Deriving A One-Period Forward Rate

Why a Forward Rate Can Be Derived From Spot Rates

A spot rate applies from the valuation date to a particular maturity. A forward rate applies to an interval that begins in the future. For example, the one-year spot rate describes accumulation from today to year 1, while the two-year spot rate describes accumulation from today to year 2. Together, those spot rates determine the rate for the one-year interval from year 1 to year 2 that makes two investment strategies accumulate to the same value.

The forward rate derived here is an implied break-even rate: it is the rate that equates the stated spot-rate accumulations under the model. It is not, by itself, a prediction of the interest rate that will actually prevail in one year.

Definition: Let $s_1$ be the one-year effective annual spot rate and $s_2$ the two-year effective annual spot rate, both observed at time 0. Let $f_{1,2}$ be the one-year effective forward rate applying from time 1 to time 2. The subscripts $1,2$ identify the start and end of that forward interval.

The rates must be quoted on a consistent basis: here, each is an effective annual rate, and the time intervals are whole years. For the usual positive accumulation factors, the rates satisfy $1+s_1>0$ and $1+s_2>0$. The derivation assumes that the same investment can be valued using either strategy, with no differences in credit risk, currency, taxes, transaction costs, or other cash-flow conditions.

The Two Investment Strategies

Start with one dollar at time 0. In the first strategy, invest it for two years at the two-year spot rate. Since $s_2$ is an effective annual rate, the two-year accumulated value is $(1+s_2)^2$.

In the second strategy, invest for one year at the one-year spot rate, then apply the one-year forward rate for the interval from year 1 to year 2. The accumulated value is $(1+s_1)(1+f_{1,2})$. Equating the two amounts gives the forward rate.

(1+s_2)^2=(1+s_1)(1+f_{1,2})

f_{1,2}=\frac{(1+s_2)^2}{1+s_1}-1

The square on $1+s_2$ matters: $s_2$ is an annual effective rate applied over two years. The one-year spot rate, in contrast, is applied once. The forward rate is then the rate for one additional year, so its accumulation factor is $1+f_{1,2}$, not $(1+f_{1,2})^2$.

Deriving the Formula From Accumulated Values

The equality is justified by comparing what the same initial amount would be worth at year 2 through the two strategies. If the amounts differed while both strategies were available on the stated terms, an investor could choose the higher accumulated value for the same initial investment. The break-even forward rate removes that difference. This argument is an accumulation comparison; it does not require forecasting a future observed rate.

Write the two-year spot accumulation.
One dollar invested from time 0 to time 2 accumulates to $(1+s_2)^2$, because $s_2$ is an effective annual rate for each of the two years.

Write the staged accumulation.
The first year at $s_1$ produces $1+s_1$ dollars at time 1. Applying the forward accumulation factor for the next year produces $(1+s_1)(1+f_{1,2})$ dollars at time 2.

Equate the time-2 values and solve.
Set $(1+s_2)^2=(1+s_1)(1+f_{1,2})$, divide by $1+s_1$, and subtract 1.

This derivation depends on matching the same initial amount, valuation date, maturity date, and annual effective convention. If a problem instead supplies nominal rates, rates with different compounding periods, or rates for different dates, the accumulation factors must first be expressed on a consistent basis. The formula above should not be applied by simply inserting quoted percentages that represent different conventions.

Worked Example: Deriving a Forward Rate From Two Spot Rates

A hypothetical investment schedule has a one-year effective annual spot rate of 3% and a two-year effective annual spot rate of 4%. Find the one-year forward rate from year 1 to year 2. Also verify the result for an initial investment of $10,000. Assume both spot rates are available at time 0, the annual effective convention applies, and there are no transaction costs or differences in investment risk.

Method. Equate the two-year accumulation under the two-year spot rate to the accumulation from investing for one year at $s_1$ and then for one year at $f_{1,2}$. This avoids treating the two-year spot rate as a one-year rate.

First calculate the two-year accumulation factor:

$$ (1+s_2)^2=(1.04)^2=1.0816 $$

The first-year accumulation factor is $1+s_1=1.03$. Therefore,

1+f_{1,2}=\frac{1.0816}{1.03}=1.0500970874

Subtract 1:

f_{1,2}=1.0500970874-1=0.0500970874

Thus the forward rate is 5.0097% effective for the year from time 1 to time 2, rounded to four decimal places as a percentage.

Verification. The direct two-year accumulation of $10,000 is

10{,}000(1.0816)=\$10{,}816.00

Under the staged strategy, the amount at year 1 is $10{,}000(1.03)=\$10{,}300.00$. Applying the unrounded forward accumulation factor gives

10{,}300(1.0500970874)=\$10{,}816.00

The two strategies agree at year 2, as required. The common mistake this calculation avoids is using $4\%$ twice as if it were the rate for the second year alone; $4\%$ is the annual effective spot rate for the full two-year investment.

Interpreting the Sign and Size

A forward rate is determined by the two accumulation factors, not by subtracting or averaging the quoted spot rates. It can be above or below either spot rate. In particular, a negative forward rate is possible even when both spot rates are positive: a lower two-year spot rate can imply that the second-year accumulation factor must be below 1 to reconcile the two strategies. A negative rate means $1+f_{1,2}<1$; it does not mean that the accumulated amount is negative.

Worked Example: A Negative Implied Forward Rate

Suppose the one-year effective annual spot rate is 5% and the two-year effective annual spot rate is 2%. Calculate the forward rate from year 1 to year 2, then check the result using an initial investment of $100,000. Assume the same annual effective convention and matched investment conditions used in the derivation.

Method. Calculate the two-year spot accumulation factor, divide by the first-year spot accumulation factor, and subtract 1. Do not infer the forward rate from the direction of the spot rates alone.

The two-year spot accumulation factor is

$$ (1.02)^2=1.0404 $$

The first-year accumulation factor is $1.05$. Hence,

1+f_{1,2}=\frac{1.0404}{1.05}=0.9908571429

Therefore,

f_{1,2}=0.9908571429-1=-0.0091428571

The implied forward rate is $-0.9143\%$ effective for one year, rounded to four decimal places as a percentage. Its accumulation factor is positive, $0.9908571429$, but less than 1.

Verification. Investing $100,000 for two years at the two-year spot rate gives

100{,}000(1.0404)=\$104{,}040.00

In the staged strategy, the amount at year 1 is $100{,}000(1.05)=\$105{,}000.00$. Applying the forward accumulation factor gives

105{,}000(0.9908571429)=\$104{,}040.00

The equal final values confirm the result. The common mistake is to conclude that a negative forward rate is impossible because the spot rates are positive. The accumulation comparison shows why that conclusion is incorrect.

Using the Formula Reliably

A useful exam method is to identify the time-0 spot rates and write the accumulation factors before substituting. This keeps the maturity information visible and makes it easier to detect a wrong exponent or an inappropriate rate convention.

Label the rates by maturity.
Identify $s_1$ as the one-year effective annual spot rate and $s_2$ as the two-year effective annual spot rate.

Write each strategy's year-2 value per dollar.
Use $(1+s_2)^2$ for the two-year spot investment and $(1+s_1)(1+f_{1,2})$ for the staged investment.

Solve and retain precision.
Compute $f_{1,2}=(1+s_2)^2/(1+s_1)-1$. Keep intermediate values unrounded, then round the final rate to the requested precision.

Check by re-accumulating.
Confirm that $(1+s_1)(1+f_{1,2})$ equals $(1+s_2)^2$, allowing for any rounding in the displayed rate.

The formula is compact, but its meaning depends on the timeline. A spot rate starts at time 0; the forward rate in this tutorial starts at time 1 and ends at time 2. A common error is to use $s_2-s_1$, or the approximation $2s_2-s_1$, as the exact answer. The exact relation uses compounded accumulation factors, so these shortcuts generally do not give the correct rate.

Worked Example: Avoiding the Arithmetic-Rate Shortcut

Suppose the one-year effective annual spot rate is 4% and the two-year effective annual spot rate is 3%. Find the one-year forward rate from year 1 to year 2, and compare it with the shortcut $2s_2-s_1$. Assume annual effective rates and the matched investment conditions stated earlier.

Method. Use the accumulation equation for the exact answer, then calculate the shortcut separately to see why it is not the governing formula.

The two-year spot accumulation factor is

$$ (1.03)^2=1.0609 $$

Divide by the first-year factor $1.04$:

1+f_{1,2}=\frac{1.0609}{1.04}=1.0200961538

Thus,

f_{1,2}=1.0200961538-1=0.0200961538

The exact implied forward rate is 2.0096%, rounded to four decimal places as a percentage. The shortcut gives

2s_2-s_1=2(3\%)-4\%=2\%

That shortcut is close here, but it is not exact: it differs from 2.0096%. The correct check is that $1.04(1.0200961538)=1.0609$. The common mistake is to replace the compounded equality with a linear relation between rates.

Common Errors and Why the Distinction Matters

Using the wrong exponent. The two-year spot rate is annual effective, so its two-year factor is $(1+s_2)^2$, not $1+2s_2$.
Applying the forward rate for two years. The rate $f_{1,2}$ covers only the interval from time 1 to time 2, so its factor is $1+f_{1,2}$.
Subtracting spot rates. The difference $s_2-s_1$ does not equate the accumulated values and is generally not the forward rate.
Interpreting the forward as a forecast. The calculation produces an implied break-even rate from the given spot rates. It does not establish what a future realized rate will be.
Rounding too early. Keep the quotient unrounded until the final rate; otherwise the year-2 check may show a small artificial discrepancy.

This derivation is useful whenever a problem asks for the rate that connects two spot maturities through a specific future one-year interval. The key is to compare values at the same date and to use the correct effective annual accumulation factor for each interval.

Key takeaway: With annual effective spot rates $s_1$ and $s_2$, the one-year forward rate from year 1 to year 2 is $f_{1,2}=(1+s_2)^2/(1+s_1)-1$. It is obtained by equating the two strategies' accumulated values at year 2.

Check Your Understanding

For each problem, use annual effective rates and keep intermediate calculations unrounded. Assume the same investment conditions used in the derivation.

The one-year spot rate is 2.5% and the two-year spot rate is 3.5%. Calculate the one-year forward rate from year 1 to year 2. (Answer: 4.5098%)
The one-year spot rate is 6% and the two-year spot rate is 4.5%. Find the implied forward rate and state whether its accumulation factor is above or below 1. (Answer: 3.0212%; above 1)
The one-year spot rate is 3% and the two-year spot rate is 2%. Calculate the forward rate and verify the accumulated value at year 2 for an initial $1,000 investment. (Answer: 1.0097%; $1,040.40)
The one-year spot rate is 4.8% and the two-year spot rate is 4%. Find the forward rate from year 1 to year 2. (Answer: 3.2061%)
The one-year spot rate is 2.2% and the two-year spot rate is 2%. Find the forward rate, then compare the year-2 accumulation for $50,000 under the two-year spot strategy and the staged strategy. (Answer: 1.8004%; both strategies accumulate to $52,020.00)

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