Yield to Maturity with Semiannual Coupons

Introduction

A bond with semiannual coupons makes two payments each year, so its yield calculation must use half-year cash-flow periods. The central task is to match the timing of the cash flows with the periodic rate used to discount them. A common annual quotation is then obtained from that periodic yield, but the quotation must not be mistaken for the rate applied in each half-year.

Assume throughout that the bond is valued on a coupon date, has no default, and pays the stated coupons and face value on schedule. This avoids accrued-interest adjustments: the price used in the equation is the amount paid on that coupon date. The yield to maturity is the single periodic discount rate that makes the present value of those promised cash flows equal to the observed price.

Definition: Let $F$ be a bond's face value, $c$ its annual coupon rate expressed as a decimal, and $n$ its number of years to maturity. For semiannual coupons, the coupon each six months is $C=Fc/2$, and the number of payment periods is $N=2n$. The semiannual periodic yield, $j$, is the effective yield per six-month period that solves the bond-price equation. The bond-equivalent annual yield is $2j$; the effective annual yield is $(1+j)^2-1$. These are different annual measures.

The Semiannual Bond-Price Equation

If the bond has $N$ remaining half-year periods, the first $N-1$ cash flows are coupons of $C$, and the last cash flow is $C+F$. Discount each payment by the number of half-year periods until it is received:

P=\sum_{k=1}^{N-1}\frac{C}{(1+j)^k}+\frac{C+F}{(1+j)^N} =C\frac{1-(1+j)^{-N}}{j}+F(1+j)^{-N}.

Here $P$ is the bond's price on the coupon date. The equation is also commonly written as the present value of all $N$ coupons plus the present value of face value. Given $P$ and the bond's terms, $j$ is generally found by numerical search or a financial calculator; there is usually no convenient algebraic formula that isolates it.

The reason for discounting the $k$th payment by $(1+j)^k$ is that $j$ is the rate for one six-month period. One dollar invested for $k$ such periods accumulates to $(1+j)^k$ dollars. Reversing that accumulation gives a present value of $(1+j)^{-k}$ dollars for each dollar due at period $k$.

Translate the coupon terms.
Divide the annual coupon rate by two when finding the coupon payment: $C=Fc/2$.

Count half-year periods.
For $n$ whole years to maturity with semiannual coupons, use $N=2n$.

Solve the price equation.
Find the per-period rate $j$ that equates the discounted cash flows to $P$.

State the annual measure.
Report $2j$ if the requested quotation is bond-equivalent annual yield, or $(1+j)^2-1$ if the requested measure is effective annual yield.

Worked Yield Calculations

Worked Example: Solving for a Premium Bond's Yield

A hypothetical bond has face value $F=\$1{,}000$, an annual coupon rate of 8%, two years to maturity, and a price of $\$1{,}037.17$ on a coupon date. Coupons are paid semiannually. Find the semiannual periodic yield and the bond-equivalent annual yield.

Method. Convert the annual coupon to a half-year payment and search for $j$ in the price equation. There are $N=2(2)=4$ periods, and each coupon is $C=\$1{,}000(0.08)/2=\$40$. The cash flows are $\$40,\$40,\$40,\$1{,}040$.

A trial rate of $j=2\%$ gives discount factors, for periods 1 through 4, of $0.980392,\ 0.961169,\ 0.942322,\ 0.923845$. The corresponding present values are $\$39.22,\ \$38.45,\ \$37.69,\ \$960.80$, totaling approximately $\$1{,}076.15$. This is above the observed price.

At $j=4\%$, the discount factors are $0.961538,\ 0.924556,\ 0.888996,\ 0.854804$. The cash-flow present values are $\$38.46,\ \$36.98,\ \$35.56,\ \$888.99$, totaling approximately $\$1{,}000.00$, below the observed price. Numerical search therefore places the solution between 2% and 4%.

Testing $j=3\%$, the discount factors are $0.970874,\ 0.942596,\ 0.915142,\ 0.888487$. The present values are $\$38.83,\ \$37.70,\ \$36.61,\ \$924.03$. Their total is $\$1{,}037.17$ to the nearest cent, matching the quoted price. Thus $j=3.000\%$ per half-year, and the bond-equivalent annual yield is $2(3.000\%)=\mathbf{6.000\%}$.

Check. The price is above face value, while the 8% annual coupon rate exceeds the 6% bond-equivalent yield. That direction is consistent with a premium price. The search avoided the mistake of treating 6% as the rate for each half-year.

Worked Example: Solving for a Discount Bond's Yield

A hypothetical bond has face value $\$5{,}000$, a 7% annual coupon rate, five years to maturity, semiannual coupons, and a coupon-date price of $\$4{,}797.23$. Find its semiannual periodic yield and its bond-equivalent annual yield.

Method. The coupon is $C=\$5{,}000(0.07)/2=\$175$, and the number of periods is $N=2(5)=10$. Search for $j$ in $P=175[1-(1+j)^{-10}]/j+5{,}000(1+j)^{-10}$.

At $j=4\%$, $(1.04)^{-10}=0.6755642$, so the coupon annuity factor is $[1-0.6755642]/0.04=8.110895$. The coupon present value is $\$175(8.110895)=\$1{,}419.41$. The face-value present value is $\$5{,}000(0.6755642)=\$3{,}377.82$. The total is $\$1{,}419.41+\$3{,}377.82=\$4{,}797.23$, matching the price. Therefore $j=4.000\%$ per half-year and the bond-equivalent annual yield is $2(4.000\%)=\mathbf{8.000\%}$.

Check. The price is below face value, and the coupon rate of 7% is below the 8% bond-equivalent yield. The discount price is consistent with those terms. The number of periods is ten—not five—because the bond pays twice a year.

Worked Example: A Par Bond and Two Annual Yield Measures

A hypothetical bond with face value $\$2{,}500$ pays a 5% annual coupon semiannually and matures in three years. Its coupon-date price is $\$2{,}500$. Find its semiannual periodic yield, bond-equivalent annual yield, and effective annual yield.

Method. The coupon is $C=\$2{,}500(0.05)/2=\$62.50$, and $N=6$. The coupon is exactly $2.5\%$ of face value each period. At $j=2.5\%$, let $v=(1+j)^{-1}$. Then the present value is $Fj(1-v^6)/j+Fv^6=F(1-v^6)+Fv^6=F$. Thus the price is $\$2{,}500$, and $j=2.500\%$ solves the equation.

The bond-equivalent annual yield is $2j=2(2.500\%)=\mathbf{5.000\%}$. The effective annual yield accounts for compounding between the two half-year periods: $(1.025)^2-1=1.050625-1=0.050625$, or $\mathbf{5.0625\%}$.

Check. The coupon rate and bond-equivalent annual yield are both 5%, so the price equals face value. The effective annual yield is slightly higher because earning 2.5% in each half-year includes compounding.

Annualization and a Common Exam Trap

A semiannual periodic yield is not itself an annual quotation. Under the bond-equivalent convention, double it: $y_{\mathrm{BE}}=2j$. This convention is a simple annualization of the six-month rate. If an effective annual yield is required instead, compound across both half-year periods: $i_{\mathrm{eff}}=(1+j)^2-1$. Use the annual measure requested; do not substitute one for the other.

Common mistake: For the first example, the correct periodic yield is 3%. A 6% bond-equivalent annual yield therefore means 3% per six months. If 6% is incorrectly used as the half-year rate, the calculated price becomes $\$40[1-(1.06)^{-4}]/0.06+\$1{,}000(1.06)^{-4}$, approximately $\$930.70$, not $\$1{,}037.17$. The correction is to divide the bond-equivalent annual quotation by two before discounting each semiannual cash flow.

A second frequent error is to use the annual coupon payment $Fc$ at every six-month date. The coupon paid each half-year is $Fc/2$. A reliable setup states the cash flow, period count, and periodic discount rate before entering numbers in a calculator. These steps align with the SOA VEE Accounting and Finance syllabus.

Key takeaway: With semiannual coupons, use half the annual coupon rate to calculate each coupon payment, twice the years to maturity to count periods, and the semiannual periodic yield to discount each payment. Annualize only after solving, using the convention specified.

Check Your Understanding

For each problem, assume valuation on a coupon date, no default, and the stated semiannual coupon schedule.

A bond with $\$1{,}000$ face value, a 4% annual coupon rate, and three years to maturity is priced at face value. Find the semiannual periodic yield and bond-equivalent annual yield. (Answer: 2.000% per half-year; 4.000% bond-equivalent annual yield)
A two-year bond with $\$1{,}000$ face value and a 10% annual coupon rate is priced at $\$1{,}074.34$. Solve for the semiannual periodic yield and bond-equivalent annual yield. (Answer: 3.000% per half-year; 6.000% bond-equivalent annual yield)
A three-year bond with $\$2{,}000$ face value and a 4% annual coupon rate is priced at $\$1{,}891.66$. Find the semiannual periodic yield. (Answer: 3.000% per half-year; 6.000% bond-equivalent annual yield)
A bond's solved semiannual periodic yield is 3.25%. Calculate its effective annual yield, to four decimal places as a percentage. (Answer: $(1.0325)^2-1=6.6056\%$)
A four-year bond with $\$1{,}000$ face value and a 6% annual coupon rate is priced at $\$1{,}035.85$. Find its semiannual periodic yield and bond-equivalent annual yield. (Answer: 2.500% per half-year; 5.000% bond-equivalent annual yield)

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