Maturity Units and Fractional Periods

Why Maturity Units Matter

A spot rate is tied to a particular maturity, and its quoted compounding convention determines how that rate is used. A maturity of 15 months can be written as 15 months, 5 quarters, or 1.25 years; these are three descriptions of the same time. The calculation must use the time unit required by the rate convention, not simply the unit in which the maturity happens to be stated.

The most common errors are using 15 as an exponent for a rate quoted per year, treating a nominal rate as an annual-effective rate, or applying a spot rate for one maturity to a different maturity. The basic discipline is to identify the actual maturity, express it in a consistent time unit, and then use the formula matching the quote.

Definition: A spot rate is an interest rate for a stated maturity $T$, measured from the valuation date. An accumulation factor describes the growth of one unit of value over that maturity; its reciprocal is the corresponding discount factor. The formula for each factor depends on both the rate convention and the maturity.

Convert the Maturity Before Calculating

When the rate convention is annual, the time variable is normally expressed in years. Under the simple calendar conversion used for a stated number of months or quarters,

T=\frac{\text{number of months}}{12} \qquad\text{or}\qquad T=\frac{\text{number of quarters}}{4}.

Thus 9 months is $9/12=0.75$ years, 5 quarters is $5/4=1.25$ years, and 18 months is $18/12=1.5$ years. The conversion changes the representation of time, not the date itself.

This conversion assumes the maturity is specified in whole months or quarters and that a year is being treated as 12 months or 4 quarters. If a problem instead gives calendar dates and specifies an actual-day or other day-count convention, use that convention to determine the year fraction. Do not silently substitute a month-count conversion for an explicitly stated day-count rule.

Quoted maturity	Equivalent time in years	Typical use
6 months	$6/12=0.5$	Annual-effective or continuous rate formulas
3 quarters	$3/4=0.75$	Annual rate formulas; 3 periods if quarterly conversion is specified
15 months	$15/12=1.25$	Annual rate formulas; 5 quarters
18 months	$18/12=1.5$	Annual rate formulas; 6 quarters

Match the Formula to the Rate Convention

Let $s_T$ denote an annual-effective spot rate for maturity $T$ years. Under annual-effective compounding, the accumulation factor is $(1+s_T)^T$, and the discount factor is its reciprocal. The exponent is the maturity in years, even when it is fractional.

A_T=(1+s_T)^T, \qquad D_T=\frac{1}{(1+s_T)^T}=(1+s_T)^{-T}.

For a nominal annual rate $i^{(m)}$ convertible $m$ times per year, the rate per conversion period is $i^{(m)}/m$. If there are $mT$ conversion periods over the maturity, the accumulation factor is $(1+i^{(m)}/m)^{mT}$. When the maturity falls between the stated conversion dates, using a fractional value of $mT$ is appropriate only when the problem's rate model allows fractional-period accumulation. A contract that prescribes a separate rule for partial periods must be valued under that rule instead.

A_T=\left(1+\frac{i^{(m)}}{m}\right)^{mT}, \qquad D_T=\left(1+\frac{i^{(m)}}{m}\right)^{-mT}.

For a continuously compounded spot rate $\delta_T$, the accumulation and discount factors are $e^{\delta_T T}$ and $e^{-\delta_T T}$. Here too, $T$ is expressed in years when $\delta_T$ is an annual force of interest.

A_T=e^{\delta_T T}, \qquad D_T=e^{-\delta_T T}.

These formulas use the rate appropriate to maturity $T$. A spot curve may have different rates at different maturities, so a rate observed for 6 months is not automatically the rate to use for 18 months.

Why Accumulation and Discount Factors Are Reciprocals

The discount factor reverses the accumulation over the same maturity under the same rate model. For example, if one unit grows by a factor $A_T$, then the amount that grows to one unit must be $1/A_T$. This gives the reciprocal relationship for each of the rate conventions above.

For annual-effective compounding, multiplying the factors makes the cancellation explicit:

(1+s_T)^T(1+s_T)^{-T} =(1+s_T)^{T-T} =(1+s_T)^0 =1.

The same reasoning applies to nominal compounding because the exponents $mT$ and $-mT$ cancel, and to continuous compounding because $\delta_T T-\delta_T T=0$. This is a useful check on an unrounded calculation. Rounded factors may not multiply to exactly 1, so report their rounded product as an approximation rather than forcing it to equal 1.

Worked Applications

Worked Example: Nine Months at an Annual-Effective Spot Rate

A hypothetical investment is valued over 9 months. The annual-effective spot rate for that exact maturity is 8%. Find the accumulation factor and the accumulated value of $10,000. Round the factor to six decimal places and the dollar amount to the nearest cent.

Method. Convert the maturity into years, then use the annual-effective formula. The rate is already specified for the 9-month maturity, so no other maturity's spot rate is needed.

The time in years is $T=9/12=0.75$. Therefore,

A_T=(1+0.08)^{0.75} =1.08^{0.75} =1.0594197\ldots

The accumulated value is

\$10{,}000(1.0594197\ldots) =\$10{,}594.197\ldots =\$10{,}594.20.

The accumulation factor rounded to six decimal places is 1.059420. The result is greater than the initial amount, as expected for a positive rate. The mistake to avoid is using an exponent of 9: the annual-effective exponent is the maturity in years, $0.75$, not the number of months.

Worked Example: Five Quarters at a Nominal Quarterly Rate

A hypothetical account earns a nominal annual rate of 7.2% convertible quarterly for 5 quarters. Find the accumulation factor and the annual-effective rate equivalent over one year. Give rates to four decimal places.

Method. A nominal rate convertible quarterly has four conversion periods per year. First find the rate per quarter, then count the quarterly periods in the 5-quarter maturity.

The quarterly rate is $0.072/4=0.018$, or 1.8%. There are exactly 5 quarterly periods, so the accumulation factor is

A=\left(1+\frac{0.072}{4}\right)^5 =1.018^5.

Building the power one step at a time,

1.018^2=1.036324, \qquad 1.018^3=1.054977832,

1.018^4=1.073967433, \qquad 1.018^5=1.093298847\ldots

Thus the accumulation factor is approximately 1.093299. For one year, there are 4 quarterly periods, so the annual-effective rate equivalent to the nominal quote is

1.018^4-1 =1.073967433-1 =0.073967433 =7.3967\%.

The five-quarter accumulation exceeds one because the rate is positive. Do not calculate $1.072^{1.25}$: that treats the nominal rate as though it were an annual-effective rate, which it is not.

Worked Example: Seven Months at a Continuous Spot Rate

For a hypothetical maturity of 7 months, the continuously compounded spot rate is 4.8% per year. Calculate the accumulation and discount factors, rounding each to four decimal places. Also check their displayed product.

Method. Express the time in years and apply the continuous-rate formulas. Since the rate is annual, first convert $7$ months to $T=7/12$ years.

\delta_T T=0.048\left(\frac{7}{12}\right)=0.028.

The accumulation factor is

A_T=e^{0.028}=1.028394\ldots\approx1.0284.

The discount factor is

D_T=e^{-0.028}=0.972388\ldots\approx0.9724.

Using the displayed four-decimal factors,

1.0284(0.9724)=1.00001616\approx1.0000 \quad\text{to four decimal places.}

The unrounded factors multiply to exactly 1 because $e^{0.028}e^{-0.028}=1$. The rounded factors are only approximations; their product must be rounded from the displayed multiplication, not assumed to equal exactly 1. A common error is to use $e^{0.048(7)}$, treating 7 months as 7 years rather than $7/12$ of a year.

Worked Example: Reading Two Maturities from a Spot Curve

A hypothetical spot curve gives annual-effective spot rates of 4% for 6 months and 8% for 18 months. Find the discount factor for each maturity, rounding to six decimal places. Explain why the two rates cannot be interchanged.

Method. Convert each date into years and pair it with the rate quoted for that maturity. Apply the annual-effective discount factor separately.

For 6 months, $T=6/12=0.5$ years, and the applicable rate is 4%:

D_{0.5}=(1.04)^{-0.5} =\frac{1}{\sqrt{1.04}} =\frac{1}{1.0198039\ldots} =0.980581\ldots \approx0.980581.

For 18 months, $T=18/12=1.5$ years, and the applicable rate is 8%:

D_{1.5}=(1.08)^{-1.5} =\frac{1}{1.08\sqrt{1.08}} =\frac{1}{1.1223689\ldots} =0.890972\ldots \approx0.890972.

The factors use both the correct maturity and its corresponding spot rate. Using 4% for 18 months or 8% for 6 months would ignore the maturity-specific information in the spot curve. The factors are both less than 1, consistent with positive rates.

Common Traps and a Reliable Procedure

A rate quote contains more than a percentage: it also has a compounding convention and an associated maturity. A calculation can use a correct-looking percentage and still be wrong if the time unit or convention is mismatched. In particular, the expression $(1+s_T)^T$ is appropriate for an annual-effective rate, not for a nominal rate merely because both are quoted per year.

Identify the maturity.
Record the date or stated duration of the cash flow, and check which spot rate applies to that maturity.

Convert the time unit.
For annual formulas, express months as months divided by 12 and quarters as quarters divided by 4, unless a specified day-count convention requires another year fraction.

Identify the rate convention.
Use annual-effective, nominally convertible, or continuous compounding as actually quoted.

Calculate and check.
Use the matching exponent and verify that accumulation and discount factors are reciprocals before rounding.

These unit and convention checks are useful throughout term-structure calculations: they keep a maturity measured in months from being confused with a rate measured per year, and keep the spot rate associated with one date from being applied to another. Consistent conversion is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Key takeaway: Convert the maturity to the time unit required by the quote, use the spot rate for that maturity, and apply the formula for its compounding convention. For annual rates, months and quarters ordinarily become fractional years before the exponent is applied.

Check Your Understanding

For each problem, identify the time unit and rate convention before calculating. Round factors to four decimal places unless otherwise stated.

A 3-quarter maturity has an annual-effective spot rate of 4%. Find its accumulation factor. (Answer: $1.04^{0.75}\approx1.0299$)
A 14-month maturity has a continuously compounded spot rate of 3.6% per year. Find its discount factor. (Answer: $e^{-0.036(14/12)}=e^{-0.042}\approx0.9589$)
A rate of 8% nominal, convertible semiannually, applies for 9 months. Under a model that permits fractional conversion periods, find the accumulation factor. (Answer: $(1+0.08/2)^{2(9/12)}=1.04^{1.5}\approx1.0606$)
A 15-month maturity has an annual-effective spot rate of 4.8%. Find the discount factor, rounding to four decimal places. (Answer: $(1.048)^{-15/12}=(1.048)^{-1.25}\approx0.9431$)
A hypothetical spot curve gives annual-effective rates of 3% at 1 year and 5% at 2 years. Find the discount factors at both maturities and state which rate applies at 18 months if no 18-month rate is supplied. (Answer: $D_1=1/1.03\approx0.9709$, $D_2=1/(1.05)^2\approx0.9070$; neither quoted rate is identified as the 18-month spot rate.)

Quoted maturity	Equivalent time in years	Typical use
6 months	\(6/12=0.5\)	Annual-effective or continuous rate formulas
3 quarters	\(3/4=0.75\)	Annual rate formulas; 3 periods if quarterly conversion is specified
15 months	\(15/12=1.25\)	Annual rate formulas; 5 quarters
18 months	\(18/12=1.5\)	Annual rate formulas; 6 quarters

What You'll Learn