Deriving The Premium-Discount Sign From Coupon And Yield

Introduction

A bond’s price is not compared with its redemption amount by looking at the coupon alone. The relevant comparison is between the coupon paid each period and the yield earned each period on the redemption amount. Expressing the price as the present value of its contractual cash flows makes that comparison precise: it determines whether the price is above, below, or equal to the amount paid at maturity.

This tutorial derives that sign result for a conventional level-coupon bond. It assumes that the bond is valued just after a coupon date, has equal payment periods and a fixed coupon, and is discounted at one yield per period. For irregular payment dates or changing coupons, the simple annuity expression used here does not apply directly.

Definition: Consider a bond with $n$ remaining equal periods. Let $R$ be its redemption amount at the end of period $n$, $C$ the coupon paid at the end of each period, and $j$ the effective yield per period. Assume $R>0$, $n$ is a positive integer, and $j>-1$. The price $P$ is the present value of the $n$ coupons and the redemption payment, discounted at $j$. Write $v_j=1/(1+j)$.

The Price-minus-Redemption Formula

The present value of the coupons is $C a_{\overline{n}|}$ at yield $j$, and the present value of redemption is $R v_j^n$. Thus

P=C a_{\overline{n}|}+R v_j^n, \qquad a_{\overline{n}|}=\frac{1-v_j^n}{j}\quad (j\ne 0).

The annuity factor $a_{\overline{n}|}$ is positive: it is the sum of the positive discount factors $v_j+v_j^2+\cdots+v_j^n$. The key is to write the redemption amount $R$ using that same factor. Since $1-v_j=jv_j$, the finite geometric sum gives $1-v_j^n=j a_{\overline{n}|}$. Consequently, for $j\ne0$,

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

This also holds at $j=0$ by direct valuation: $P=nC+R$, so $P-R=nC$, which agrees with $(C-Rj)a_{\overline{n}|}$ when the annuity factor is interpreted as $n$. The derivation explains the result: relative to paying $R$ today, the bond’s coupon stream adds value, while discounting its future redemption subtracts value. Their net amount is the per-period difference $C-Rj$, multiplied by a positive factor.

Theorem: For a level-coupon bond with positive redemption amount and equal periods, the sign of $P-R$ is the sign of $C-Rj$. Therefore:

If $C>Rj$, then $P>R$: the bond is at a premium to redemption.
If $C<Rj$, then $P<R$: the bond is at a discount to redemption.
If $C=Rj$, then $P=R$: the bond is at par with redemption.

Equivalently, compare the coupon rate per period, $C/R$, with the yield per period $j$. A coupon rate above the yield implies a premium; one below the yield implies a discount. The comparison is about the bond’s price relative to its redemption amount, not about whether its price is above or below some other market price.

Worked Applications

Worked Example: An Annual Coupon Bond at a Lower Yield

A hypothetical bond has redemption amount $R=\$1{,}000$, pays an annual coupon of $C=\$60$, and has five annual periods remaining. Its annual effective yield is $j=4\%$. Find the price and state the premium or discount relative to redemption.

Method. Use the price-minus-redemption formula. The payments and yield are both annual, so no rate conversion is needed.

First compare the coupon with interest at the yield on redemption:

Rj=1{,}000(0.04)=\$40, \qquad C-Rj=60-40=\$20.

The annuity factor is

v_j^5=\frac{1}{1.04^5}=0.82192711, \qquad a_{\overline{5}|}=\frac{1-(1.04)^{-5}}{0.04}=4.45182233.

Therefore,

P-R=20(4.45182233)=\$89.03645, \qquad P=1{,}000+89.03645=\$1{,}089.03645.

To check by present-value components, the coupon value is $60(4.45182233)=\$267.10934$, and the redemption value is $1{,}000(0.82192711)=\$821.92711$. Their sum is approximately $\$1{,}089.03645$, consistent with the difference calculation. Rounded to the nearest cent, the price is $\$1{,}089.04$, a premium of $\$89.04$. The method avoids assuming that a coupon bond is worth its redemption amount merely because redemption occurs at maturity.

Worked Example: A Semiannual Bond at a Higher Yield

A hypothetical bond redeems for $\$5{,}000$ in two years and pays $\$180$ every six months. The annual nominal yield is $8\%$ convertible semiannually. Find its price and discount relative to redemption.

Method. Convert the yield to the six-month period used by the cash flows, then compare the six-month coupon with yield interest on redemption. There are $n=2(2)=4$ periods, and the yield per period is $j=0.08/2=0.04$.

Rj=5{,}000(0.04)=\$200, \qquad C-Rj=180-200=-\$20.

Thus the price must be below redemption. The relevant discount and annuity factors are

v_j^4=\frac{1}{1.04^4}=0.85480419, \qquad a_{\overline{4}|}=\frac{1-(1.04)^{-4}}{0.04}=3.62989521.

The price difference and price are

P-R=(-20)(3.62989521)=-\$72.59790, \qquad P=5{,}000-72.59790=\$4{,}927.40210.

As a check, the coupon present value is $180(3.62989521)=\$653.38114$, and the redemption present value is $5{,}000(0.85480419)=\$4{,}274.02095$. Their sum is $\$4{,}927.40209$, differing only by rounding in the displayed factors. To the nearest cent, the price is $\$4{,}927.40$, or $\$72.60$ below redemption.

Worked Example: Comparing Coupon and Yield with Nonpar Redemption

A hypothetical bond pays an annual coupon of $\$64$, redeems for $\$1{,}080$ after three years, and has an annual effective yield of $5.5\%$. Determine whether it sells above or below redemption and find the price to the nearest cent.

Method. Compare the coupon with yield interest on the actual redemption amount. Do not substitute a face amount for redemption when the two differ.

Rj=1{,}080(0.055)=\$59.40, \qquad C-Rj=64-59.40=\$4.60.

Because this difference is positive, the price is above redemption. For three annual periods,

v_j^3=\frac{1}{1.055^3}=0.85161366, \qquad a_{\overline{3}|}=\frac{1-(1.055)^{-3}}{0.055}=2.69793347.

Then

P-R=4.60(2.69793347)=\$12.41049, \qquad P=1{,}080+12.41049=\$1{,}092.41049.

For a present-value check, the coupons contribute $64(2.69793347)=\$172.66774$, and redemption contributes $1{,}080(0.85161366)=\$919.74275$. These sum to approximately $\$1{,}092.41049$. The price is $\$1{,}092.41$, a premium of $\$12.41$. Using the actual redemption amount is essential: the yield interest to compare with the coupon is $\$59.40$ per year.

Rate Conventions and the Comparison

The formula uses one period as the unit for both the coupon $C$ and the yield $j$. If coupons are paid $m$ times per year and the yield is quoted as a nominal annual rate $i^{(m)}$ convertible $m$ times, then the yield per coupon period is $j=i^{(m)}/m$. If instead an effective annual yield $i$ is given, the equivalent per-period yield is $j=(1+i)^{1/m}-1$. In either case, use the coupon amount paid per period and the matching periodic yield in $C-Rj$.

The comparison can also be expressed as a rate test: compare $C/R$, the coupon rate per payment period, with $j$. It is not enough to compare two percentages that are stated over different time units. For example, a 6% annual coupon rate paid semiannually corresponds to a coupon of $0.06R/2$ each half-year, or a 3% coupon rate per half-year. If the semiannual yield is 4%, the bond is at a discount because $3\%<4\%$.

Common Exam Trap

A frequent error is to compare the annual coupon rate directly with a periodic yield. Consider a $\$1{,}000$ bond with a 6% annual coupon rate paid semiannually, four half-year periods remaining, and an 8% nominal annual yield convertible semiannually.

Incorrect calculation: comparing the annual coupon rate of 6% with the semiannual yield of 4% suggests a premium because $6\%>4\%$. Those rates have different period units, so that conclusion is unsupported.

Correction: the coupon per half-year is $1{,}000(0.06)/2=\$30$, while the yield interest per half-year on redemption is $1{,}000(0.08/2)=\$40$. Thus $C-Rj=30-40=-\$10$, and

P-R=(-10)a_{\overline{4}|} =(-10)\left(\frac{1-(1.04)^{-4}}{0.04}\right) =-\$36.29895.

The bond is at a discount, with price approximately $\$963.70$. The correction is to put the coupon and yield on the same payment-period basis before comparing them.

When the Sign Rule Applies

The sign result is useful because it answers the premium-or-discount question without first calculating the entire price. Its assumptions matter: the coupon must be level, the payments must occur at equal periods, and the same yield must discount all remaining cash flows. The standard formula also assumes valuation just after a coupon date, so that the next coupon is one full period away.

For a bond with irregular payment dates, the price is still found by discounting each cash flow at its appropriate time, but the coupon cash flows need not form the annuity used in this derivation. Similarly, if coupons vary, a single $C$ cannot be factored out. In those settings, calculate the price from the specified cash flows rather than applying this sign rule without checking its assumptions.

For the conventional bond considered here, the rule is direct: compare each period’s coupon $C$ with $Rj$, the yield interest on redemption for that period. The resulting sign gives the price’s position relative to redemption, while the magnitude of $(C-Rj)a_{\overline{n}|}$ gives the dollar difference.

Check Your Understanding

For each bond, use the stated payment period consistently. Round dollar amounts to the nearest cent.

A bond redeems for $\$2{,}000$, pays $\$140$ annually for four years, and has a 5% annual effective yield. Find its price and state the premium or discount. (Answer: Premium of $\$141.84$; price $\$2{,}141.84$.)
A bond redeems for $\$2{,}000$, pays $\$36$ every six months for two years, and has an 8% nominal annual yield convertible semiannually. Find its price and state the amount below or above redemption. (Answer: Discount of $\$159.72$; price $\$1{,}840.28$.)
A bond redeems for $\$1{,}500$, pays an annual coupon of $\$90$ for four years, and has a 5% annual effective yield. Show the comparison that determines the sign, then find the price. (Answer: $C-Rj=\$15>0$, so it is at a premium of $\$53.19$; price $\$1{,}553.19$.)
A bond pays an annual coupon of $\$70$, redeems for $\$1{,}250$ after four years, and has a 5% annual effective yield. Determine whether it is at a premium or discount and calculate its price. (Answer: $C-Rj=\$7.50>0$, so it is at a premium of $\$26.59$; price $\$1{,}276.59$.)

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