Forward-Rate Calculation Problems With Unequal Intervals

Why Unequal Forward Intervals Need Care

A forward rate applies to an investment period that begins in the future. When the start and end dates are not equally spaced, the interval length determines how an accumulation factor must be annualized. If the quoted compounding convention also changes, the same interval accumulation may have different rate quotations, even though its financial value is unchanged.

The reliable approach is to work first with accumulation factors. Find the accumulation to the interval's end and divide by the accumulation to its start. Then express that interval factor using the requested rate convention. This separates the time interval from the quotation convention and makes the result easy to check.

Definition: Let $i_t$ be the annual effective spot rate for an investment from time 0 to time $t$, measured in years. Its accumulation factor is $C(t)=(1+i_t)^t$. For an interval from $t_1$ to $t_2$, where $0\leq t_1<t_2$, the implied forward accumulation factor is $$ F(t_1,t_2)=\frac{C(t_2)}{C(t_1)}. $$ The interval length is $\Delta=t_2-t_1$. This ratio assumes that the quoted spot rates and dates are consistent with the same no-arbitrage accumulation curve.

The spot rate $i_{t_2}$ applies from time 0 to $t_2$, not just from $t_1$ to $t_2$. Therefore, the rate for the forward interval is generally not the difference between the two spot rates. The accumulation ratio, rather than a rate subtraction, isolates the future interval.

Expressing the Interval Factor Under Different Conventions

Once $F(t_1,t_2)$ is known, match it to the accumulation formula for the requested convention. For an annual effective forward rate $f$, the interval has accumulation $(1+f)^\Delta$. For a nominal annual rate $j^{(m)}$, convertible $m$ times per year, the interval has accumulation $(1+j^{(m)}/m)^{m\Delta}$, provided the interval contains $m\Delta$ conversion periods. For a constant force of interest $\delta$, the interval accumulation is $e^{\delta\Delta}$.

F(t_1,t_2) =(1+f)^\Delta =\left(1+\frac{j^{(m)}}{m}\right)^{m\Delta} =e^{\delta\Delta}

These expressions describe equivalent ways to represent the same interval accumulation; they do not assert that the interest rate is constant across all maturities. A quoted forward rate represents a constant rate over its stated interval. When using a nominal rate, the usual periodic-compounding interpretation requires conversion dates to align with the interval. Do not apply the nominal-rate formula blindly to an interval that does not contain an integer number of conversion periods; first clarify the quotation convention.

Key formulas: If an interval accumulation factor $F$ is given, its equivalent annual effective forward rate is $f=F^{1/\Delta}-1$. Its equivalent nominal annual rate convertible $m$ times per year is $j^{(m)}=m(F^{1/(m\Delta)}-1)$, when the interval contains $m\Delta$ conversion periods. Its equivalent force of interest is $\delta=\ln(F)/\Delta$.

Why the Accumulation Ratio Works

The forward accumulation formula follows directly from investing over successive time periods. Suppose one dollar invested at time 0 accumulates to $C(t_1)$ at time $t_1$. If that amount is then invested from $t_1$ to $t_2$, let the interval accumulation factor be $F$. The total accumulation at $t_2$ is $C(t_1)F$. It must equal the accumulation $C(t_2)$ for the full investment from time 0, so:

C(t_1)F=C(t_2) \quad\Longrightarrow\quad F=\frac{C(t_2)}{C(t_1)}.

This identity is a no-arbitrage consistency condition: under the same accumulation curve, investing straight to $t_2$ and investing first to $t_1$, then over the forward interval, give the same accumulated value. It also explains why unequal intervals cause no special difficulty once the dates are used correctly. The interval length enters when $F$ is converted into an annual rate.

Worked Example: Effective Forward Rate Across Unequal Maturities

The annual effective spot rate to year 2 is 4%, and the annual effective spot rate to year 5 is 5%. Find the annual effective forward rate from year 2 to year 5. Check the result using accumulated values. Give the rate to four decimal places as a percentage.

Method. Calculate the two spot accumulation factors, divide to obtain the accumulation over the three-year forward interval, and annualize over three years.

At year 2, $C(2)=(1.04)^2=1.0816$. At year 5, $C(5)=(1.05)^5=1.2762815625$. Thus:

F(2,5)=\frac{1.2762815625}{1.0816} \approx 1.1799940478.

The interval length is $5-2=3$ years. If $f$ is the annual effective forward rate, then $(1+f)^3=1.1799940478$. Therefore:

f=(1.1799940478)^{1/3}-1 \approx 0.0567200 =5.6720\%.

Check. Using the unrounded forward rate, $1.0816(1+f)^3\approx1.0816(1.1799940478)=1.2762815625$, which reproduces the year-5 accumulation. The common mistake avoided here is annualizing over five years, the full maturity, rather than over the three-year forward interval.

Changing from a Nominal Forward Rate to a Spot Rate

A forward quote and a spot quote can use different compounding conventions. Convert the forward quote to an interval accumulation factor before combining it with a spot accumulation factor. Then, if required, convert the resulting total accumulation into the requested spot-rate convention.

Find the starting accumulation.
Use the spot rate and maturity at the forward interval's starting date.

Find the interval accumulation.
Use the forward rate, its compounding frequency, and the number of conversion periods in the interval.

Combine the factors.
Multiply the starting accumulation by the forward interval accumulation to obtain the accumulation at the ending date.

Convert to the requested spot quote.
For an annual effective spot rate at time $t$, solve $C(t)=(1+i_t)^t$.

Worked Example: Semiannual Forward Quote and Annual Effective Spot Rate

The annual effective spot rate to year 0.5 is 3%. From year 0.5 to year 2.5, the nominal annual forward rate convertible semiannually is 5.8%. Find the annual effective spot rate to year 2.5. Give the result to four decimal places as a percentage.

Method. Combine the starting spot accumulation with the semiannual forward accumulation. The forward interval is 2 years, so it contains $2(2)=4$ semiannual periods.

The starting accumulation is $C(0.5)=(1.03)^{0.5}$. Each half-year accumulation under the forward quote is $1+0.058/2=1.029$, so the interval factor is $(1.029)^4=1.121144263$ (using unrounded powers). Hence:

C(2.5)=(1.03)^{0.5}(1.029)^4 \approx 1.1378371.

Let $i_{2.5}$ denote the annual effective spot rate to year 2.5. Since $(1+i_{2.5})^{2.5}=C(2.5)$:

i_{2.5} =(1.1378371)^{1/2.5}-1 \approx 0.0530089 =5.3009\%.

Check. Raising $1.0530089$ to the power 2.5 gives approximately 1.1378371, matching the accumulation formed from the spot and forward pieces. A common error is to use five semiannual periods because year 2.5 is five half-years from time 0. The forward quote applies only from year 0.5 to year 2.5, which contains four half-year periods.

Changing a Force Quote into a Nominal Quote

A force of interest is continuous-compounding notation. If a constant force $\delta$ applies over an interval of length $\Delta$, its accumulation factor is $e^{\delta\Delta}$. To report an equivalent nominal rate convertible $m$ times per year, equate that factor to the nominal accumulation over the same interval. The conversion frequency changes the rate quotation, not the accumulated value.

Worked Example: Force Forward Rate and Quarterly Equivalent

The annual effective spot rate to year 1 is 3%. From year 1 to year 3, the force of interest is 0.04 per year. Find (a) the annual effective spot rate to year 3 and (b) the equivalent nominal annual forward rate convertible quarterly. Give both rates to four decimal places as percentages.

Method. First accumulate over the two-year forward interval using the force. Then combine it with the year-1 spot accumulation. Convert the same forward factor to quarterly nominal form for part (b).

The forward interval has length $3-1=2$, so its accumulation factor is $e^{0.04(2)}=e^{0.08}\approx1.0832870677$. The starting accumulation is $C(1)=1.03$. Therefore:

C(3)=1.03e^{0.08} \approx 1.1157856797.

For the annual effective spot rate $i_3$, solve $(1+i_3)^3=C(3)$:

i_3=(1.1157856797)^{1/3}-1 \approx 0.0371946 =3.7195\%.

For an equivalent nominal annual forward rate $j^{(4)}$ convertible quarterly, there are $4(2)=8$ quarters in the interval. Equate the accumulation factors and solve:

\left(1+\frac{j^{(4)}}{4}\right)^8=e^{0.08}, \qquad j^{(4)}=4\left(e^{0.08/8}-1\right) =4(e^{0.01}-1) \approx 0.0402007 =4.0201\%.

Check. The quarterly nominal quote gives $(1+0.0402007/4)^8\approx1.0832871$, the same forward accumulation as the force quote. Multiplying by 1.03 reproduces the year-3 accumulation. The common mistake avoided is to report the force, 4%, as the equivalent quarterly nominal rate; those quotations differ even though their interval accumulation is equal.

Common Errors and a Reliable Check

Unequal intervals create several predictable errors. A calculation should preserve both the dates and the rate convention at every step.

Using the wrong interval length. The exponent for annual effective rates is $t_2-t_1$, not $t_2$, unless the forward interval actually begins at time 0.
Subtracting spot rates. The forward rate comes from a ratio of accumulation factors; spot-rate differences do not generally produce it.
Miscounting nominal periods. For a nominal rate convertible $m$ times yearly, count $m(t_2-t_1)$ periods in the forward interval. Do not count from time 0 unless the forward interval starts there.
Mixing rates before converting to factors. A nominal rate, an annual effective rate, and a force are not directly interchangeable. Convert each to an accumulation factor over the relevant duration.
Rounding too early. Retain full calculator precision through the accumulation and rate conversion; round only the final reported rate.

A useful final test is to reconstruct the ending accumulation in two ways. Calculate it directly from the ending spot rate, and calculate it as the starting spot accumulation multiplied by the forward interval factor. The two values should agree, subject only to displayed rounding.

Key takeaway: For an unequal forward interval, first calculate $F(t_1,t_2)=C(t_2)/C(t_1)$. Then express that same factor under the requested convention using the actual interval length. Checking the accumulated values at both endpoints catches most timing and compounding errors. This calculation discipline is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the forward interval before applying a rate formula. Keep intermediate values unrounded and check the requested convention.

The annual effective spot rates to years 1 and 4 are 2% and 3%, respectively. Find the annual effective forward rate from year 1 to year 4. (Answer: 3.3355%)
The annual effective spot rate to year 2 is 2.5%. From year 2 to year 3.5, the nominal annual forward rate convertible semiannually is 4.8%. Find the accumulation factor from time 0 to year 3.5. (Answer: 1.1281000)
From year 0.5 to year 2.5, a constant force of interest of 0.025 applies. Find the equivalent annual effective forward rate for this interval. (Answer: 2.5315%)
The annual effective spot rates to years 1.5 and 4.5 are 3.2% and 4%, respectively. Find the annual effective forward rate from year 1.5 to year 4.5. (Answer: 4.4023%)
The forward accumulation factor from year 1.2 to year 3.7 is 1.10. Express this as a nominal annual forward rate convertible monthly. (Answer: 3.8185%)

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