Accumulating Interim Cash Flows Along a Forward Curve

Why Interim Proceeds Need a Common Date

An investment may generate proceeds before its final evaluation date. A receipt at year 1 and a receipt at year 3 cannot simply be added to a balance measured at year 4: the earlier receipt has more time to earn interest. To compare them consistently, accumulate every receipt to the same terminal date, using the rates that apply over the intervening periods.

When those period rates are forward rates, each receipt follows a different part of the forward curve. A receipt at year 2 is accumulated only through the rates for years 3 onward; it does not earn interest for years 1 and 2. Once all proceeds have been accumulated to the terminal date, their total can be compared with the initial investment to calculate an equivalent annual return.

Definition: Let $f_j$ be the effective forward rate for the year from time $j-1$ to time $j$. A cash receipt $C_k$ at time $k$, accumulated to terminal time $T$, has terminal value $$ C_k\prod_{j=k+1}^{T}(1+f_j). $$ If $k=T$, the product is empty and equals 1, so the receipt is already at the terminal date. The total terminal wealth from receipts $C_1,\ldots,C_T$ is $$ W_T=\sum_{k=1}^{T}C_k\prod_{j=k+1}^{T}(1+f_j). $$

This method assumes that the applicable one-year rates are known and that each receipt can actually be reinvested at the rate used for its remaining period. If the curve is being used as an assumed reinvestment schedule, the result is conditional on that assumption. If actual reinvestment rates differ, use those realized rates to calculate actual terminal wealth. This calculation is an accumulation to a focal date; it is not a present-value calculation.

From Terminal Wealth to an Annualized Return

Suppose an initial investment $I_0$ is made at time 0, and its proceeds are reinvested so that they produce terminal wealth $W_T$ at time $T$. The equivalent annual effective return $i_{\mathrm{eq}}$ is the single annual rate that would turn $I_0$ into $W_T$ over $T$ years:

I_0(1+i_{\mathrm{eq}})^T=W_T, \qquad i_{\mathrm{eq}}=\left(\frac{W_T}{I_0}\right)^{1/T}-1.

The annualization assumes the period from initial investment to terminal wealth is $T$ years. It summarizes the total growth into one equivalent annual rate; it does not mean each receipt earned that same rate. In particular, the rate is based on the accumulated terminal wealth, not on the sum of the receipts without regard to their dates.

Why the Accumulation Formula Works

Over one year, a balance grows by the factor $1+f_j$ for the forward rate applying in that year. For a receipt at time $k$, the first year in which it can earn interest is the interval from $k$ to $k+1$, with factor $1+f_{k+1}$. It continues to earn the factors for each later interval through the year ending at $T$. Multiplication of successive accumulation factors therefore gives its terminal value.

For example, a receipt at time $k$ has value $C_k(1+f_{k+1})$ at time $k+1$. At time $k+2$, that balance is multiplied by $1+f_{k+2}$, giving $C_k(1+f_{k+1})(1+f_{k+2})$. Repeating this step through time $T$ gives the product in the definition. Since the accumulated receipts are all measured at the same date, they can then be added.

Set the timeline.
Mark the initial investment, each receipt date, the terminal date, and the forward rate applying to each interval.

Accumulate each receipt separately.
For a receipt at time $k$, multiply by the factors for intervals after $k$, stopping at $T$.

Add at the focal date.
Sum the terminal values to obtain $W_T$. Do not add cash amounts that remain at different dates.

Annualize if required.
Use $i_{\mathrm{eq}}=(W_T/I_0)^{1/T}-1$, then check that the rate reproduces $W_T$ when applied for $T$ years.

Worked Examples

Worked Example: Three-Year Terminal Wealth and Annual Return

An investment of $5,000 produces receipts of $1,000 at year 1 and $2,000 at year 2. The terminal date is year 3. The forward rates for the intervals ending in years 1, 2, and 3 are 2%, 4%, and 3%, respectively. Assume the receipts are reinvested at the applicable forward rates. Find terminal wealth and the equivalent annual return on the initial $5,000.

Method. Accumulate each receipt from its own date to year 3. The year-1 receipt uses the rates for years 2 and 3; the year-2 receipt uses only the rate for year 3.

The year-1 receipt accumulates to:

1{,}000(1.04)(1.03) =1{,}000(1.0712) =\$1{,}071.20.

The year-2 receipt accumulates to:

2{,}000(1.03)=\$2{,}060.00.

Thus terminal wealth is:

W_3=1{,}071.20+2{,}060.00=\$3{,}131.20.

The annualized equivalent return is:

i_{\mathrm{eq}} =\left(\frac{3{,}131.20}{5{,}000}\right)^{1/3}-1 =(0.62624)^{1/3}-1 \approx -0.1444.

So the equivalent annual return is approximately $-14.44\%$ per year, rounded to two decimal places. Check: $5{,}000(1-0.1444)^3$ is approximately $3,131, consistent with the calculated terminal wealth, subject to rounding. The common mistake this method avoids is applying the year-1 rate to a receipt that does not exist until year 1.

Worked Example: Several Receipts With Different Reinvestment Periods

A fund begins with $8,000 and receives $1,800 at the end of year 1, $2,500 at the end of year 2, and $3,000 at the end of year 3. Its terminal date is year 4. The forward rates for years 1 through 4 are 3%, 4%, 2%, and 5%. Assuming these are the reinvestment rates actually earned, calculate terminal wealth and the equivalent annual return on the initial investment.

Method. Match each receipt to the factors after its receipt date. The year-1 receipt earns the rates for years 2, 3, and 4:

1{,}800(1.04)(1.02)(1.05) =1{,}800(1.11384) =\$2{,}004.912.

The year-2 receipt earns the rates for years 3 and 4:

2{,}500(1.02)(1.05) =2{,}500(1.071) =\$2{,}677.50.

The year-3 receipt earns only the year-4 rate:

3{,}000(1.05)=\$3{,}150.00.

Add the terminal amounts:

W_4=2{,}004.912+2{,}677.50+3{,}150.00 =\$7{,}832.412 \approx \$7{,}832.41.

Using the unrounded terminal amount for the return calculation:

i_{\mathrm{eq}} =\left(\frac{7{,}832.412}{8{,}000}\right)^{1/4}-1 \approx -0.00528 =-0.528\%.

The equivalent annual return is approximately $-0.528\%$, to three significant figures. The negative result is possible because the accumulated receipts total less than the initial investment. A useful check is that applying the resulting annual factor for four years to $8,000 reproduces approximately $7,832.41. The common mistake avoided here is giving every receipt the same accumulation period.

Worked Example: Including a Terminal-Date Receipt

An investor contributes $10,000 at time 0. The investment pays $1,500 at year 1 and $4,000 at year 3, and a final $6,500 at year 4. The forward rates for years 1 through 4 are 2%, 3%, 4%, and 5%. Assume the first two payments are reinvested at the applicable rates. Find terminal wealth and the equivalent annual return over four years.

Method. Accumulate the year-1 and year-3 receipts to year 4. The year-4 payment is already at the terminal date and earns no additional interest in this calculation.

For the year-1 receipt, use the factors for years 2, 3, and 4:

1{,}500(1.03)(1.04)(1.05) =1{,}500(1.12476) =\$1{,}687.14.

For the year-3 receipt, use only the year-4 factor:

4{,}000(1.05)=\$4{,}200.00.

The final payment is $6,500 at year 4. Therefore:

W_4=1{,}687.14+4{,}200.00+6{,}500.00 =\$12{,}387.14.

The equivalent annual return is:

i_{\mathrm{eq}} =\left(\frac{12{,}387.14}{10{,}000}\right)^{1/4}-1 \approx 0.0550 =5.50\%.

Thus terminal wealth is $12,387.14 and the equivalent annual return is approximately 5.50%. Check: $10{,}000(1.0550)^4$ is approximately $12,387. The common mistake avoided is accumulating the terminal-date payment by an extra year.

Forward Rates, Realized Rates, and Common Traps

A forward curve specifies rates associated with future periods under the term-structure framework. For an accumulation calculation, those rates determine terminal wealth only if the proceeds are in fact reinvested at those rates. If the reinvestment rates actually earned are different, the realized terminal wealth must be computed using the realized rates for the relevant periods instead. The distinction is important: an implied rate is not automatically a realized investment outcome.

Another common error is to average the forward rates and apply that average to every receipt. This generally fails because the receipts have different remaining accumulation periods, and because successive interest factors multiply. The relevant accumulation factor for a particular receipt is the product of the factors for exactly the intervals between its receipt date and the terminal date.

Common exam trap: A receipt at time $k$ does not earn the rate for the interval ending at time $k$. It is received at the end of that interval. Its first accumulation factor is $1+f_{k+1}$, not $1+f_k$. Also, do not include a factor for a period after the stated terminal date.

Before calculating, check the timeline and ask three questions: when is each cash flow received, which intervals remain after that date, and are the stated forward rates assumed to be the rates actually earned? After calculating, check that terminal wealth is the sum of amounts all valued at the same date and that the annualized return reproduces that wealth when applied to the initial investment.

These steps are aligned with the SOA Exam FM (Financial Mathematics) syllabus: they connect dated cash flows, reinvestment assumptions, forward rates, and return measurement without treating a curve-implied accumulation as automatically realized.

Check Your Understanding

For each problem, accumulate the receipts to the stated terminal date before calculating any annualized return.

An initial investment of $6,000 produces receipts of $1,200 at year 1 and $2,500 at year 2. The terminal date is year 3, and the forward rates for years 1, 2, and 3 are 2%, 3%, and 4%. Find terminal wealth and the equivalent annual return. (Answer: $3,885.44; approximately $-13.5\%$ per year.)
An investment of $5,000 produces $600 at year 1 and $1,200 at year 3, with terminal date year 4. The forward rates for years 1 through 4 are 1%, 2%, 3%, and 4%. Find terminal wealth. (Answer: $1,903.57.)
An investment of $9,000 produces $2,000 at year 1, $1,500 at year 2, and $5,000 at year 4. The terminal date is year 4, and the forward rates for years 1 through 4 are 2%, 2.5%, 3%, and 4%. Find the equivalent annual return. (Answer: terminal wealth $8,802.76; approximately $-0.552\%$ per year.)
An investment of $10,000 produces $3,000 at year 1 and $4,000 at year 2, with terminal date year 3. The curve-implied forward rates for years 1, 2, and 3 are 2%, 5%, and 4%, but actual reinvestment rates for years 2 and 3 are 4% and 6%. Assuming the year-1 receipt is reinvested at both actual rates and the year-2 receipt at the year-3 actual rate, find actual terminal wealth and the equivalent annual return. (Answer: $7,547.20; approximately $-8.95\%$ per year.)

What You'll Learn