per year to three decimal places. The other quadratic root is negative and cannot equal $1+i_c$ for a positive yield.

Yield to maturity. The annual coupon is 4% of the $1,000 maturity redemption amount. At a 4% annual effective yield, the present value is

40a_{\overline{4}|}+1000v^4 = 40\left(\frac{1-(1.04)^{-4}}{0.04}\right) +\frac{1000}{1.04^4} =1000.

Therefore the yield to maturity is 4.000% per year. As a check on the call calculation, at $i_c\approx5.94229\%$, $40/(1+i_c)+1080/(1+i_c)^2\approx37.756+962.244=1000.000$. Under these terms, the call-date assumption gives a higher yield because the $1,040 call amount exceeds the $1,000 maturity amount and is received earlier. This comparison is conditional on the specified scenario; it does not predict whether the issuer will call the bond.

Worked Example: A Below-Par Call Amount Produces a Lower Yield to Call

A hypothetical bond costs $1,000, pays $60 annually, is callable at the end of year 2 for $980, and matures at the end of year 4 for $1,000. Find both yields and explain why their ordering differs from the previous example.

Yield to call. The two coupons are $60 each, and the year-2 payment also includes the $980 call amount. Thus the final payment in this scenario is $1,040. Let $i_c$ be the yield to call and $x=1+i_c$:

1000=\frac{60}{x}+\frac{1040}{x^2}, \qquad 1000x^2-60x-1040=0.

The positive root is

x=\frac{60+\sqrt{4{,}163{,}600}}{2000} \approx\frac{60+2040.4901}{2000} =1.0502451.

Therefore $i_c\approx0.0502451$, or 5.025% per year to three decimal places.

Yield to maturity. The $60 annual coupon is 6% of the $1,000 maturity redemption amount. At a 6% annual effective yield, the present value of the maturity-scenario cash flows is

60a_{\overline{4}|}+1000v^4 = 60\left(\frac{1-(1.06)^{-4}}{0.06}\right) +\frac{1000}{1.06^4} =1000.

So the yield to maturity is 6.000% per year. The yield-to-call equation checks at approximately $60/1.0502451+1040/(1.0502451)^2=57.129+942.871=1000.000$. Here the call amount is below the maturity amount, and the bond is assumed to be redeemed sooner under the call scenario. Both features affect the present value and help explain why the yield to call is lower in this example. The comparison comes from the cash flows, not from a rule that yield to call must always be above or below yield to maturity.

Worked Example: A One-Year Call and a Two-Year Maturity

A hypothetical bond costs $1,000 and pays $50 at the end of each year. It may be called at the end of year 1 for $1,030; if it is not called, it matures at the end of year 2 for $1,000. Calculate both yields.

Yield to call. Under the call-date assumption, the only payment is the year-1 coupon plus the call amount: $50+$1,030=$1,080. If $i_c$ is the yield to call, then

1000=\frac{1080}{1+i_c}, \qquad 1+i_c=1.08, \qquad i_c=0.08.

The yield to call is 8.000% per year.

Yield to maturity. Under the maturity-date assumption, the year-1 payment is $50 and the year-2 payment is $50+$1,000=$1,050. At 5%,

\frac{50}{1.05}+\frac{1050}{1.05^2} =47.61905+952.38095 =1000.

Therefore the yield to maturity is 5.000% per year. The check shows that the assumed cash flows in each scenario discount to the same purchase price. The yields differ because the first scenario uses a one-year horizon and a $1,030 call amount, whereas the second uses a two-year horizon and a $1,000 maturity amount.

A Common Exam Trap

A frequent error is to omit the scheduled coupon on the redemption date. In the first example, an incorrect yield-to-call equation would be

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

This equation treats $1,040 as the entire year-2 payment, even though the bond also pays the $40 year-2 coupon. In fact, the equation above has a 4.000% root: at 4%, $40/1.04+1040/1.04^2=38.4615+961.5385=1000$. That is not the stated call scenario. The correct final payment is $40+$1,040=$1,080, so the correct equation is $1000=40/(1+i_c)+1080/(1+i_c)^2$, giving 5.942%.

Another related error is to use the call amount but continue adding coupons after the assumed call date. Those later coupons are not part of the call scenario: if the bond is redeemed at year 2, the year-3 and year-4 cash flows in the maturity scenario are excluded. Keep each yield calculation attached to one redemption date and its corresponding redemption amount.

Key takeaway: First choose the redemption scenario. Then include every coupon up to that date, add the applicable redemption amount to the coupon paid on that date, and discount only those scenario cash flows to the purchase price. Yield to call and yield to maturity are conditional calculations; neither alone says which redemption will occur.

Check Your Understanding

For each problem, use annual effective yields and assume purchase occurs on a coupon date. Report yields to three decimal places where needed.

A hypothetical bond is purchased for $1,000, pays $35 annually for two years, and redeems for $1,000 at maturity. Find its yield to maturity. (Answer: 3.500% per year)
A hypothetical bond costs $980, pays $40 annually, and is callable at the end of year 2 for $1,010. Find the yield to call, including the year-2 coupon. (Answer: approximately 5.571% per year)
A hypothetical bond costs $1,000, pays $50 annually, may be called after year 1 for $1,020, and matures after year 3 for $1,000. Find both yields and identify which is higher. (Answer: yield to call $=7.000\%$; yield to maturity $=5.000\%$ per year; yield to call is higher)
A hypothetical bond costs $1,000, pays $40 annually, may be called after year 1 for $1,040, and matures after year 3 for $1,000. Write the two yield equations and calculate both yields. (Answer: yield to call: $1000=1080/(1+i_c)$, so $i_c=8.000\%$; yield to maturity: $1000=40/1.04+40/1.04^2+1040/1.04^3$, so $i_m=4.000\%$)
A hypothetical bond costs $1,000, pays $50 annually, may be called after year 2 for $1,020, and matures after year 3 for $1,000. Set up and solve both yield equations, then compare the results. (Answer: yield to call solves $1000=50/(1+i_c)+1070/(1+i_c)^2$, giving approximately 5.971%; yield to maturity is 5.000% per year; yield to call is higher)

What You'll Learn