Why Compare Two Yields?
A callable bond gives its issuer the right, subject to the bond’s terms, to repay the bond before its stated maturity. That possibility creates two distinct cash-flow scenarios for an investor: the bond remains outstanding until maturity, or it is called on an eligible date. Yield to maturity and yield to call quantify those scenarios separately. Comparing them is useful only when the assumed payment dates and redemption amounts are kept explicit.
A yield is the rate that makes the present value of an assumed set of future bond payments equal to the bond’s observed price. Yield to maturity uses payments through the stated maturity date. Yield to call instead assumes payments stop at a specified call date and includes the call redemption amount on that date. Neither calculation, by itself, predicts which scenario will occur.
Set Up the Cash Flows Before Solving
For a bond with annual coupon payments, let \(C\) be the coupon each year, \(F\) the maturity redemption amount, \(K\) the call redemption amount, \(P\) the observed price, and \(n\) the number of years until maturity. If a specified call occurs after \(m\) years, the yield equations are:
Here \(j\) and \(j_c\) are the annual yield rates for the respective scenarios. The coupon on the call date is included if the bond’s terms provide for that coupon to be paid before redemption. The formula assumes the stated coupons and redemption amount are paid as scheduled under the scenario being valued, and that one yield rate is used to discount all payments in that scenario.
For semiannual coupons, use a rate \(r\) per half-year and count the number of half-year periods. If the annual yield is quoted as a nominal rate convertible twice a year, it is \(2r\). Do not substitute an annual rate directly into an equation whose periods are half-years.
Why the Yield Equation Works
The equation is a present-value balance. Under a level yield \(j\), a payment \(C\) one year from now has present value \(C(1+j)^{-1}\), a payment two years from now has present value \(C(1+j)^{-2}\), and so on. Adding the coupon present values gives the annuity factor; adding the final redemption present value gives the full price equation:
For YTC, the same reasoning ends at the call date: there are \(m\) coupon payments and the call redemption amount is paid at time \(m\). Replacing the maturity cash flows with those call-scenario cash flows gives the YTC equation. This derivation also shows why the call amount is not an extra payment in addition to the maturity redemption amount: in the call scenario, it replaces the later principal payment.
Worked Examples
Worked Example: Compare YTM and YTC for an Annual-Pay Bond
A hypothetical bond has a price of $981.67, a $50 annual coupon, and a $1,000 maturity redemption in two years. It can be called immediately after the first annual coupon for $1,020. Calculate its YTM and its YTC for that call date. Report yields to two decimal places.
Method. First value the maturity scenario, then replace the second-year coupon and maturity redemption with the first-year coupon and the call redemption for the call scenario. The call payment is $1,020, not $1,000.
YTM. The price is given rounded to cents. The unrounded price consistent with a 6% YTM is:
This rounds to the stated $981.67. Therefore the YTM is 6.00% to two decimal places. The two-year cash flow in this equation is $1,050, consisting of the $50 coupon plus the $1,000 maturity redemption.
YTC. If the bond is called after one year, its only assumed payment is the $50 coupon plus the $1,020 call redemption, or $1,070 total. Thus:
The YTC is 9.00% to two decimal places. As a check, discounting the $1,070 call-date payment at the unrounded rate gives approximately $981.67. The YTC exceeds the YTM here because the assumed call payment is relatively large and arrives earlier than the maturity cash flows. Common mistake avoided: using $1,000 instead of the $1,020 call redemption would omit the call premium and give a different yield.
Worked Example: Compare Yields With Semiannual Coupons
A hypothetical four-year bond has a $1,000 maturity redemption and pays $40 every six months. Its price is set to the value that gives a 4.00% nominal annual YTM convertible semiannually. It is callable after four half-year periods for $1,030. Calculate its YTM and YTC, reporting the nominal annual yields convertible semiannually to two decimal places.
Method. Work first in half-year periods. The YTM per half-year is \(r=0.02\), because the stated nominal annual yield is 4.00% convertible twice a year. There are eight periods to maturity, so the price is:
The discount factor is \((1.02)^{-8}\approx0.853490\), and therefore \(a_{\overline{8}|\,0.02}\approx7.3255\). The coupon present value is \(40(7.3255)\approx$293.02\); the maturity redemption present value is \(1{,}000(0.853490)\approx$853.49\). Hence \(P\approx$1,146.51\), and the YTM is 4.00% nominal convertible semiannually.
For the call scenario there are four coupons, followed by the $1,030 call redemption in period four. Solve:
This equation can be solved with a financial calculator or by numerical trial. The positive solution is approximately \(r=0.009856\) per half-year. The corresponding nominal annual YTC is \(2r\approx0.01971\), or 1.97%.
Check. At \(r=0.009856\), \((1+r)^{-4}\approx0.96153\) and \(a_{\overline{4}|\,r}\approx3.9034\). The coupon present value is approximately \(40(3.9034)=$156.14\), and the call redemption present value is approximately \(1{,}030(0.96153)=$990.37\). Their sum is approximately $1,146.51, matching the price. The YTC is below the YTM because the bond is bought at a premium and, in the call scenario, the principal is repaid earlier for less than the purchase price. Common mistake avoided: treating 4.00% as the half-year rate would use a rate twice as large as the stated nominal yield convention allows.
Worked Example: Identify the Scenario Behind Each Yield
A hypothetical bond costs $920.15, pays a $60 annual coupon, has a $1,000 maturity redemption in five years, and is callable after three years for $1,025. Calculate YTM and YTC to three decimal places. Use the price \(P=$920.1458\) before rounding.
Method. Use five annual coupons and the $1,000 maturity redemption for YTM. For YTC, use three annual coupons and the $1,025 call redemption. Each equation is solved for its own annual yield.
For YTM, the price equation is:
At \(j=0.08\), the discount factor is \((1.08)^{-5}\approx0.680583\), and \(a_{\overline{5}|\,0.08}\approx3.992710\). The coupon present value is \(60(3.992710)\approx$239.56\); the redemption present value is \(1{,}000(0.680583)\approx$680.58\). Their sum is approximately $920.15, so the YTM is 8.000%.
For the three-year call scenario, solve:
Numerical solution gives \(j_c\approx0.099645\), or a YTC of 9.965%. At this rate, the coupon present value is approximately $149.30 and the call redemption present value is approximately $770.85. Their sum is approximately $920.15, confirming the solution to rounding.
The YTC is greater than the YTM in this example. That comparison follows from the specified cash flows and price; it is not a universal rule for callable bonds. Common mistake avoided: discounting the $1,025 call amount for five years, or including coupons after the call date, would value a different scenario.
Interpreting the Comparison
YTM and YTC answer different conditional questions. YTM asks what yield is implied if the bond remains outstanding through maturity and pays its stated maturity redemption. A specified YTC asks what yield is implied if the bond is called on that date and pays that call redemption. A callable bond’s actual cash flows may follow neither scenario exactly if it is called on a different eligible date or remains outstanding longer than the chosen call date.
| Feature | Yield to maturity | Yield to call |
|---|---|---|
| Final assumed payment date | Stated maturity | Specified call date |
| Final principal amount | Maturity redemption | Call redemption |
| Coupons included | Through maturity | Through the call date |
| Interpretation | Yield under the maturity scenario | Yield under the specified call scenario |
The relative size of the two yields depends on the bond price, coupon schedule, timing, and redemption amounts together. A call premium can raise the call-scenario cash flow, while an earlier payment date shortens the period over which the investor receives coupons and has capital invested. These effects can work in different directions. Therefore, do not infer which yield is larger from the word “callable” alone.
The calculation also assumes that the quoted price is the price used in the yield equation and that all stated payments occur as assumed. If a problem specifies a settlement convention, accrued interest, or a different yield quotation, use that convention rather than silently changing the price or rate basis. The formulas here apply to level coupon payments at regular intervals; irregular payment schedules require discounting each dated cash flow at its appropriate time.
Common Exam Traps
- Using maturity cash flows in a YTC equation. Stop coupons at the specified call date and use the call redemption amount there.
- Confusing a call premium with the maturity redemption. The call amount replaces, rather than supplements, the later maturity principal payment in the call scenario.
- Mixing annual and semiannual units. With semiannual payments, count half-year periods and use a rate per half-year; double that rate only when reporting a nominal annual yield convertible semiannually.
- Treating either yield as a forecast. Each is a yield implied by an assumed set of cash flows, not proof that the issuer will follow that scenario.
- Rounding too early. Keep calculator precision while solving and checking the equation, then round the final yield as requested.
Check Your Understanding
For each problem, identify the correct scenario cash flows before solving or comparing yields.
- A $1,000 bond pays a $50 annual coupon, matures in four years for $1,000, and sells for $1,000. It is callable after two years for $1,000, immediately after that year’s coupon. Find the YTM and the YTC for that call date. (Answer: YTM = 5.00%; YTC = 5.00%.)
- A bond sells at par, has a 5.00% coupon rate paid semiannually, and can be called after two years for more than its $1,000 face amount. Without solving for an exact YTC, compare YTC with YTM, assuming the call date follows a coupon payment. (Answer: YTM is 5.00% nominal convertible semiannually; YTC is greater than 5.00% nominal convertible semiannually.)
- A zero-coupon bond costs $760, pays $1,000 at maturity in five years, and may instead be called after three years for $900. Compute the annual effective YTM and YTC, to two decimal places. (Answer: YTM \(=(1000/760)^{1/5}-1\approx5.64\%\); YTC \(=(900/760)^{1/3}-1\approx5.81\%\).)
- A bond has $1,000 face value, a $20 coupon every half-year, and six half-years to maturity. Its price is \(20a_{\overline{6}|\,0.02}+1{,}000(1.02)^{-6}\). It is callable after four half-years for $1,015. State its YTM and write the equation whose solution gives its nominal annual YTC convertible semiannually. (Answer: YTM = 4.00% nominal convertible semiannually; solve \(P=20a_{\overline{4}|\,r}+1{,}015(1+r)^{-4}\), then report YTC as \(2r\).)
- A callable bond’s price is $980. It pays annual coupons of $40, has a $1,000 maturity redemption in three years, and can be called after one year for $1,010 immediately after the coupon. Write both yield equations and state which cash flows are included in each. (Answer: YTM solves \(980=40a_{\overline{3}|\,j}+1{,}000(1+j)^{-3}\); YTC solves \(980=(40+1{,}010)(1+j_c)^{-1}\). The maturity scenario has three coupons and maturity redemption; the call scenario has one coupon and call redemption.)