Reinvestment At Implied Forward Rates

Why Rollover Accumulation Needs Care

An investment may be described as being rolled over: it earns interest during one interval, and its accumulated value is reinvested for the next interval. When the rates for those intervals are implied by a term structure, the calculation links forward rates to an accumulated value. The key is to match each rate to its own time interval and compounding convention.

A computed rollover value is not automatically a promise about what an investor will actually earn. An implied forward rate is derived from current spot rates. It represents the rate consistent with those current rates under the specified no-arbitrage model; it is not, by itself, a forecast of a future market rate. To treat a computed value as locked in, the relevant forward rates must be contractually available, or the investment must otherwise earn those rates with the assumed timing and terms.

Definition: A rollover investment places an initial amount into an investment for one interval, then reinvests the accumulated amount for a later interval, repeating as needed. If $f_{s,t}$ is an annual effective forward rate for the interval from time $s$ to time $t$, the modelled accumulation factor for that interval is $(1+f_{s,t})^{t-s}$. This convention assumes the annual effective rate applies over the interval, including when its length is fractional.

The Rollover Accumulation Formula

Suppose an initial amount $P$ is invested from time 0 to time $T$ through successive intervals with boundaries $0=t_0<t_1<\cdots<t_n=T$. For each interval, use the forward rate that applies to those exact start and end times. Under the annual effective convention in the definition, the amount at time $T$ is

P_T=P\prod_{k=1}^{n}(1+f_{t_{k-1},t_k})^{t_k-t_{k-1}}.

If a rate is quoted as a single effective rate for its full interval, rather than as an annual effective rate, use the interval accumulation factor directly instead of raising it to the interval length. The quote’s convention determines the factor; do not apply an annual-rate exponent to a rate that already represents the whole interval.

Model assumptions: The formula assumes the stated forward rates apply to the specified intervals, interest is reinvested at each boundary, and no cash is withdrawn. For a deterministic, no-arbitrage comparison with locking in an investment, the rates and cash flows must also be comparable: in particular, ignore transaction costs, taxes, default risk, and differences in borrowing and lending terms. An actual rollover at future market rates can have a different outcome if those rates differ from the implied forwards.

Why the Forward Factors Multiply

The multiplication follows from carrying each interval’s ending value into the next interval as its new principal. For two intervals, the first grows $P$ to $P(1+f_{t_0,t_1})^{t_1-t_0}$. The second applies its factor to that entire amount, not just to the original principal:

P_{t_2} = \left[P(1+f_{t_0,t_1})^{t_1-t_0}\right] (1+f_{t_1,t_2})^{t_2-t_1}.

Repeating the same reasoning for each interval gives the product formula. The formula is also consistent with the product identity for chained forward rates: the product of the interval accumulation factors equals the accumulation factor implied by the spot rate for the full term. Thus, in the same no-arbitrage model, rolling through the implied forward rates has the same terminal value as investing for the full term at the corresponding spot rate.

Worked Example: Rolling Through Three Annual Forward Rates

A hypothetical investor places $12,000 at time 0. The annual effective implied forward rates are 3% for year 1, 4% for year 2, and 5% for year 3. Find the modelled accumulated value at time 3. Assume each rate applies over its stated one-year interval and that the full amount is reinvested at each year-end.

Method. Apply each one-year accumulation factor in sequence. Since each interval is one year, its factor is $1+f$.

After year 1: $12{,}000(1.03)=12{,}360$.

After year 2: $12{,}360(1.04)=12{,}854.40$.

After year 3: $12{,}854.40(1.05)=13{,}497.12$.

Equivalently, the total factor is $1.03(1.04)(1.05)=1.12476$, so $12{,}000(1.12476)=13{,}497.12$. The check agrees with the sequential calculation. The modelled time-3 value is $13,497.12. This calculation avoids the mistake of applying each rate to the initial $12,000 instead of reinvesting the accumulated balance.

Fractional Intervals and Annual Effective Rates

A forward rate can apply to a period shorter or longer than one year. If $f_{s,t}$ is an annual effective rate for an interval of length $t-s$ years, its accumulation factor is $(1+f_{s,t})^{t-s}$. For example, a half-year interval at a 2% annual effective rate has factor $(1.02)^{0.5}$, not $1.02$. The exponent records the length of the interval in years.

Worked Example: A Half-Year Followed by a Longer Interval

A hypothetical $10,000 investment earns an annual effective implied forward rate of 2% from time 0 to time 0.5, then 4% from time 0.5 to time 2. Find the time-2 accumulated value. Assume fractional-year accumulation uses annual effective rates and the full balance is reinvested at time 0.5.

Method. The first interval lasts 0.5 years, and the second lasts $2-0.5=1.5$ years. Apply each annual effective rate for its own interval length.

At time 0.5:

10{,}000(1.02)^{0.5} = 10{,}000(1.009950494) \approx 10{,}099.50494.

From time 0.5 to time 2: the interval factor is $(1.04)^{1.5}\approx 1.060596059$. Therefore,

10{,}000(1.02)^{0.5}(1.04)^{1.5} \approx 10{,}000(1.071149514) \approx 10{,}711.49514.

Rounding only the final amount to the nearest cent gives $10,711.50. The factor check confirms that the total accumulation factor is approximately 1.071149514. A common error is to apply the 4% rate for just one year, or to multiply by 1.04 without accounting for the 1.5-year interval. Neither uses the stated annual effective convention correctly.

Checking a Rollover Against Spot Rates

When the interval forward rates are implied by spot rates, the product of their accumulation factors must match the spot accumulation factor over the full term. For annual effective spot rates, let $i_t$ denote the spot rate for a term of $t$ years. For adjacent integer-year maturities, the forward factor from year $t-1$ to year $t$ is the ratio of the two spot accumulation factors:

1+f_{t-1,t} = \frac{(1+i_t)^t}{(1+i_{t-1})^{t-1}}.

This is the product identity for chained forward rates expressed as a single adjacent-interval factor. Multiplying those ratios cancels the intermediate spot factors, leaving the full-term spot accumulation factor.

Worked Example: Deriving Rollover Rates from Spot Rates

The annual effective spot rates for one, two, and three years are 3%, 4%, and 5%, respectively. A hypothetical investor places $8,000 at time 0 and rolls it over at the implied one-year forward rates. Find the time-3 value. Assume each rollover rate is available for its interval and ignore costs and credit risk.

Method. The first-year forward rate equals the one-year spot rate, 3%. Derive each later adjacent forward factor by taking the ratio of the corresponding spot accumulation factors.

First interval: the factor is $1.03$, so the balance at time 1 is $8{,}000(1.03)=8{,}240$.

Second interval: the forward factor is

\frac{(1.04)^2}{1.03} = \frac{1.0816}{1.03} \approx 1.050097087.

The implied rate is approximately 5.0097%. Applying its factor gives $8{,}240(1.050097087)\approx 8{,}652.80$, which also equals $8{,}000(1.04)^2$.

Third interval: its forward factor is

\frac{(1.05)^3}{(1.04)^2} = \frac{1.157625}{1.0816} \approx 1.070289386.

The implied rate is approximately 7.0289%. The time-3 balance is $8{,}652.80(1.070289386)\approx 9{,}261.00$. As a check, the three factors telescope to $(1.05)^3=1.157625$, and $8{,}000(1.157625)=9{,}261.00$. The modelled accumulated value is $9,261.00. The check catches errors in a derived forward rate or in the interval to which it applies.

What the Calculated Value Does—and Does Not—Mean

The rollover calculation is useful for comparing an investment strategy with a full-term investment on a consistent term structure. If the rates are the implied forwards and the model assumptions hold, the product identity gives equal terminal values. That equality is a no-arbitrage relation, not evidence that future short-term rates are known.

In a real investment decision, an investor may roll funds over at future market rates that have not been fixed today. Those realized rates may be higher or lower than the current implied forward rates. The formula then remains a calculation of the value if the specified rates apply, not a guarantee of the eventual balance. A forward contract or another arrangement may lock in a rate, but its terms and risks must be considered separately.

Before calculating, write down the interval boundaries, identify whether each rate is annual effective or applies to a whole interval, and state whether the question asks for an implied-rate value or an actual guaranteed investment value. These checks make the calculation’s meaning as important as its arithmetic.

Key takeaway: Multiply the accumulation factors for the successive intervals, using each implied forward rate over its specified length. Under consistent spot and forward rates, the result matches accumulation at the full-term spot rate. It describes an actual locked-in return only when the relevant rates and investment terms are genuinely secured.

Check Your Understanding

For each problem, use the stated annual effective convention. Treat the amounts and rates as hypothetical, and retain precision until the final monetary answer.

A $15,000 deposit is rolled over at 3% for year 1 and 4% for years 2 and 3. Find its value at time 3. (Answer: $16,710.72)
A $20,000 investment earns implied annual effective forward rates of 1.5%, 2%, and -1% over years 1, 2, and 3, respectively. Find its time-3 value. (Answer: $20,498.94)
A $5,000 investment earns an annual effective forward rate of 2% from time 0 to 0.25, then 3% from time 0.25 to 1. Find its time-1 value, to the nearest cent. (Answer: $5,137.45)
The one-year and two-year annual effective spot rates are 4% and 5%. A $12,000 investment is rolled over at the implied one-year rate and then the implied forward rate from year 1 to year 2. Find the time-2 value. (Answer: $13,230.00)
A $10,000 investment earns an annual effective forward rate of 3% from time 0 to time 2 and 4% from time 2 to time 3. Find its time-3 value. (Answer: $11,033.36)

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