Forward Rates At Fractional Maturities

Why Fractional Maturities Require Care

A forward rate is tied to a particular interval: it describes the rate implied between a start date and an end date by the spot curve. Those dates need not fall on whole years. If the interval begins at 0.5 years and ends at 1.5 years, the accumulation to each date must be calculated using the spot rate for that maturity and the stated spot-rate convention. The forward rate is then annualized over the length of the interval.

This distinction matters because a spot rate is not itself an accumulation factor. Under the annual effective spot-rate convention used here, a spot rate for maturity \(t\) years gives an accumulation factor of \((1+i_t)^t\). The exponent is the maturity in years, including any fractional part. Do not replace that convention with simple-interest prorating or with a different compounding convention.

Definition: Let \(i_t\) be the annual effective spot rate for maturity \(t\) years. Under the stated annual effective convention, the accumulation factor from time 0 to time \(t\) is \((1+i_t)^t\). For \(0\leq s<t\), the annual effective forward rate from time \(s\) to time \(t\), denoted here by \(f_{s,t}\), is the annual effective rate which, applied for \(t-s\) years, produces the accumulation implied between those dates.

Formula for a Fractional-Maturity Forward Rate

The forward interval has length \(t-s\), not \(t\). Equating the accumulation from time 0 to time \(t\) with the accumulation to time \(s\) followed by the forward accumulation gives

(1+i_s)^s(1+f_{s,t})^{t-s}=(1+i_t)^t.

Solving for the annual effective forward rate gives

f_{s,t} = \left( \frac{(1+i_t)^t}{(1+i_s)^s} \right)^{1/(t-s)}-1.

The formula uses each spot rate at its own maturity. The end-date accumulation factor is divided by the start-date accumulation factor to isolate the accumulation over the forward interval. The exponent \(1/(t-s)\) converts that interval accumulation into an annual effective rate. If \(t-s=1\), the exponent is 1; for any other interval length, it is generally not.

Condition: This formula assumes that the quoted spot rates are annual effective rates and that a maturity of \(t\) years corresponds to accumulation \((1+i_t)^t\). If rates are quoted under another convention, first express the dated accumulation factors under the applicable convention. The formula does not justify interpolating between observed spot rates.

Worked Calculations

Worked Example: A One-Year Forward Interval With Half-Year Endpoints

A hypothetical spot curve gives annual effective spot rates of 2.5% at 0.5 years and 3.5% at 1.5 years. Find the annual effective forward rate from 0.5 to 1.5 years. Use the stated annual effective spot-rate convention.

Method. Calculate the accumulation factors to 0.5 and 1.5 years, divide the latter by the former, and annualize over the interval. Here the interval is \(1.5-0.5=1\) year.

The accumulation to 1.5 years is

(1.035)^{1.5}=1.05295774.

The accumulation to 0.5 years is

(1.025)^{0.5}=1.01242284.

Thus the implied accumulation factor from 0.5 to 1.5 years is

\frac{1.05295774}{1.01242284}=1.04003653.

Because this factor covers one year, the annual effective forward rate is \(1.04003653-1=0.04003653\), or 4.0037% to four decimal places. The check is that the start-date factor multiplied by the forward factor recovers the end-date factor: \(1.01242284(1.04003653)\approx1.05295774\).

Common mistake avoided. The spot rates are not simply subtracted to obtain a forward rate. The forward rate comes from the ratio of dated accumulation factors; subtracting \(2.5\%\) from \(3.5\%\) would give 1%, which does not reproduce the required accumulation.

Worked Example: Fractional Start and End Dates

A hypothetical annual effective spot curve has \(i_{0.25}=4.0\%\) and \(i_{1.25}=5.0\%\). Calculate the annual effective forward rate from 0.25 to 1.25 years. Round the final rate to four decimal places.

Method. Apply each rate for its full maturity under the annual effective convention. The forward interval is \(1.25-0.25=1\) year, even though neither endpoint is a whole-year maturity.

The end-date and start-date accumulation factors are

(1.05)^{1.25}=1.06288590, \qquad (1.04)^{0.25}=1.00985342.

Their ratio is the implied accumulation over the interval:

\frac{1.06288590}{1.00985342}=1.05251496.

The interval is one year, so the annual effective forward rate is \(1.05251496-1=0.05251496\), or 5.2515% to four decimal places. As a check, multiplying the start-date factor by the implied forward factor gives \(1.00985342(1.05251496)\approx1.06288590\), the end-date accumulation factor.

Common mistake avoided. Using only the one-year difference between the maturities does not mean that the spot rates are applied for one year. The start-date spot factor uses exponent 0.25, and the end-date spot factor uses exponent 1.25; the one-year length is used only when annualizing the ratio.

Worked Example: A Forward Rate Over a Non-Integer Number of Years

A hypothetical annual effective spot curve gives \(i_{0.5}=3.0\%\) and \(i_{2}=4.0\%\). Find the annual effective forward rate from 0.5 to 2 years. Round the final rate to four decimal places.

Method. First find the ratio of accumulation factors, then take the power \(1/(2-0.5)=1/1.5\) to convert the 1.5-year accumulation into an annual effective rate.

The end-date factor is

$$ (1.04)^2=1.08160000. $$

The start-date factor is

(1.03)^{0.5}=1.01488916.

The implied accumulation over the 1.5-year forward interval is

\frac{1.08160000}{1.01488916}=1.06573\ \text{(approximately)}.

Using the unrounded accumulation factors in the annualization,

f_{0.5,2} = \left( \frac{(1.04)^2}{(1.03)^{0.5}} \right)^{1/1.5}-1 \approx 0.0433548.

The annual effective forward rate is therefore 4.3355% to four decimal places. As a check, accumulating at this rate for 1.5 years gives approximately \(1.06573\); multiplying by the start-date factor gives approximately \(1.08160\), the end-date factor.

Common mistake avoided. The ratio \(1.06573\) is the accumulation over 1.5 years, not an annual rate. Reporting \(6.573\%\) as the annual effective forward rate would fail to annualize over the actual interval.

Using Discount Factors

A discount factor \(D(t)\) is the present value at time 0 of one unit payable at time \(t\). When spot rates and discount factors refer to the same dates and convention, \(D(t)=1/(1+i_t)^t\). Therefore, the forward accumulation factor from \(s\) to \(t\) can also be written as \(D(s)/D(t)\).

(1+f_{s,t})^{t-s} = \frac{D(s)}{D(t)}, \qquad f_{s,t} = \left(\frac{D(s)}{D(t)}\right)^{1/(t-s)}-1.

This is the same relationship expressed using discount factors rather than spot rates. It is useful when the curve data are supplied as discount factors; there is no need to convert them to spot rates first. The maturities and the interval length must still be measured in consistent year units.

Worked Example: Forward Rate From Discount Factors

A hypothetical term structure has \(D(0.5)=0.9800\) and \(D(1.5)=0.9400\). Find the annual effective forward rate from 0.5 to 1.5 years. Round the final rate to four decimal places.

Method. Divide the start-date discount factor by the end-date discount factor to obtain the one-year forward accumulation factor:

\frac{D(0.5)}{D(1.5)} = \frac{0.9800}{0.9400} = 1.04255319.

The interval is one year, so the annual effective forward rate is \(1.04255319-1=0.04255319\), or 4.2553% to four decimal places. The check is \(D(1.5)(1.04255319)\approx0.9800=D(0.5)\): investing one unit at the forward rate from 0.5 to 1.5 years gives the same time-0 value relationship encoded by the two discount factors.

Common mistake avoided. Reversing the ratio would give \(D(1.5)/D(0.5)<1\), which is a discount factor for the interval rather than its accumulation factor. For a positive forward accumulation, use the start-date discount factor divided by the end-date discount factor.

Derivation and Practical Checks

The formula follows directly from the meaning of the forward rate. Start with one unit at time 0. Accumulating it to time \(s\) gives \((1+i_s)^s\). If that amount then earns the annual effective forward rate for \(t-s\) years, its value at time \(t\) is \((1+i_s)^s(1+f_{s,t})^{t-s}\). But investing the original unit directly to time \(t\) under the spot curve gives \((1+i_t)^t\). Equating the two values and solving gives

(1+i_s)^s(1+f_{s,t})^{t-s}=(1+i_t)^t \quad\Longrightarrow\quad f_{s,t} = \left( \frac{(1+i_t)^t}{(1+i_s)^s} \right)^{1/(t-s)}-1.

This is an accumulation identity, not an assumption that the forward rate will actually be earned in the future. It also explains why the interval endpoints matter: changing either endpoint changes one or both dated accumulation factors, and changing the interval length changes the annualization exponent.

Before accepting a calculation, check these items:

Use the spot rate associated with each date: \(i_s\) at the start and \(i_t\) at the end.
Use \(s\) and \(t\), not the interval length, as the exponents on the spot accumulation factors.
Use \(t-s\) as the interval length and \(1/(t-s)\) as the annualization exponent.
For discount factors, use \(D(s)/D(t)\) as the forward accumulation factor.
Substitute the result into the accumulation identity to check that it recovers the end-date factor.

A tempting shortcut is to interpolate spot rates linearly between maturities and use the interpolated rate as a forward rate. That is not the calculation here. A forward rate is derived from the supplied dated spot factors. Interpolation is an additional modeling choice and requires an explicit rule; it cannot be inferred from the forward-rate formula.

Key takeaway: For fractional endpoints, preserve the annual effective spot convention at each maturity: use \((1+i_s)^s\) and \((1+i_t)^t\), form their ratio, and annualize over exactly \(t-s\) years.

Check Your Understanding

For each problem, assume the stated spot rates are annual effective and use the accumulation convention described above.

A hypothetical curve has \(i_{0.25}=2.0\%\) and \(i_{1.25}=3.0\%\). Find the annual effective forward rate from 0.25 to 1.25 years, to four decimal places. (Answer: 3.2515%)
A hypothetical curve has \(i_{0.5}=2.5\%\) and \(i_{1.5}=3.5\%\). Find the annual effective forward rate from 0.5 to 1.5 years, to four decimal places. (Answer: 4.0037%)
A hypothetical curve has \(i_{0.5}=3.0\%\) and \(i_{2}=4.0\%\). Find the annual effective forward rate from 0.5 to 2 years, to four decimal places. (Answer: 4.3355%)
A hypothetical curve has \(i_{0.75}=3.2\%\) and \(i_{1.25}=3.8\%\). Find the annual effective forward rate from 0.75 to 1.25 years, to four decimal places. (Answer: 4.7065%)
A hypothetical term structure gives \(D(0.5)=0.9800\) and \(D(1.5)=0.9400\). Find the annual effective forward rate from 0.5 to 1.5 years, to four decimal places. (Answer: 4.2553%)

What You'll Learn