Why Above-Market Coupons Produce Premium Prices

Introduction

A bond’s coupon is fixed by its contract, while the required yield reflects the return an investor demands for the bond’s cash flows. If each coupon is larger than the return required on the bond’s face value for that period, the coupons are relatively generous. Their present value, together with the present value of the face value repaid at maturity, makes the bond worth more than face value.

The key comparison is between two amounts stated for the same period: the coupon payment and the required yield per period multiplied by face value. The period must match the coupon frequency. An annual yield cannot be compared directly with a semiannual coupon amount without first converting the yield to a semiannual basis.

Definition: Consider a fixed-cash-flow coupon bond with face value $F$, coupon payment $C$ each period, $n$ remaining coupon periods, and required yield $i$ per period. For $n\geq 1$, the coupon is above market when $C>iF$. Under the standard pricing model described below, its price $P$ is greater than $F$. If $n=0$, there are no remaining coupons and the price is $F$, not a premium.

Model and Price Comparison

Use the standard fixed-cash-flow model: the bond pays $C$ at the end of each of the next $n$ periods and repays $F$ with the final coupon. The required yield $i$ is an effective rate per coupon period, remains constant over the bond’s remaining term, and is positive. Assume the bond is valued immediately after a coupon date, so there is no accrued interest. Also assume the promised payments are made in full and on schedule; the model does not adjust for default, taxes, transaction costs, or embedded options.

Under these assumptions, discount each promised payment at $i$. The present value of the coupons is $C a_{\overline{n}|}$, where $a_{\overline{n}|}$ is the present value of an annuity-immediate of 1 per period at rate $i$. The present value of the principal is $Fv^n$, where $v=1/(1+i)$.

P=C a_{\overline{n}|}+Fv^n,\qquad a_{\overline{n}|}=\frac{1-v^n}{i}.

To compare the price with face value, express $F$ as the present value of a level annuity paying $iF$ each period, plus the present value of the principal. The annuity is worth $(iF)a_{\overline{n}|}$, and the principal is worth $Fv^n$. Since $i a_{\overline{n}|}=1-v^n$, these values add to $F$. Comparing the bond’s actual coupon with that reference coupon gives the price difference.

F=(iF)a_{\overline{n}|}+Fv^n

P-F=(C-iF)a_{\overline{n}|}.

This is the central present-value comparison. For $i>0$ and $n\geq1$, $a_{\overline{n}|}$ is positive because it is the sum of positive discount factors $v+v^2+\cdots+v^n$. Therefore, if $C>iF$, then $P-F>0$, and so $P>F$. If there are no periods remaining, the annuity factor is zero: there are no future coupons to give the bond extra value, and the maturity payment is $F$.

Key result: With at least one coupon period remaining, a positive required yield per period, and the fixed-cash-flow assumptions above, $C>iF$ implies $P>F$. The comparison uses the coupon payment and required return per period, not unmatched annual and semiannual figures.

Worked Present-Value Comparisons

In each example, first identify the period used for both the coupon and the yield. Then calculate the bond’s present value at that required yield and check whether it exceeds face value.

Worked Example: Annual Coupons Above the Required Return

A bond has face value $\$1{,}000$, pays an annual coupon of $\$80$, and has three annual coupon periods remaining. The required yield is 5% effective annually. Find its price and verify whether it sells at a premium.

Method and period check. Both the coupon and yield are annual. Here $F=\$1{,}000$, $C=\$80$, $i=0.05$, and $n=3$. The required return on face value per year is $iF=0.05(\$1{,}000)=\$50$. Since $\$80>\$50$, the coupon is above market.

Coupon present value. First calculate $v=1/1.05=0.9523809524$, and then $v^3=1/(1.05)^3=0.8638375985$. Thus $a_{\overline{3}|}=(1-v^3)/0.05=(1-0.8638375985)/0.05=2.7232480294$. The present value of the coupons is $\$80(2.7232480294)=\$217.8598424$.

Principal present value and price. The present value of the $\$1{,}000$ maturity payment is $\$1{,}000(0.8638375985)=\$863.8375985$. Therefore, $P=\$217.8598424+\$863.8375985=\$1{,}081.6974409$, or $\$1{,}081.70$ to the nearest cent.

Check and common mistake avoided. The price exceeds face value by $\$1{,}081.6974409-\$1{,}000=\$81.6974409$, consistent with the above-market coupon. Comparing the coupon rate with the required yield alone suggests the same direction here, but the cash-flow comparison is $\$80$ against $iF=\$50$; it makes the relevant dollar amounts and period explicit.

Worked Example: Matching a Semiannual Coupon to a Periodic Yield

A bond has face value $\$5{,}000$, a 6% annual coupon rate, and eight semiannual coupon periods remaining. Its required yield is 4% nominal annually, convertible semiannually. Calculate its price, rounding only the final answer to the nearest cent, and determine whether it is a premium bond.

Method and period check. The semiannual coupon is $C=0.06(\$5{,}000)/2=\$150$. The required yield per semiannual period is $i=0.04/2=0.02$, not 4%. Therefore the required return on face value per period is $iF=0.02(\$5{,}000)=\$100$. Since $\$150>\$100$, the coupon is above market.

Coupon present value. The discount factor for eight periods is $v^8=1/(1.02)^8=0.853490371$ (shown to nine decimal places). The annuity factor is $a_{\overline{8}|}=(1-0.853490371)/0.02=7.32548145$. The coupon present value is $\$150(7.32548145)=\$1{,}098.8222175$.

Principal present value and price. The maturity payment has present value $\$5{,}000(0.853490371)=\$4{,}267.451855$. Adding the unrounded displayed components gives $P=\$1{,}098.8222175+\$4{,}267.451855=\$5{,}366.2740725$, or $\$5{,}366.27$ to the nearest cent.

Check and common mistake avoided. The calculated price is $\$366.2740725$ above face value, as expected when $C>iF$. A common error is to use the 4% nominal annual yield directly against the semiannual coupon. That would compare mismatched periods; the correct periodic yield is 2%, making the required return $\$100$ each half-year.

Worked Example: Checking a Premium by Discounting Each Cash Flow

A two-year bond has face value $\$1{,}000$, pays annual coupons of $\$120$, and has a required annual yield of 10%. Find its price by discounting each payment separately, and explain why the result is above face value.

Method and comparison. There are two annual periods, so $F=\$1{,}000$, $C=\$120$, $i=0.10$, and $n=2$. The required return on face value each year is $iF=0.10(\$1{,}000)=\$100$. The coupon exceeds that amount by $\$20$.

Discount each payment. The first coupon has present value $\$120/1.10=\$109.0909091$. In year two, the bond pays both the $\$120$ coupon and $\$1{,}000$ principal, so that payment has present value $\$1{,}120/(1.10)^2=\$1{,}120/1.21=\$925.6198347$. Hence $P=\$109.0909091+\$925.6198347=\$1{,}034.7107438$, or $\$1{,}034.71$ to the nearest cent.

Check and common mistake avoided. The result is $\$34.7107438$ above face value. The reference bond paying $\$100=iF$ each year would have present value $\$100/1.10+\$1{,}100/1.21=\$90.9090909+\$909.0909091=\$1{,}000$. The actual bond pays $\$20$ more in each of its two coupon periods, whose present value is $\$20(1/1.10+1/1.21)=\$34.7107438$. The mistake to avoid is discounting the final coupon but forgetting that principal is paid at maturity as well.

What the Comparison Does—and Does Not—Say

The condition $C>iF$ answers a specific pricing question: at the stated required yield and under the fixed-cash-flow assumptions, is the present value greater than face value? It does not mean the bond’s price is set by comparing a coupon with face value alone. The timing of each payment matters because payments at different dates receive different discount factors.

It also does not mean that every observed market price must follow this comparison under every circumstance. The result assumes a known periodic required yield and promised cash flows paid in full. If payment timing, the yield basis, or the expected cash flows differ, the model inputs must be adjusted before applying the calculation. In particular, the coupon frequency and the yield frequency must be consistent.

A useful verification is to compare the bond with a reference cash-flow pattern that pays $iF$ in each coupon period and returns $F$ at maturity. Discounted at $i$, that reference pattern is worth exactly $F$. An above-market coupon replaces each reference coupon with a larger payment, while leaving the maturity principal unchanged. Each larger coupon has a positive present value, so the total price is above $F$. This reasoning is the present-value basis for the premium conclusion.

Common Exam Trap: Mixing Annual and Periodic Amounts

Suppose a $\$5{,}000$ bond pays $\$150$ every six months and has a required yield of 4% nominal annually, convertible semiannually. A wrong calculation might set the required return to $0.04(\$5{,}000)=\$200$ per period and conclude that $\$150<\$200$. This applies an annual rate to a half-year period.

The correction is to convert the required yield to the coupon period: $i=0.04/2=0.02$. The required return per half-year is $iF=0.02(\$5{,}000)=\$100$, so $\$150>\$100$. The coupon is above market on the matched-period basis. Always identify the coupon amount per period and yield per period before comparing them.

Key takeaway: For at least one remaining coupon period, compare $C$ with $iF$, using the same period for both. If $C>iF$, discounting the promised cash flows at $i$ gives $P>F$, provided the fixed-cash-flow model assumptions hold.

Check Your Understanding

For each problem, keep the coupon and required yield on the same period basis. Prices should be rounded to the nearest cent where requested.

A bond with face value $\$800$ pays annual coupons at a 7.5% rate. It has two annual periods remaining and a required annual yield of 5%. Find its price and state whether it is above face value. (Answer: $\$837.19$; above face value.)
A $\$2{,}000$ bond pays quarterly coupons at a 12% annual coupon rate, has four quarterly periods remaining, and has a required yield of 4% nominal annually convertible quarterly. Find the coupon per quarter and the required return on face value per quarter. Is the coupon above market? (Answer: Coupon $\$60$; required return $\$20$; yes.)
A bond has face value $\$1{,}500$, annual coupons of $\$90$, three years remaining, and a 5% effective annual required yield. Calculate its price using the price-difference comparison and round to the nearest cent. (Answer: $\$1{,}540.85$.)
A bond has face value $\$1{,}200$, an annual coupon of $\$100$, and a required annual yield of 6%, but its maturity payment is due today and no coupon payments remain. What is its value under the stated model? (Answer: $\$1{,}200$; there is no remaining coupon annuity.)
A bond has face value $\$2{,}500$, pays annual coupons of $\$175$, has two annual periods remaining, and a required annual yield of 6%. Find its price from the present-value comparison and round to the nearest cent. (Answer: $\$2{,}545.83$.)

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