Solving For A Coupon Rate At A Target Yield

Why Solve for a Coupon Rate?

A bond’s price depends on its coupons, redemption payment, maturity, and yield. Often the price and target yield are known, while the coupon is the quantity to determine. This is useful when assessing what coupon a hypothetical bond would need to offer to meet a specified price at a specified yield.

The key is to distinguish three quantities: the coupon payment each period, the annual coupon rate applied to face value, and the yield used to discount the cash flows. The coupon rate is not the yield. A target yield determines the discount factors; the price equation then determines the coupon that is consistent with that yield.

Definition: For a conventional bond with level coupons, let $P$ be its price at a coupon date, $F$ its face value, $R$ its redemption value at maturity, $t$ its term in years, and $m$ the number of coupon payments per year. If the annual nominal coupon rate is $q$, the coupon payment each period is $C=Fq/m$. If the target nominal yield convertible $m$ times per year is $i^{(m)}$, the yield per coupon period is $j=i^{(m)}/m$, and the number of coupon periods is $n=mt$.

Assume coupons are paid at the end of each period, the redemption payment is made with the final coupon, the target yield is constant over the term, and the price is measured on a coupon date. This setup excludes accrued interest, irregular payment dates, and default or call contingencies. Under these assumptions, the present-value equation is

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

If the target is specified as an effective annual yield $i$, rather than a nominal yield convertible $m$ times per year, use the equivalent periodic rate $j=(1+i)^{1/m}-1$. The discounting must match the yield convention in the problem.

Rearranging the Price Equation

The coupon payment appears only in the first term, so the equation can be rearranged directly. Once $C$ is found, multiply by the number of payments per year and divide by face value to obtain the annual nominal coupon rate.

C=\frac{P-R(1+j)^{-n}}{a_{\overline{n}|}}, \qquad q=\frac{mC}{F} =\frac{m\left[P-R(1+j)^{-n}\right]}{F a_{\overline{n}|}}.

The numerator $P-R(1+j)^{-n}$ is the part of the price that must be supplied by the present value of the coupons. Dividing it by the annuity factor gives the level coupon payment that has that present value.

Useful condition: Since $a_{\overline{n}|}>0$ for a positive yield, a nonnegative coupon requires $P\ge R(1+j)^{-n}$. If the price is below the present value of redemption alone, the equation requires a negative coupon. That is an algebraic result, not a conventional positive-coupon bond.

Why the Rearrangement Works

Each coupon is discounted separately at the target periodic yield. The first coupon, paid one period from now, has present value $C(1+j)^{-1}$; the second has present value $C(1+j)^{-2}$; and the final coupon has present value $C(1+j)^{-n}$. Because every payment is the same amount $C$, $C$ factors out of their sum:

\begin{aligned} P &=\sum_{k=1}^{n} C(1+j)^{-k}+R(1+j)^{-n}\\ &=C\sum_{k=1}^{n}(1+j)^{-k}+R(1+j)^{-n}\\ &=C a_{\overline{n}|}+R(1+j)^{-n}. \end{aligned}

Subtracting the discounted redemption value leaves the present value attributable to coupons. Dividing by the sum of the coupon discount factors gives the one level payment $C$ that produces that present value. Finally, $mC$ is the annual coupon amount, so $mC/F$ is the annual coupon rate as a proportion of face value.

Worked Applications

Worked Example: Annual Coupons and a One-Year Bond

A hypothetical bond has face value $F=\$1{,}000$, redemption value $R=\$1{,}100$, and one year remaining. It pays one coupon at maturity. Its price on the coupon date is $P=\$1{,}040$. Determine the annual coupon rate that gives a target effective annual yield of 10%. Assume a single year-end coupon and redemption payment, with no accrued interest.

Method. There is one payment period, so $m=1$, $n=1$, and the periodic yield is $j=0.10$. Avoid treating the price as if it were all coupon value: first discount the known redemption payment.

The one-period discount factor and annuity factor are

(1+j)^{-1}=\frac{1}{1.10}=0.9090909, \qquad a_{\overline{1}|}=0.9090909.

The present value of redemption is $\$1{,}100(0.9090909)=\$1{,}000$. Therefore the present value required from the coupon is $\$1{,}040-\$1{,}000=\$40$. The coupon payment is

C=\frac{\$40}{0.9090909}=\$44.

There is one coupon per year, so the annual coupon rate is $q=1(\$44)/\$1{,}000=0.044$, or 4.4%.

Check. The coupon’s present value is $\$44/1.10=\$40$, and redemption’s present value is $\$1{,}100/1.10=\$1{,}000$. Their sum is $\$1{,}040$, the stated price. The calculation avoids confusing the coupon rate with the yield: the coupon rate is 4.4%, while the target yield is 10%.

Worked Example: Semiannual Coupons and a Two-Year Term

A hypothetical two-year bond has face value $F=\$1{,}000$, redemption value $R=\$1{,}100$, and a price of $\$1{,}011.3512374$ at a coupon date. Coupons are paid semiannually. Find the annual nominal coupon rate that gives a target nominal yield of 10% convertible semiannually. Assume level end-of-period coupons and redemption with the final coupon.

Method. Convert the quoted yield to a yield per half-year and count the half-year periods. A 10% nominal yield convertible semiannually means $j=0.10/2=0.05$, not 10% per half-year. There are $n=2(2)=4$ periods.

The relevant discount factor and annuity factor are

(1.05)^{-4}=0.822702475, \qquad a_{\overline{4}|}=\frac{1-0.822702475}{0.05}=3.5459505.

The present value of redemption is $\$1{,}100(0.822702475)=\$904.9727225$. The price remaining to be supplied by coupons is $\$1{,}011.3512374-\$904.9727225=\$106.3785149$, approximately. Thus

C=\frac{\$106.3785149}{3.5459505}\approx \$30.0000 \quad\text{per half-year}.

There are two coupons per year. The annual coupon amount is $2(\$30)=\$60$, so the annual nominal coupon rate is $\$60/\$1{,}000=0.06$, or 6.0%.

Check. The coupon present value is approximately $\$30(3.5459505)=\$106.378515$. The redemption present value is approximately $\$904.972722$. Together they give approximately $\$1{,}011.351237$, agreeing with the stated price to the displayed precision. The per-period coupon is $\$30$; the annual coupon rate is 6%, not 3% or 30%.

Worked Example: A Price with a Redemption Premium

A hypothetical bond pays annual coupons, has face value $F=\$1{,}000$, redemption value $R=\$1{,}440$, and matures in two years. Its price is $\$1{,}061.111111$. Determine the annual coupon rate for a target effective annual yield of 20%. Assume two equal year-end coupons and redemption with the second coupon.

Method. The annual target yield is also the periodic yield because payments are annual. Hence $j=0.20$, $m=1$, and $n=2$. Calculate the redemption present value before solving for the coupon.

(1.20)^{-1}=\frac{5}{6}, \qquad (1.20)^{-2}=\frac{25}{36}, \qquad a_{\overline{2}|}=\frac{5}{6}+\frac{25}{36}=\frac{55}{36}.

The redemption present value is $\$1{,}440(25/36)=\$1{,}000$. Thus the coupon present value must be $\$1{,}061.111111-\$1{,}000=\$61.111111$, to the displayed precision. Solving gives

C=\frac{\$61.111111}{55/36}\approx \$40.0000.

With annual payments, $q=C/F=\$40/\$1{,}000=0.04$, so the required annual coupon rate is 4.0%.

Check. The coupons have present value $\$40(55/36)=\$61.111111$. Redemption has present value $\$1{,}000$. The total is $\$1{,}061.111111$, as stated. The price exceeds face value even though the coupon rate is below the yield because redemption is substantially above face value; the redemption amount, not face value, is the maturity cash flow in this problem.

Common Mistake: Confusing Coupon Rate and Coupon Payment

For $m$ payments per year, a nominal annual coupon rate $q$ produces a payment of $Fq/m$ each period. For example, a 6% annual coupon rate on $\$1{,}000$ face value with semiannual payments means $C=\$1{,}000(0.06)/2=\$30$ every six months.

A wrong calculation would use $C=\$1{,}000(0.06)=\$60$ every half-year, and then describe the result as a 6% annual coupon rate. That doubles the intended annual coupon amount: two payments of $\$60$ total $\$120$, or 12% of face value per year. The correction is to divide the annual coupon amount by the payment frequency before using it in the bond price equation. Conversely, after solving for a periodic coupon, multiply by $m$ before dividing by $F$ to report the annual coupon rate.

Key takeaway: Discount the redemption value at the target periodic yield, subtract it from the price, and divide the remainder by the coupon annuity factor to find the periodic coupon. Convert that payment to an annual coupon rate using $q=mC/F$.

Check Your Understanding

For each problem, assume a conventional bond priced on a coupon date, level end-of-period coupons, and redemption paid with the final coupon. Report the annual coupon rate unless the question asks otherwise.

A one-year bond has face value $\$2{,}000$, redemption value $\$2{,}100$, price $\$2{,}080$, and an annual target yield of 5%. It pays one coupon at maturity. Find the coupon payment and annual coupon rate. (Answer: $\$84$ coupon; 4.2% annual coupon rate.)
A two-year bond has face value and redemption value of $\$1{,}000$, pays semiannually, and has a target nominal yield of 10% convertible semiannually. Its price is $\$964.540495$. Find the annual nominal coupon rate. (Answer: 8.0%.)
A two-year bond has face value $\$1{,}000$, redemption value $\$1{,}210$, annual coupons, and a target effective annual yield of 10%. Its price is $\$1{,}086.776860$. Find the annual coupon rate. (Answer: 5.0%.)
A three-year bond has face value $\$5{,}000$, redemption value $\$5{,}500$, and annual coupons. Its target effective annual yield is 5%, and its price is approximately $\$5{,}295.7564$. Find the annual coupon payment and coupon rate. (Answer: $\$200$ annually; 4.0%.)
A two-year bond has face value $\$1{,}000$, redemption value $\$1{,}000$, annual coupons, a target effective annual yield of 5%, and a price of $\$900$. Can it have a nonnegative level coupon? (Answer: No; the present value of redemption alone is approximately $\$907.03$, already greater than the price.)

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