Nominal Compounding Quotes for Zero-Coupon Bonds

Why the Compounding Quote Matters

A zero-coupon bond makes no interim coupon payments: its cash flow is a single redemption amount at maturity. To price it from a yield quote, the key is to interpret the quote’s compounding convention correctly. A nominal annual yield convertible several times per year is not itself the rate applied in each period. It must first be divided by the number of conversion periods per year.

This tutorial considers a bond with a known redemption amount, a known term in whole years, and a nominal annual yield convertible $m$ times per year. The bond is assumed to pay the stated redemption amount at maturity, with no interim cash flows or default. The quoted yield is assumed to apply over the full term. Under these assumptions, the price is the redemption amount discounted by the periodic accumulation factor.

Definition: A nominal annual yield $i^{(m)}$ convertible $m$ times per year is a quoted annual rate for which the rate applied in each conversion period is $i^{(m)}/m$. Over one year, the accumulation factor is $\left(1+i^{(m)}/m\right)^m$. The corresponding effective annual rate is $\left(1+i^{(m)}/m\right)^m-1$, not $i^{(m)}$ in general.

From the Quote to a Zero-Coupon Price

Let $R$ be the redemption amount, $t$ the term in years, and $m$ the number of conversion periods per year. Assume $t$ is a positive whole number and that the bond’s maturity falls exactly at the end of a conversion period. There are $mt$ periods to maturity. The periodic rate is $i^{(m)}/m$, so the accumulation factor from the purchase date to maturity is $\left(1+i^{(m)}/m\right)^{mt}$.

\text{Price}=\frac{R}{\left(1+i^{(m)}/m\right)^{mt}}

The expression assumes a positive accumulation factor, so $1+i^{(m)}/m>0$. In the usual positive-yield setting this condition is automatically satisfied. If the maturity does not fall on a conversion date or the term is not a whole number of years, the period count and timing convention require further care; the formula above is for the stated whole-year case.

Why the Formula Holds

The formula follows from equating the bond’s price at time zero to the present value of its sole future payment. At each conversion date, an amount grows by the factor $1+i^{(m)}/m$. Applying that factor successively over $mt$ periods gives the maturity value of one unit invested at purchase.

Identify the periodic rate.
The nominal annual quote is divided by its conversion frequency, giving $i^{(m)}/m$ for each period.

Count the periods.
With $m$ periods per year for $t$ years, there are $mt$ periods before redemption.

Accumulate the purchase price.
If the purchase price is $P$, its value at maturity under the quoted yield is $P\left(1+i^{(m)}/m\right)^{mt}$.

Set the maturity value equal to redemption.
Since the only maturity cash flow is $R$, solve $P\left(1+i^{(m)}/m\right)^{mt}=R$ for $P$.

Dividing both sides of the maturity-value equation by the accumulation factor gives the displayed price formula. This is the same present-value principle used with effective annual yields, but the rate and period count must follow the nominal quote’s conversion frequency.

Worked Pricing Examples

Worked Example: A Five-Year Bond with Semiannual Conversion

A hypothetical zero-coupon bond will redeem for $10,000 in five years. Its yield is quoted as 4% nominal per year convertible semiannually. Find its price.

Method. Convert the nominal quote to a half-year rate, count the half-year periods, and discount the redemption amount. This avoids treating the 4% quote as an effective annual rate.

The number of conversion periods per year is $m=2$, and the annual nominal yield is $i^{(2)}=0.04$. Thus the periodic rate is

\frac{i^{(2)}}{2}=\frac{0.04}{2}=0.02

There are $2(5)=10$ half-year periods to maturity. The accumulation factor over those periods is

(1.02)^{10}=1.218994419995

Therefore,

P=\frac{10{,}000}{1.218994419995}=8{,}203.483\ldots

Rounded to the nearest cent, the bond price is $8,203.48. Check: accumulating this price for ten periods at 2% gives approximately $8{,}203.48(1.02)^{10}=\$10{,}000.00$, subject to the displayed price rounding.

Worked Example: Comparing Semiannual and Monthly Quotes

Two hypothetical zero-coupon bonds each redeem for $1,000 in two years. Bond A has a 6% nominal annual yield convertible semiannually; Bond B has a 6% nominal annual yield convertible monthly. Price both bonds and compare them.

Method. Apply each quote using its own conversion frequency. Although the nominal figures are both 6%, the periodic rates and the resulting annual accumulation factors differ.

For Bond A, $m=2$, so the periodic rate is $0.06/2=0.03$, and the number of periods is $2(2)=4$. Its accumulation factor is

$$ (1.03)^4=1.12550881 $$

Its price is

P_A=\frac{1{,}000}{1.12550881}=888.487\ldots=\$888.49

For Bond B, $m=12$, so the periodic rate is $0.06/12=0.005$, and the number of periods is $12(2)=24$. Its accumulation factor is

(1.005)^{24}=1.12715978\ldots

Thus,

P_B=\frac{1{,}000}{1.12715978\ldots}=887.186\ldots=\$887.19

Bond B costs approximately $\$888.49-\$887.19=\$1.30$ less. The check is that the monthly quote produces a slightly larger two-year accumulation factor, so the same redemption amount has a slightly lower present value. The common mistake is to use four periods for both bonds just because both quotes are stated per year; the conversion frequency determines the period count.

Worked Example: Distinguishing a Nominal Quote from an Effective Rate

A hypothetical zero-coupon bond redeems for $10,000 in two years. Its yield is quoted as 8% nominal per year convertible quarterly. Find the price, and compare it with the price that would result if 8% were instead an effective annual yield.

Method. First use the nominal quote exactly as stated. Then calculate the alternative price under the distinct effective-rate assumption. This comparison exposes the error of treating the two types of quote as interchangeable.

For the nominal quote, $m=4$. The rate per quarter is

\frac{0.08}{4}=0.02

Two years contain $4(2)=8$ quarters. The accumulation factor and price are

(1.02)^8=1.171659381\ldots

P_{\text{nominal}}=\frac{10{,}000}{1.171659381\ldots} =8{,}534.903\ldots=\$8{,}534.90

For comparison only, if 8% were an effective annual yield, the two-year accumulation factor would be $(1.08)^2=1.1664$. The corresponding price would be

P_{\text{effective}}=\frac{10{,}000}{1.1664} =8{,}573.388\ldots=\$8{,}573.39

The stated nominal quarterly quote implies an effective annual rate of $(1.02)^4-1=0.08243216$, or 8.243216%. It therefore gives a larger two-year accumulation factor and a lower price than an 8% effective annual yield. The check is consistent: the nominal quote with quarterly conversion is the higher effective rate in this comparison, so its price must be lower.

Common Errors and a Reliable Check

Most mistakes with nominal compounding quotes come from applying a correct present-value idea with the wrong rate or number of periods. A useful check is to write the periodic rate and period count separately before entering anything into a calculator.

Using the nominal annual rate as the period rate. For a 6% nominal quote convertible monthly, the monthly rate is $0.06/12=0.005$, not 0.06.
Using years as the exponent. With quarterly conversion over three years, there are $4(3)=12$ periods, not three.
Calling the nominal rate an effective annual rate. A 6% nominal quote convertible monthly has an effective annual rate of $(1.005)^{12}-1$, not 6%.
Using a coupon-bond cash-flow pattern. A zero-coupon bond has one redemption payment in this model. There are no coupon payments to add to the present value.
Reversing the accumulation factor. A future redemption is discounted by dividing by the factor, or equivalently multiplying by its reciprocal. Multiplying the redemption by the accumulation factor moves in the wrong direction.

Key takeaway: For a nominal annual yield $i^{(m)}$ convertible $m$ times per year, use $i^{(m)}/m$ per period and $mt$ periods over $t$ whole years. Discount the single redemption amount by $\left(1+i^{(m)}/m\right)^{mt}$.

Check Your Understanding

For each problem, assume the quoted yield applies over the full term, the bond has no interim payments or default, and redemption occurs at the stated maturity. Round dollar answers to the nearest cent.

A zero-coupon bond redeems for $2,000 in three years at 10% nominal per year convertible semiannually. Find its price. (Answer: $1,492.43)
A bond redeems for $5,000 in nine months at 6% nominal per year convertible monthly. Find its price. (Answer: $4,780.52)
A bond redeems for $10,000 in two years at 8% nominal per year convertible quarterly. Find its price. (Answer: $8,534.90)
A bond redeems for $5,000 in one year at 12% nominal per year convertible quarterly. Find the periodic rate, number of periods, and price. (Answer: 3% per quarter, 4 periods, $4,442.44)
Two bonds each redeem for $10,000 in two years. Bond A yields 4% nominal convertible annually; Bond B yields 4% nominal convertible quarterly. Find both prices and identify the less expensive bond. (Answer: Bond A, $9,245.56; Bond B, $9,234.83; Bond B is less expensive.)

What You'll Learn