Premium Or Discount Relative To Redemption Value

Why the Benchmark Matters

A bond can be issued with a par value of $1,000 and promise to repay $1,200 at maturity. Those are different amounts with different roles: the par value may determine the coupons, while the redemption value is the amount due at the end. If a bond price is described as a premium or discount, the benchmark must therefore be stated rather than assumed.

In this tutorial, premium and discount are measured against the bond’s redemption value. A bond priced above its redemption value is at a premium relative to redemption; one priced below it is at a discount relative to redemption. A comparison with par can also be useful, but it answers a different question when par and redemption value are not equal.

Definition: Let $P$ be a bond’s price at a coupon date and $R$ its stated redemption value at maturity. The bond is at a premium relative to redemption if $P>R$, at a discount relative to redemption if $P<R$, and at redemption value if $P=R$. If the par value $F$ differs from $R$, the comparison of $P$ with $F$ must be reported separately.

The distinction is not merely terminology. A bond can be above par but below redemption, or below par but above redemption. Calling either bond simply a “premium bond” or “discount bond” without naming the benchmark can conceal the relevant comparison.

Bond Price and Redemption Value

Assume a bond is valued on a coupon date, has $n$ remaining coupon periods, pays level coupons of $C$ at the end of each period, and repays $R$ with the final coupon. Let $j$ be the effective yield per coupon period and $v=1/(1+j)$. Assume the stated cash flows will be paid as scheduled and that the same yield is used to discount each period. The price is the present value of all remaining promised payments:

P=\sum_{k=1}^{n} C v^k+R v^n

The final cash flow is $C+R$, which is why the last coupon appears in the sum and the redemption amount appears as a separate term. This formula also makes clear why the coupon base and redemption benchmark must not be confused: the coupon $C$ may be calculated from par, but the final principal cash flow is $R$.

Why the present-value formula holds: A dollar payable at the end of period $k$ has a present value of $v^k$ dollars at the stated yield, because $v^k(1+j)^k=1$. Thus a payment of $X_k$ at time $k$ has present value $X_kv^k$. Adding the present values of all the coupon and redemption payments gives the bond price.

For a bond paying coupons as a percentage of par, the coupon amount and redemption amount should be recorded separately. For example, if par is $F$, the annual coupon rate is $c$, and coupons are paid $m$ times per year, then the coupon per period is $C=Fc/m$. If the annual nominal yield convertible $m$-thly is $i^{(m)}$, the yield per coupon period is $j=i^{(m)}/m$. Neither expression requires the redemption value to equal par.

This setup assumes valuation on a coupon date and a fixed number of regular periods. For a bond valued between coupon dates or with irregular cash-flow dates, discount each scheduled cash flow for its actual time to payment using the applicable yield convention. The classification itself remains a direct comparison of the resulting price with the stated redemption value.

Worked Examples With Nonpar Redemption

Worked Example: Above Par but Below Redemption

A hypothetical bond has par value $1,000, pays an annual coupon rate of 2% of par, and repays $1,200 at the end of two years. Coupons are annual. The annual effective yield is 5%. Find the price and classify it relative to both par and redemption.

Method. First determine the actual cash flows, then discount each one at the annual yield. The annual coupon is $C=1{,}000(0.02)=\$20$. The cash flows are $20 at the end of year 1 and $1,220 at the end of year 2.

\begin{aligned} P&=\frac{20}{1.05}+\frac{1{,}220}{1.05^2}\\ &=19.047619+1{,}106.575964\\ &=1{,}125.623583 \end{aligned}

Rounded to the nearest cent, the price is $1,125.62. It is $125.62 above par, but $74.38 below the $1,200 redemption value. Therefore, it is a discount relative to redemption, despite being above par.

Common mistake avoided. Comparing the price only with $1,000 would lead to the wrong redemption-based classification. The calculation is checked by comparing the computed price separately with each benchmark: $1{,}000<1{,}125.62<1{,}200$.

Worked Example: Below Par but Above Redemption

A hypothetical bond has par value $1,000, pays annual coupons equal to 12% of par, and repays $800 after two years. Coupons are annual, and the annual effective yield is 10%. Find the price and state its classification relative to par and redemption.

Method. The annual coupon is $C=1{,}000(0.12)=\$120$. The cash flows are $120 at the end of year 1 and $920 at the end of year 2. Discount both at 10% per year.

\begin{aligned} P&=\frac{120}{1.10}+\frac{920}{1.10^2}\\ &=109.090909+760.330579\\ &=869.421488 \end{aligned}

Rounded to the nearest cent, the price is $869.42. It is $130.58 below par but $69.42 above the $800 redemption value. The bond is therefore at a premium relative to redemption, even though its price is below par.

Common mistake avoided. A price below par does not establish a discount relative to redemption when $R<F$. The check is $800<869.42<1{,}000$, which confirms the two different classifications.

Worked Example: Semiannual Coupons and a Redemption Discount

A hypothetical bond has par value $1,000, a nominal annual coupon rate of 10% convertible semiannually, and redemption value $900 one year from now. It has two coupon periods remaining. Its nominal annual yield is 10% convertible semiannually. Calculate the price and classify it against both benchmarks.

Method. The coupon per half-year is $C=1{,}000(0.10)/2=\$50$. The yield per half-year is $j=0.10/2=0.05$. The payments are $50 in six months and $950 in one year, the latter consisting of the final coupon plus redemption.

\begin{aligned} P&=\frac{50}{1.05}+\frac{950}{1.05^2}\\ &=47.619048+861.678005\\ &=909.297053 \end{aligned}

Rounded to the nearest cent, the price is $909.30. This is $9.30 above redemption but $90.70 below par. Thus the bond is at a premium relative to redemption and at a discount relative to par.

Common mistake avoided. Equal nominal coupon and yield rates do not, by themselves, imply a price equal to par when the redemption amount differs from par. The computed price is checked against both values: $900<909.30<1{,}000$.

A Reliable Classification Process

Use a deliberate sequence whenever a problem involves a nonpar redemption amount. The coupon rate may be quoted on par, and the yield may be quoted in a compounding convention that needs conversion to the coupon period. Neither detail changes which amount is used as the redemption benchmark.

Identify the three amounts.
Record par $F$, redemption value $R$, and the price $P$ sought or given. Do not treat $F$ and $R$ as interchangeable.

Construct the promised cash flows.
Calculate coupons from the stated coupon basis and add $R$ to the final coupon. The last payment is $C+R$ when coupons are level.

Use a consistent yield period.
Convert the yield to an effective rate per payment period if needed, and discount each payment for its number of periods.

Compare with each benchmark independently.
Use $P$ versus $R$ for the redemption-based classification, and $P$ versus $F$ only when the question asks for a par comparison.

A useful way to present a conclusion is explicit: “The price is a discount relative to the $1,200 redemption value, although it is above the $1,000 par value.” This wording prevents a reader from having to infer which benchmark is intended. If a problem or market convention specifically asks for a premium or discount relative to par, make that comparison and label it as such.

Common Benchmark Errors

Automatically using par. Par often determines the coupon, but it need not be the maturity payment. Compare with the specified redemption amount when classifying relative to redemption.
Using the coupon rate as the price classification. A coupon rate describes the contractual coupon calculation; it does not define whether the price is above or below redemption.
Forgetting to add redemption to the final coupon. If the last coupon is $C$, the final cash flow is $C+R$, not $R$ alone.
Comparing only once. When $F\ne R$, state both comparisons if both are relevant. A bond can be a premium relative to one benchmark and a discount relative to the other.
Ignoring the yield period. For semiannual coupons, a nominal annual yield convertible semiannually must be divided by two to obtain the effective yield per half-year.

Key takeaway: Par, coupon basis, and redemption value are distinct bond quantities. Calculate the price from the promised cash flows, then compare that price with the redemption value to determine a redemption-based premium or discount. If par differs from redemption, report the par comparison separately.

Check Your Understanding

For each problem, identify the relevant redemption-based classification. Where useful, also compare the price with par. Round dollar amounts to the nearest cent.

A zero-coupon bond has par value $1,000 and repays $1,100 in two years. The annual effective yield is 5%. Find its price and classify it relative to par and redemption. (Answer: Price $997.73; discount relative to the $1,100 redemption value and below par.)
A bond with par value $1,000 pays annual coupons of 4% of par and repays $900 in two years. The annual effective yield is 5%. Find its price and classify it relative to redemption. (Answer: Price $890.70; discount relative to redemption.)
A bond has par value $2,000, pays annual coupons of 6% of par, and repays $2,200 in three years. The annual effective yield is 4%. Find its price and classify it relative to both benchmarks. (Answer: Price approximately $2,288.80; premium relative to both redemption and par.)
A bond has par value $1,000, a nominal annual coupon rate of 8% convertible semiannually, and redemption value $950 after four half-year periods. The nominal annual yield is 6% convertible semiannually. Find its price and compare it with par and redemption. (Answer: Price approximately $992.75; premium relative to redemption and discount relative to par.)
A bond has par value $1,000, pays annual coupons of 7% of par, and repays $800 in three years. The annual effective yield is 6%. Find its price and state both classifications. (Answer: Price approximately $858.81; premium relative to redemption and discount relative to par.)

What You'll Learn