Implied Forward Rates Versus Realized Reinvestment Rates

Why an Implied Rate Is Not a Realized Rate

A spot curve can imply the rate that is consistent with investing over a later interval, as determined by rates available today. That implied rate is useful for comparing dated cash flows and describing a no-arbitrage relationship. It does not, by itself, tell us what rate will actually be available when the later interval begins.

The distinction matters whenever money will be received before the end of an investment horizon and then reinvested. If the realized reinvestment rate differs from the rate implied by today’s spot curve, the terminal amount differs too. The comparison must use compound accumulation factors over the same dates, not a simple subtraction of annual rates.

Definition: Let $i_t$ be the annual effective spot rate for an investment from time 0 to time $t$. For $0 \leq m < n$, the annual effective forward rate $f_{m,n}$ is the rate for the interval from time $m$ to time $n$ implied by the current spot rates: $$ (1+i_n)^n=(1+i_m)^m(1+f_{m,n})^{n-m}. $$ The forward rate is an implied rate under the pricing assumptions used to construct the spot curve. It is not a guarantee that an investor will later be able to reinvest at that rate.

This definition uses annual effective rates and time measured in years. The interval length is $n-m$, so the forward accumulation factor is $(1+f_{m,n})^{n-m}$. If the dates are not whole years, the same principle applies with consistent time units and an appropriate effective-rate convention.

Deriving the Forward Rate

The forward-rate equation follows by comparing two ways to describe accumulation from time 0 to time $n$. Investing for the full $n$ years at the $n$-year spot rate produces the factor $(1+i_n)^n$. Alternatively, invest for $m$ years at the $m$-year spot rate, then use the implied forward rate for the remaining $n-m$ years. Equating the two factors gives the definition above.

(1+i_n)^n=(1+i_m)^m(1+f_{m,n})^{n-m}

Solving for the forward rate gives

f_{m,n}= \left( \frac{(1+i_n)^n}{(1+i_m)^m} \right)^{1/(n-m)}-1.

This calculation requires spot rates for the two relevant maturities and consistent compounding and timing conventions. It is an algebraic implication of those rates; it does not require a claim that future interest rates are known. Under idealized no-arbitrage assumptions, the relationship corresponds to the rate that can be locked in for the future interval through suitable lending and borrowing transactions. In practice, transaction costs, credit risk, liquidity, and investment constraints can affect whether that theoretical rate is directly available.

Identify the interval.
Use $m$ as the start time and $n$ as the end time, so the forward interval lasts $n-m$ years.

Convert each spot rate to an accumulation factor.
Calculate $(1+i_m)^m$ and $(1+i_n)^n$; do not compare the quoted annual rates alone.

Find the implied interval factor and annual rate.
Divide the longer-maturity accumulation factor by the shorter-maturity factor, then take the $(n-m)$th root and subtract 1.

Compare with the realized rate separately.
Accumulate the money available at time $m$ using the rate actually earned over the interval, then compare terminal values.

Worked Comparisons

A reliable comparison starts with a specific amount available at the forward start date. If $C$ is available at time $m$, the two terminal values at time $n$ are

C(1+f_{m,n})^{n-m} \quad\text{and}\quad C(1+r)^{n-m},

where $r$ is the annual effective rate actually earned over the interval, assuming that rate is constant for the whole interval. The difference is the difference between these accumulated values. It is not generally equal to $C(n-m)(f_{m,n}-r)$, because interest compounds.

Worked Example: A Realized Rate Below the Implied Forward Rate

The annual effective spot rate is 4% for two years and 5% for four years. An investor will have $20,000 available at year 2. Find the implied annual forward rate for years 2 to 4. If the investor instead earns an actual annual effective rate of 5% over those two years, find the shortfall at year 4 relative to earning the implied rate.

Method and assumption. Use the spot-rate accumulation factors to derive $f_{2,4}$. Assume the stated actual rate of 5% applies to both years of reinvestment, with annual compounding.

The two-year spot accumulation factor and four-year spot accumulation factor are

(1.04)^2=1.0816, \qquad (1.05)^4=1.21550625.

The accumulation factor implied for years 2 to 4 is their ratio:

\frac{1.21550625}{1.0816} =1.1238038554\ \text{(approximately)}.

This factor covers two years, so take its square root to find the annual forward accumulation factor. Equivalently,

1+f_{2,4} = \sqrt{\frac{(1.05)^4}{(1.04)^2}} = \frac{(1.05)^2}{1.04} = \frac{1.1025}{1.04} = 1.0600961538.

Thus $f_{2,4}=0.0600961538$, or 6.0096% to four decimal places.

At the implied rate, the year-4 value of $20,000 is

20{,}000(1.1238038554) \approx \$22{,}476.08.

At the actual rate of 5%, it is

20{,}000(1.05)^2 = 20{,}000(1.1025) = \$22{,}050.00.

The shortfall is $\$22{,}476.08-\$22{,}050.00=$ $426.08. This result is consistent with the accumulation factors: the implied two-year factor exceeds the actual factor by approximately $1.1238038554-1.1025=0.0213038554$, and $20{,}000(0.0213038554)\approx \$426.08$. The common mistake avoided here is treating the 5% actual rate as if it were the forward rate simply because it matches one of the spot rates.

Worked Example: Comparing Terminal Values From a One-Year Forward Start

The one-year spot rate is 2%, and the three-year spot rate is 3%. An investor expects to have $15,000 at year 1. Calculate the implied annual forward rate for years 1 to 3. Then find the terminal-value shortfall if the actual annual effective reinvestment rate is 3% for both years.

Method and assumption. Divide the three-year spot accumulation factor by the one-year spot accumulation factor. Compare the resulting implied terminal value with two years of accumulation at the stated realized rate.

The spot factors are

(1.02)^1=1.02, \qquad (1.03)^3=1.092727.

Therefore, the implied two-year accumulation factor is

\frac{1.092727}{1.02} = 1.0713009804\ \text{(approximately)}.

There are two years in the forward interval, so

1+f_{1,3} = \sqrt{1.0713009804} \approx 1.0350367, \qquad f_{1,3}\approx 0.0350367.

The implied forward rate is 3.5037% to four decimal places. The year-3 value at that implied rate is

15{,}000(1.0713009804) \approx \$16{,}069.51.

The actual two-year accumulation factor at 3% is $(1.03)^2=1.0609$, so the realized terminal value is

15{,}000(1.0609)=\$15{,}913.50.

The shortfall relative to the implied-rate value is approximately $\$16{,}069.51-\$15{,}913.50=$ $156.01. The check using unrounded factors is $15{,}000(1.0713009804-1.0609)\approx \$156.01$. This avoids rounding the forward rate too early and then allowing that rounding to change the dollar comparison.

Worked Example: A Realized Rate Above the Implied Rate

The two-year spot rate is 3%, and the four-year spot rate is 4%. An investor has $12,000 at year 2 and earns an actual annual effective rate of 5.5% for years 2 to 4. Calculate the implied forward rate and the excess of the realized year-4 value over the value at the implied rate.

Method and assumption. Derive the two-year forward factor from the two spot factors. Assume the realized 5.5% rate applies throughout the two-year reinvestment period.

The spot accumulation factors are

(1.03)^2=1.0609, \qquad (1.04)^4=1.16985856.

The implied two-year forward factor is

\frac{1.16985856}{1.0609} = \left(\frac{(1.04)^2}{1.03}\right)^2 \approx 1.1027038929.

Taking the square root gives the annual forward accumulation factor:

1+f_{2,4} = \frac{1.0816}{1.03} = 1.0500970874, \qquad f_{2,4}\approx 5.0097\%.

At the implied rate, the year-4 value is

12{,}000(1.1027038929) \approx \$13{,}232.45.

At the actual 5.5% rate, the factor is $(1.055)^2=1.113025$, giving

12{,}000(1.113025)=\$13{,}356.30.

The realized value exceeds the implied-rate value by approximately $\$13{,}356.30-\$13{,}232.45=$ $123.85. The factor check gives $12{,}000(1.113025-1.1027038929)\approx \$123.85$. The sign is sensible: the actual rate, 5.5%, is above the implied forward rate, so the actual terminal amount is larger.

Interpretation and Common Exam Traps

A forward rate summarizes what today’s spot rates imply for a later interval under the specified pricing framework. It should not be described as a known future rate or as a guaranteed forecast. The future rate available to reinvest cash may be higher or lower. That difference creates reinvestment-rate risk for an investor who must place interim cash flows again.

One trap is to average spot rates to estimate a forward rate. For example, averaging the 4% two-year spot rate and the 5% four-year spot rate gives 4.5%, but that is not the rate implied for years 2 to 4. The spot rates apply to different full investment periods, so their compound accumulation factors must first be placed on a consistent timeline. In the worked example, the correct forward rate is 6.0096%, not 4.5%.

A second trap is to subtract annual rates and multiply the difference by the principal and number of years. The dollar effect depends on compounded factors. With $C$ available at time $m$, the terminal-value difference is

C\left((1+r)^{n-m}-(1+f_{m,n})^{n-m}\right),

where the sign depends on whether the realized rate $r$ is above or below the implied rate. This expression assumes a single constant realized rate for the interval. If rates vary within the interval, the realized accumulation factor must reflect each period’s rate; it cannot be replaced by one rate unless that rate is an equivalent rate calculated from the full accumulation.

This distinction is useful when interpreting spot-curve calculations, evaluating a planned reinvestment, or checking whether a stated terminal value is consistent with an assumed rate. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus: calculate the forward rate from spot rates, keep the implied and realized rates conceptually separate, and measure the financial consequence using the correct dated accumulation factors.

Key takeaway: Spot rates imply a forward accumulation factor through a ratio of spot accumulation factors. A realized reinvestment rate is a separate input. Compare their accumulated values over the same interval; do not treat the implied rate as a guaranteed future outcome.

Check Your Understanding

For each problem, use annual effective rates and round final money amounts to the nearest cent and rates to four decimal places.

The one-year spot rate is 2%, and the two-year spot rate is 3%. Find the implied one-year forward rate for year 1 to year 2. (Answer: 4.0098%)
The two-year spot rate is 2.5%, and the four-year spot rate is 3.5%. Find the implied annual forward rate for years 2 to 4. (Answer: 4.5098%)
An investor has $5,000 at year 1. The one-year spot rate is 2%, the three-year spot rate is 3%, and the actual reinvestment rate is 2.5% for two years. Find the shortfall at year 3 relative to the amount accumulated at the implied forward rate. (Answer: $103.38 shortfall)
The two-year spot rate is 4%, and the four-year spot rate is 5%. An amount of $10,000 is available at year 2 and earns an actual 6.5% annual effective rate for two years. Find the excess of its actual year-4 value over the value at the implied forward rate. (Answer: $104.21 excess)
The one-year spot rate is 1.5%, and the three-year spot rate is 2.5%. An investor has $20,000 at year 1 and earns an actual 2% annual effective rate for two years. Find the implied forward rate and the shortfall in terminal value at year 3. (Answer: implied rate 3.0037%; shortfall $411.52)

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