Fractional-Term Zero-Coupon Bonds

Introduction

A zero-coupon bond makes a single payment at redemption and no interim coupon payments. Its price is therefore the present value of that one future amount. The calculation is direct when the term is a whole number of years, but care is needed when redemption is, for example, six months or one year and three months away. The yield quote tells you how interest accumulates; the term must be measured in units consistent with that quote.

This tutorial considers a deterministic bond with a known redemption amount and a stated yield convention. The bond is assumed to pay the full redemption amount on the stated date, with no default, transaction costs, taxes, or interim cash flows. There is no mortality assumption or probability distribution in these examples. The calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: A zero-coupon bond with price $P$, redemption value $R$, and time to redemption $t>0$ pays $R$ at time $t$ and makes no other payments. If the accumulation factor from time 0 to time $t$, under the stated yield convention, is $a(t)$, then its price is $P=R/a(t)$.

Fractional Terms and the Yield Convention

A fractional-year term does not, by itself, determine how to calculate interest. First identify the quoted convention. With an effective annual rate $i$, the annual accumulation factor is $1+i$. With a nominal annual rate $i^{(m)}$ convertible $m$ times per year, the rate per conversion period is $i^{(m)}/m$. With a force of interest $\delta$, accumulation is continuous. Each convention has its own accumulation factor for a fractional term.

For a nominal rate, the formula below uses the standard compound accumulation extension to a fractional number of conversion periods: a fractional period contributes the corresponding fractional power of the periodic accumulation factor. If a contract or problem instead specifies a different rule for an incomplete conversion period, such as simple interest for that part, use that stated rule. Do not silently switch conventions.

Yield quote	Accumulation to time $t$	Zero-coupon price
Effective annual rate $i$	$(1+i)^t$	$R(1+i)^{-t}$
Nominal annual rate $i^{(m)}$, convertible $m$ times per year	$(1+i^{(m)}/m)^{mt}$	$R(1+i^{(m)}/m)^{-mt}$
Force of interest $\delta$	$e^{\delta t}$	$Re^{-\delta t}$

In the nominal-rate row, $mt$ is the number of conversion periods, which need not be an integer. For example, a term of $0.375$ years under a quarterly convention has $4(0.375)=1.5$ quarterly periods. The exponent is $1.5$, not $0.375$ and not $2$.

Key result: For a fixed positive accumulation factor per conversion period, the fractional-period compound model uses the same factor raised to the fractional number of periods. Thus, with nominal rate $i^{(m)}$, the accumulation over $t$ years is $(1+i^{(m)}/m)^{mt}$. This formula assumes the specified compound accumulation convention applies through the fractional period.

Why the Price Formula Holds

The price is the amount at time 0 that accumulates to the redemption value at time $t$. Under an effective annual rate, the accumulation factor is $(1+i)^t$, including when $t$ is fractional. Equating the accumulated price with the payment due at redemption gives

$$ P(1+i)^t=R. $$

Dividing both sides by $(1+i)^t$ gives $P=R(1+i)^{-t}$. The same reasoning applies to another convention: replace the effective annual accumulation factor with the factor specified by that convention. In particular, for nominal rate $i^{(m)}$, the accumulation factor is $(1+i^{(m)}/m)^{mt}$, so

P\left(1+\frac{i^{(m)}}{m}\right)^{mt}=R \quad\Longrightarrow\quad P=R\left(1+\frac{i^{(m)}}{m}\right)^{-mt}.

The equation also provides a useful check: accumulating the calculated price to the redemption date must recover exactly $R$, apart from rounding.

Pricing with a Fractional-Year Term

Worked Example: Effective Annual Yield over One and a Half Years

A hypothetical zero-coupon bond redeems for $13,310 in $1.5$ years. Its yield is $21\%$ effective annually. Find its price to the nearest cent. Assume the redemption is certain and there are no other cash flows.

Method. The yield is effective annual, so use $P=R(1+i)^{-t}$. The term is already expressed in years; do not round $1.5$ to a whole number.

Substitute $R=13{,}310$, $i=0.21$, and $t=1.5$:

P=13{,}310(1.21)^{-1.5} =\frac{13{,}310}{1.21^{1.5}}.

Since $1.21=1.1^2$, we have $1.21^{1.5}=(1.1^2)^{1.5}=1.1^3=1.331$. Therefore

P=\frac{13{,}310}{1.331}=10{,}000.

The price is $10,000.00, to the nearest cent. Check: $10{,}000(1.21)^{1.5}=10{,}000(1.331)=13{,}310$, the stated redemption value. A common error would be to use two full years, which would discount for longer than the bond actually has to run.

Worked Example: A Fraction of a Semiannual Conversion Period

A hypothetical zero-coupon note pays $10,500 in $0.25$ years. Its yield is $20.5\%$ nominal annually, convertible semiannually. Find the price to the nearest cent. Assume compound accumulation applies to a fractional conversion period and the payment is certain.

Method. For $m=2$, the rate per semiannual period is $0.205/2=0.1025$. The number of semiannual periods is $mt=2(0.25)=0.5$. Thus the accumulation factor over the term is $(1.1025)^{0.5}$. Because $1.1025=1.05^2$, that factor is $1.05$.

P=10{,}500(1.1025)^{-0.5} =\frac{10{,}500}{1.05} =10{,}000.

The price is $10,000.00, to the nearest cent. Check: $10{,}000(1.05)=10{,}500$. This result depends on applying the nominal quote to $0.5$ of a semiannual period, as specified. It does not use one quarter of the quoted annual rate as a simple-interest discount.

Worked Example: Continuous Yield over a Fractional Term

A hypothetical zero-coupon bond redeems for $12,000 in $0.75$ years and has a force of interest of $4\%$. Find its price to the nearest cent. Assume the force is constant throughout the term and the redemption is certain.

Method. With a constant force $\delta$, accumulation over $t$ years is $e^{\delta t}$. Here, $\delta t=0.04(0.75)=0.03$.

P=12{,}000e^{-0.04(0.75)} =12{,}000e^{-0.03}.

Using $e^{-0.03}\approx0.9704455335$,

P\approx12{,}000(0.9704455335) =11{,}645.346402.

The price is $11,645.35, to the nearest cent. Check: multiplying the unrounded price by $e^{0.03}$ returns $12,000. The exponent is the force multiplied by the elapsed time; using $e^{-0.04}$ would incorrectly discount for a full year.

Solving for a Yield at a Fractional Term

The same price equation can be rearranged to solve for the yield, provided the convention and the term are known. For $P>0$, $R>0$, and $t>0$, the effective annual yield, nominal annual yield convertible $m$ times per year, and force of interest implied by the price are

i=\left(\frac{R}{P}\right)^{1/t}-1, \qquad i^{(m)}=m\left[\left(\frac{R}{P}\right)^{1/(mt)}-1\right], \qquad \delta=\frac{\ln(R/P)}{t}.

These are separate answers to separate yield conventions. They should not be treated as interchangeable quoted rates. In particular, $i^{(m)}$ depends on the conversion frequency $m$; a nominal rate is incomplete without that frequency.

Worked Example: Solving for a Nominal Yield with a Fractional Number of Periods

A hypothetical zero-coupon bond costs $10,000 and pays $12,100 in $0.5$ years. Find its nominal annual yield convertible quarterly. Assume the price is observed at time 0, redemption is certain, and the stated nominal convention applies to fractional periods.

Method. The number of quarterly periods is $mt=4(0.5)=2$. Let $j=i^{(4)}$. Equate the accumulated price to the redemption amount:

10{,}000\left(1+\frac{j}{4}\right)^2=12{,}100.

Divide by $10,000 to obtain the required accumulation factor:

\left(1+\frac{j}{4}\right)^2=1.21.

Taking the positive square root gives $1+j/4=1.1$, so $j/4=0.1$ and $j=0.40$. The nominal annual yield convertible quarterly is 40.0%. Check: the quarterly rate is $10\%$, so two quarters accumulate $10,000 by $1.1^2=1.21$, producing $12,100. The common mistake avoided here is solving for an effective annual rate and reporting it as the requested nominal rate.

Common Fractional-Term Traps

A reliable calculation begins by writing down the quote, the term in years, and the number of conversion periods when the quote is nominal. Then use one accumulation factor throughout the calculation. The following errors are especially easy to make:

Using the wrong exponent. For a nominal rate convertible quarterly, the exponent is $4t$, not $t$. A $0.5$-year term contains two quarterly periods.
Rounding the term to a whole year or period. A term of $0.75$ years is not one year. Preserve the fractional term in the exponent.
Using a different convention for a partial period without being told to. Under the fractional-power compound model, $k$ conversion periods, including a fractional part, use $(1+i^{(m)}/m)^k$. Simple-interest prorating is a different assumption.
Mixing annual and periodic rates. Use $i^{(m)}/m$ as the periodic rate; do not put the unconverted nominal annual rate directly into the periodic accumulation factor.
Rounding before the final answer. Keep the accumulation factor or logarithm at full calculator precision, then round the final money amount to cents or the yield to the requested precision.

For instance, in the semiannual example above, an incorrect calculation might use the year fraction $0.25$ as the exponent:

P_{\text{wrong}}=10{,}500(1.1025)^{-0.25} \approx \$10{,}246.95.

This uses one quarter of a conversion period, although $0.25$ years is half of a semiannual period. The correct exponent is $mt=2(0.25)=0.5$, giving $P=10{,}500(1.1025)^{-0.5}=\$10{,}000.00$. Accumulating that correct price by $1.1025^{0.5}=1.05$ recovers the $10,500 redemption value.

Key takeaway: Convert the fractional term according to the stated yield convention before discounting. Use $t$ as the exponent for an effective annual rate, $mt$ for a nominal rate convertible $m$ times per year, and $\delta t$ in the exponential for a constant force of interest.

Check Your Understanding

For each problem, use only the yield convention stated. Assume each hypothetical redemption is certain and there are no interim cash flows.

A zero-coupon bond redeems for $12,100 in $0.5$ years and yields $21\%$ effective annually. Find its price to the nearest cent. (Answer: $11,000.00)
A note pays $10,500 in $0.25$ years and yields $20.5\%$ nominal annually, convertible semiannually. Under the fractional-power compound convention, find its price to the nearest cent. (Answer: $10,000.00)
A bond redeems for $10,000 in $0.5$ years and has a constant force of interest of $6\%$. Find its price to the nearest cent. (Answer: $9,704.46)
A zero-coupon bond costs $8,000 and redeems for $10,648 in $1.5$ years. Find its effective annual yield. (Answer: $21\%$)
A zero-coupon bond costs $10,000 and redeems for $12,100 in $0.5$ years. Find its nominal annual yield convertible quarterly. (Answer: $40\%$)

Yield quote	Accumulation to time \(t\)	Zero-coupon price
Effective annual rate \(i\)	\((1+i)^t\)	\(R(1+i)^{-t}\)
Nominal annual rate \(i^{(m)}\), convertible \(m\) times per year	\((1+i^{(m)}/m)^{mt}\)	\(R(1+i^{(m)}/m)^{-mt}\)
Force of interest \(\delta\)	\(e^{\delta t}\)	\(Re^{-\delta t}\)

What You'll Learn