Calculating Yield to First Call

What Yield to First Call Measures

A callable bond gives its issuer the right to redeem the bond before its stated maturity, subject to the terms of the bond. Yield to first call is the yield implied by the purchase price if the bond is assumed to be redeemed on its first eligible call date at the stated call price. It is a conditional calculation: it does not establish that the issuer will call the bond.

The calculation uses the same present-value principle as other bond-yield calculations. The key is to use the cash flows for the assumed redemption scenario: include each coupon paid before the first call date, and include both the coupon due on the call date and the call price if the bond pays that coupon. The call price, not automatically the face amount, is the principal redemption cash flow in this scenario.

Definition: The yield to first call is the interest rate that equates a bond’s purchase price to the present value of its contractual coupon payments through its first eligible call date and the assumed redemption payment at that date. The calculation assumes redemption at the stated call price on that date and uses a specified yield convention.

The result depends on the assumed date and amount of redemption, the timing of coupons, and the yield convention. If the bond is not called on that date, the actual investor’s realized return can differ. This tutorial focuses on the single scenario in which redemption occurs at the first eligible call date.

Set Up the First-Call Cash Flows

Suppose a bond is purchased at time 0, makes coupon payments at the ends of periods, and is first callable after $n$ coupon periods. Let $P$ be its purchase price, $C_k$ the coupon paid at the end of period $k$, and $K$ the call price paid at the first call date. If the coupon due on that date is paid along with redemption, the assumed cash flow at time $n$ is $C_n+K$.

For an effective yield $i$ per period, the yield equation is the present value of every assumed payment:

P=\sum_{k=1}^{n-1}\frac{C_k}{(1+i)^k}+\frac{C_n+K}{(1+i)^n}.

For level coupons $C$, the same equation can be written using the annuity-immediate factor $a_{\overline{n}|}$:

P=C\,a_{\overline{n}|}+K(1+i)^{-n}, \qquad a_{\overline{n}|}=\frac{1-(1+i)^{-n}}{i}.

Here the annuity factor accounts for $n$ coupons, including the coupon at time $n$, while the call-price term accounts only for the redemption principal. If coupons are not level, use the individual $C_k$ terms instead. If the bond’s terms do not provide a coupon on the redemption date, do not include one.

Locate the first eligible call date.
Count the coupon periods from the purchase date to the first date on which the bond may be called under its terms.

List payments through that date.
Include each coupon that is due before redemption. At the call date, include the coupon if payable and add the stated call price.

Use the stated yield convention.
For an effective annual yield, discount annual cash flows by powers of $1+i$. For a nominal annual yield convertible semiannually, discount each half-year cash flow using the nominal rate divided by 2.

Solve and check.
Find the rate that makes the present value equal to the purchase price, then substitute it back into the equation to confirm the price.

Why the Yield Equation Works

The equation follows from the definition of a yield as the rate at which the assumed future payments have a present value equal to the amount invested. A payment $X_k$ received $k$ periods from now has present value $X_k/(1+i)^k$ when the effective rate per period is $i$. This is because accumulating that present value for $k$ periods gives $X_k$:

\frac{X_k}{(1+i)^k}(1+i)^k=X_k.

Adding the present values of all the assumed payments gives the total amount that would accumulate to those payments under that yield. Equating that sum to $P$ gives the yield equation. The equation does not apply unchanged if the payment dates or yield period differ from the periods used in the powers; those periods must first be aligned.

Worked Example: Solve an Exact Annual Yield to First Call

A bond is purchased for $100. It pays a $10 coupon at the end of each year and is first callable at the end of year 2 for $100. Assume both coupons are paid, including the coupon at the call date. Find the effective annual yield to first call.

Method. Write the two cash flows under the first-call assumption. The year-1 payment is $10. The year-2 payment is the $10 coupon plus the $100 call price, or $110. Thus:

100=\frac{10}{1+i}+\frac{110}{(1+i)^2}.

Set $v=1/(1+i)$. Then $100=10v+110v^2$, or, after dividing by 10, $10=v+11v^2$. Rearranging gives:

11v^2+v-10=0 \qquad\Longrightarrow\qquad (11v-10)(v+1)=0.

The positive discount factor is $v=10/11$. Therefore $1+i=1/v=11/10$, so $i=0.10$, or an effective annual yield of 10%.

Check. At 10%, the present values are $10/1.10=9.090909$ and $110/1.10^2=90.909091$. Their sum is $100.000000, equal to the purchase price.

Common mistake avoided. The year-2 payment is not just the $100 call price. Because the coupon is due on that date, the correct payment is $10+$100=$110.

Solving When the Yield Is Not an Exact Root

With several cash flows, the yield equation may not factor into a convenient expression. A numerical solver, financial calculator, or careful trial-and-check method can be used. For a conventional sequence of positive payments, increasing the yield decreases each payment’s present value, so the calculated price falls as the trial yield rises. This gives a practical way to bracket a solution: test rates on either side of the rate that equates present value to price, then refine the rate.

Do not round intermediate present values too early. Keep enough digits when evaluating the equation, and report the yield at the requested precision. The check is always to substitute the reported yield into the same first-call cash-flow equation and see whether it reproduces the price to the stated precision.

Worked Example: Find an Annual Yield from a Rounded Price

A bond pays a $6 coupon annually and is first callable at the end of year 3 for $102. Its purchase price is $99.01. The coupon is paid on the call date. Find the effective annual yield to first call, to two decimal places.

Method. The assumed payments are $6 at the end of year 1, $6 at the end of year 2, and $108 at the end of year 3. The final $108 consists of the $6 coupon and $102 call price. The equation is:

99.01=\frac{6}{1+i}+\frac{6}{(1+i)^2}+\frac{108}{(1+i)^3}.

This cubic equation can be solved numerically. Testing $i=7\%$ gives:

\begin{aligned} \frac{6}{1.07}&=5.607477,\\ \frac{6}{1.07^2}=\frac{6}{1.1449}&=5.240632,\\ \frac{108}{1.07^3}=\frac{108}{1.225043}&\approx 88.160170,\\ \text{present value}&\approx 5.607477+5.240632+88.160170\\ &=99.008279. \end{aligned}

The computed price at 7% is $99.008279, which rounds to the stated $99.01. Solving with the unrounded price implied by the quoted amount gives a yield just below 7%; to two decimal places, the yield is 7.00%.

Check. Substitution of 7% reproduces $99.01 to the nearest cent. The small difference between $99.008279 and $99.01 is due to the purchase price being stated only to the nearest cent.

Common mistake avoided. Discounting only three coupons, or using $102 rather than $108 at year 3, omits a payment due under the stated terms and produces an incorrect yield.

Quoted Yields with Semiannual Coupons

Some bonds pay coupons semiannually and quote yield as a nominal annual rate convertible twice per year. If that quoted yield is $j^{(2)}$, the rate used to discount each half-year period is $j^{(2)}/2$. The exponent is the number of half-year periods, not the number of years. A first call date three years after purchase is six half-year periods away.

This convention is different from an effective annual yield. Do not divide an effective annual yield by 2 to obtain its half-year rate; instead, use the effective-rate conversion appropriate to that yield convention. In the example below, the requested yield is specifically a nominal annual rate convertible semiannually.

Worked Example: Calculate a Nominal Yield to a Semiannual Call Date

A hypothetical bond is purchased for $95.5482. It pays a coupon of $3 every six months and is first callable in three years for $101. Coupons are paid on all six coupon dates, including the call date. Find the nominal annual yield convertible semiannually, to two decimal places.

Method. There are six half-year periods. Let $r$ be the yield per half-year period, so the nominal annual yield is $j^{(2)}=2r$. The payment in period 6 is the $3 coupon plus the $101 call price, or $104. The yield equation is:

95.5482 =3\left(\frac{1-(1+r)^{-6}}{r}\right)+101(1+r)^{-6}.

Solving gives $r\approx0.0400$, or about 4.00% per half-year. Therefore:

j^{(2)}=2r\approx2(0.0400)=0.0800.

The nominal annual yield convertible semiannually is 8.00%.

Check. At $r=0.04$, $(1.04)^6=1.2653190185$, so $(1.04)^{-6}\approx0.790314528$. The coupon present value is $3(1-0.790314528)/0.04\approx15.726411$, and the call-price present value is $101(0.790314528)\approx79.821767$. Their sum is approximately $95.548178, which rounds to the stated $95.5482.

Common mistake avoided. Using three periods because the call date is three years away would discount the six semiannual payments incorrectly. The exponent and periodic yield must use the same half-year unit.

Common Traps and a Reliable Check

A useful final check is to list the assumed cash flows with their dates before entering anything into a calculator. This catches most structural errors before they affect the numerical solution. The following distinction is especially important: the call price is the redemption principal, while the coupon due at redemption is a separate payment.

Key takeaway: Yield to first call is found from the purchase price and the cash flows through the first eligible call date, assuming redemption at the stated call price. Include the final coupon only when it is payable, match the yield period to the payment periods, and verify the result by discounting the cash flows back to the purchase price.

For instance, if a bond pays a $9 annual coupon and is first callable after two years for $104, then at an effective annual yield of 8% the first-call price implied by those cash flows is:

\frac{9}{1.08}+\frac{9+104}{1.08^2} =\frac{9}{1.08}+\frac{113}{1.08^2} \approx 8.333333+96.879287 =\$105.21.

An arithmetic slip in this check can be misleading: the second discounted payment is about $96.8793, so the total is about $105.21, not $105.30. A correct cash-flow setup still needs an accurate evaluation.

Do not substitute face value for call price. Use the stated call price for the assumed redemption date.
Do not omit or double-count the call-date coupon. Check whether the bond pays it on that date, then include it exactly once.
Do not confuse yield conventions. Use annual periods for an effective annual yield and the corresponding number of coupon periods for a periodic or nominal quote.
Do not interpret the result as a forecast. It is the yield conditional on redemption at the specified first call date and price.

Check Your Understanding

For each problem, use the stated first-call scenario and yield convention. Answers are rounded as indicated.

A bond costs $100, pays a $10 annual coupon, and is first callable after two years for $100. The coupon is paid on the call date. Find the effective annual yield to first call. (Answer: 10.00%)
A zero-coupon bond costs $100 and is first callable in two years for $121. Find the effective annual yield to first call. (Answer: 10.00%)
A bond costs $100, pays $2.50 every six months, and is first callable after two years for $100. All four coupons, including the call-date coupon, are paid. Find the nominal annual yield convertible semiannually. (Answer: 5.00%)
A bond costs $100.6612, pays a $8 annual coupon, and is first callable after two years for $105. The coupon is paid on the call date. Find the effective annual yield to first call, to two decimal places. (Answer: 10.00%)
A bond costs $92.9747, pays a $4 annual coupon, and is first callable after three years for $98. The coupon is paid on the call date. Find the effective annual yield to first call, to two decimal places. (Answer: 6.00%)

What You'll Learn