Realized Return Over a Fixed Investment Horizon

What a Fixed-Horizon Realized Return Measures

An investment may produce value in more than one way: an initial outlay may be followed by interim receipts and a final sale or maturity payment. To summarize the result over a fixed horizon, first determine the total wealth available at the horizon, including the accumulated value of reinvested receipts. Then find the annual effective rate that would turn the initial outlay into that same terminal wealth over the same length of time.

This rate is an equivalent summary of the outcome. It need not be the rate actually earned on every part of the investment. In particular, interim receipts may have been reinvested at rates different from the equivalent rate. The calculation is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $P>0$ be an initial outlay, let $T>0$ be the investment horizon measured in years, and let $W_T>0$ be the total terminal wealth at time $T$. The fixed-horizon realized return $r$ is the annual effective rate satisfying $$ P(1+r)^T=W_T. $$ Therefore, $$ r=\left(\frac{W_T}{P}\right)^{1/T}-1. $$

When there are no interim receipts, $W_T$ is simply the terminal value. When receipts are reinvested, each receipt contributes its accumulated value at time $T$. If receipt $C_j$ arrives at time $t_j$, denote its accumulation from $t_j$ to $T$ by $A_j(t_j,T)$. Then, if $M_T$ is the net terminal payment,

W_T=M_T+\sum_j C_j A_j(t_j,T).

For example, if a receipt is reinvested for $n$ years at a constant annual effective rate $q$, its accumulation factor is $(1+q)^n$. If it passes through several periods with annual effective reinvestment rates $q_1,\ldots,q_k$, over respective durations $n_1,\ldots,n_k$, its factor is $\prod_{h=1}^k(1+q_h)^{n_h}$. These factors describe the actual accumulation of the receipts; the equivalent rate $r$ is calculated only after the terminal wealth has been assembled.

Why the Equivalent-Rate Formula Works

By definition, an annual effective rate $r$ accumulates one dollar for one year to $1+r$. Under the equivalent-rate model, it accumulates $P$ dollars for $T$ years to $P(1+r)^T$. Requiring that this model reproduce the observed terminal wealth gives the defining equation. Dividing by $P$ and taking the positive $T$th root gives the formula.

P(1+r)^T=W_T \quad\Longrightarrow\quad (1+r)^T=\frac{W_T}{P} \quad\Longrightarrow\quad r=\left(\frac{W_T}{P}\right)^{1/T}-1.

For $T>0$, the function $P(1+r)^T$ is strictly increasing for $r>-1$. Thus, for positive $P$ and $W_T$, there is exactly one equivalent rate greater than $-1$. If $W_T<P$, that rate is negative; if $W_T=P$, it is zero; and if $W_T>P$, it is positive.

Set the horizon and outlay.
Record $P$ and $T$, using years as the time unit for an annual effective rate.

Build terminal wealth.
Accumulate each interim receipt from its own receipt date to time $T$, and include net terminal proceeds.

Solve for the equivalent rate.
Use $r=(W_T/P)^{1/T}-1$, and round only after the calculation.

Check the result.
Substitute the unrounded rate into $P(1+r)^T$ and verify that it reproduces $W_T$.

Worked Calculations

Worked Example: Terminal Value With No Interim Receipts

An investor pays \$12,000 for an asset and holds it for four years. At the end of year 4, the asset is sold for \$15,000. There are no interim receipts or additional investment costs. Find the fixed-horizon realized return.

Method. With no interim receipts, the terminal wealth is the sale proceeds: $W_4=\$15{,}000$. The initial outlay is $P=\$12{,}000$, and the horizon is $T=4$ years.

r=\left(\frac{15{,}000}{12{,}000}\right)^{1/4}-1 =(1.25)^{1/4}-1 =1.057371\ldots-1 =0.057371\ldots

The annual effective equivalent return is 5.737%, rounded to three decimal places as a percentage. To check, $12{,}000(1.057371\ldots)^4=\$15{,}000$. This verifies that the equivalent annual rate reproduces the observed terminal value.

Common mistake avoided: The total gain is \$3,000, or 25% of the outlay, but 25% is the gain over four years, not the annual effective return. The fourth root converts the four-year accumulation factor into its one-year equivalent.

Worked Example: Receipts Reinvested at Different Rates

An investment costs \$20,000 and lasts three years. It pays \$1,200 at the end of year 1 and another \$1,200 at the end of year 2. The first receipt is reinvested for two years at an annual effective rate of 4%; the second is reinvested for one year at an annual effective rate of 5%. At the end of year 3, the investment is sold for \$21,000. Find the fixed-horizon realized return.

Method. Accumulate each receipt only over the time it is actually reinvested. The year-1 receipt accumulates for two years:

1{,}200(1.04)^2 =1{,}200(1.0816) =\$1{,}297.92.

The year-2 receipt accumulates for one year:

1{,}200(1.05)=\$1{,}260.

Add these accumulated receipts to the year-3 sale proceeds:

W_3=1{,}297.92+1{,}260+21{,}000 =\$23{,}557.92.

Now use $P=\$20{,}000$ and $T=3$:

r=\left(\frac{23{,}557.92}{20{,}000}\right)^{1/3}-1 =(1.177896)^{1/3}-1 =1.0560934\ldots-1.

The fixed-horizon realized return is 5.609%, rounded to three decimal places as a percentage. The check is $20{,}000(1.0560934\ldots)^3=\$23{,}557.92$, up to the displayed precision.

The 4% and 5% rates are the actual reinvestment rates for the two receipts. The 5.609% result is the single annual effective rate equivalent to the entire investment outcome over three years; it is not a claim that either receipt earned 5.609%.

Common mistake avoided: Adding the receipts at face value gives \$23,400, which ignores the interest earned after each receipt arrives. Using that incorrect total would give $(23{,}400/20{,}000)^{1/3}-1$, about 5.373%, instead of the correct 5.609%. Each receipt must be carried forward from its own date.

Worked Example: Net Sale Proceeds and Reinvested Receipts

A hypothetical asset requires an initial outlay of \$10,000 and is held for three years. It pays \$800 at the end of year 1, reinvested for two years at 3% effective annually, and \$1,000 at the end of year 2, reinvested for one year at 4% effective annually. The asset sells at the end of year 3 for \$11,250, with a \$250 sale fee paid at that time. Find the fixed-horizon realized return.

Method. First calculate the net terminal sale proceeds:

11{,}250-250=\$11{,}000.

The year-1 receipt accumulates for two years:

800(1.03)^2=800(1.0609)=\$848.72.

The year-2 receipt accumulates for one year:

1{,}000(1.04)=\$1{,}040.

Therefore, total terminal wealth is

W_3=11{,}000+848.72+1{,}040=\$12{,}888.72.

With $P=\$10{,}000$ and $T=3$,

r=\left(\frac{12{,}888.72}{10{,}000}\right)^{1/3}-1 =(1.288872)^{1/3}-1 =0.0882695\ldots.

The realized return is 8.827%, rounded to three decimal places as a percentage. As a check, $10{,}000(1.0882695\ldots)^3$ is approximately \$12,888.72. The fee is deducted at the time it is paid; the reinvested receipts are separately accumulated to that same terminal date.

Common mistake avoided: Using the gross sale price of \$11,250 overstates terminal wealth by \$250. All terminal costs or charges must be reflected in the net amount available at the horizon.

Interpretation and Scope

The calculation has two distinct stages. First, the reinvestment assumptions determine the terminal value of each interim receipt. Second, the ratio of total terminal wealth to initial outlay determines the equivalent annual effective rate over the fixed horizon. Keeping those stages separate prevents an assumed reinvestment rate from being mistaken for the realized return.

A useful way to organize the cash flows is to put them all at time $T$: reinvest each receipt forward to $T$, include net proceeds payable at $T$, and then compare the resulting amount with the original outlay. Do not accumulate an interim receipt for the full horizon, because it was not received at time 0. Do not discount a terminal receipt backward when the calculation asks for terminal wealth.

The formula assumes that the stated amounts and reinvestment rates are applied over the stated periods, and that the initial outlay and terminal wealth are measured on a consistent basis. If a receipt is not reinvested, its value at the horizon is its amount held, unless a separate return on that holding is specified. Include fees or other deductions at the dates they apply. This fixed-horizon measure summarizes a particular investment outcome; it does not by itself describe the timing or risk of the cash flows.

Key takeaway: Find the value of every cash flow at the fixed terminal date before solving for the annual effective equivalent rate. The rate is $r=(W_T/P)^{1/T}-1$, where $W_T$ includes accumulated interim receipts and net terminal proceeds.

Check Your Understanding

For each problem, calculate terminal wealth first, then find the fixed-horizon realized return. Round rates to three decimal places as percentages.

An outlay of \$8,000 produces a single terminal payment of \$9,680 after two years. Find the realized return. (Answer: 10.000%)
An outlay of \$12,000 is held for three years. It produces receipts of \$600 at the end of year 1 and \$600 at the end of year 2. The first is reinvested for two years at 5% effective annually, the second for one year at 4%, and the terminal proceeds are \$13,000. Find the realized return. (Answer: 5.983%)
An outlay of \$25,000 is held for four years. A \$1,000 receipt at the end of year 1 is reinvested for three years at 6%; a \$1,500 receipt at the end of year 3 is reinvested for one year at 2%; terminal proceeds are \$27,000. Find the realized return. (Answer: 4.419%)
An outlay of \$10,000 produces total terminal wealth of \$11,500 after 2.5 years, including any receipts already reinvested. Find the annual effective equivalent return. (Answer: 5.750%)
An outlay of \$30,000 is held for three years. A \$900 receipt at the end of year 1 is reinvested for two years at 3.5%; a \$1,100 receipt at the end of year 2 is reinvested for one year at 4%; net terminal proceeds are \$32,500. Find the realized return. (Answer: 4.878%)

What You'll Learn