Reinvestment of Proceeds After Early Maturity or Sale

Why Early Maturity or Sale Changes the Return Calculation

An investment can end before the date at which the investor wants to measure performance. A bond might mature early in a portfolio’s planned holding period, or an asset might be sold before the investor’s horizon. In either case, the proceeds do not disappear: they must be held, deposited, or invested in something else until the horizon. The return over the full horizon therefore depends on both the original investment’s cash flows and what happens to proceeds received early.

The key timing question is: for how long does each amount earn a return? A payment received at time $t$ has only $T-t$ years to accumulate to a horizon at time $T$. Using the original asset’s return for that entire horizon, or compounding early proceeds from the original investment date, generally gives the wrong terminal value.

Definition: The realized annual effective return over a horizon is the annual effective rate $R$ that equates the original investment at time 0 to the value at time $T$ of all investment-related cash flows, after accounting for their actual reinvestment or replacement-asset outcomes. In the single-initial-outlay setting, if the original outlay is $P_0$ and terminal wealth is $W_T>0$, then $P_0(1+R)^T=W_T$.

Here, $T$ is measured in years and may be fractional. The rate $R$ is an annualized summary of the result over the whole horizon. It is not necessarily the yield earned by the original asset, nor the rate earned on proceeds after they are received.

Building Terminal Wealth From the Timeline

Assume that the investor makes one initial outlay $P_0$, has no external contributions or withdrawals, and measures performance at time $T$. Suppose a payment $C_j$ is received at time $t_j$ and held in an account earning an annual effective reinvestment rate $r_j$ until the horizon. Its value at $T$ is $C_j(1+r_j)^{T-t_j}$. A sale or maturity payment is treated in the same way if it is deposited and left to accumulate.

W_T=\sum_j C_j(1+r_j)^{T-t_j}

This expression applies when the listed cash flows are deposited and accumulated under the stated rates. If proceeds instead purchase a replacement investment, do not also treat the purchase price as a deposit earning interest. Use the replacement investment’s later cash flows, and accumulate any of those payments that arrive before $T$. This distinction prevents counting the same proceeds twice.

Once $W_T$ is known, the annual effective return is obtained by solving $P_0(1+R)^T=W_T$. The exponent is the full time from the original outlay to the measurement horizon, not the time from maturity or sale to the horizon.

R=\left(\frac{W_T}{P_0}\right)^{1/T}-1

These formulas assume that stated reinvestment rates apply over the specified periods and that the stated cash flows are the complete investment cash flows relevant to the calculation. Ignore taxes, transaction costs, and default unless the problem supplies them. If there are external investor contributions or withdrawals, the single-outlay equation does not directly describe the investor’s return; those flows must be handled separately.

Why the Annualized-Return Formula Holds

The formula follows by asking what constant annual effective rate would turn the original outlay into the calculated terminal wealth. If the original outlay earns a constant annual effective rate $R$ for $T$ years, its accumulated value is $P_0(1+R)^T$. By definition, the equivalent constant rate must produce the same amount as the actual cash-flow and reinvestment path:

$$ P_0(1+R)^T=W_T $$

Divide by $P_0$, take the positive $T$-th root, and subtract 1. Since $P_0>0$, $W_T>0$, and $T>0$, the result is well-defined:

(1+R)^T=\frac{W_T}{P_0} \qquad\Longrightarrow\qquad 1+R=\left(\frac{W_T}{P_0}\right)^{1/T} \qquad\Longrightarrow\qquad R=\left(\frac{W_T}{P_0}\right)^{1/T}-1

This is an equivalent-rate calculation, not a claim that the investor actually earned the same rate in every year. In particular, it does not replace the cash-flow accumulation step: the formula for $R$ is only as accurate as the terminal wealth $W_T$ supplied to it.

Worked Applications

Worked Example: Proceeds From an Early Maturity

An investor pays $8,000 for an asset that makes no interim payments and matures after 2 years for $9,000. The investor’s measurement horizon is 5 years after the original purchase. The maturity proceeds are deposited at an annual effective reinvestment rate of 3% for the remaining time. Find terminal wealth and the realized annual effective return over the 5-year horizon.

Method. First accumulate the maturity payment from time 2 to time 5. Then annualize terminal wealth over the full 5 years from the original outlay.

There are $5-2=3$ years between maturity and the horizon. Thus,

W_5=9{,}000(1.03)^3 =9{,}000(1.092727) =\$9{,}834.543

Using the original outlay and the full horizon,

R=\left(\frac{9{,}834.543}{8{,}000}\right)^{1/5}-1 =0.042156\ldots

Rounded to two decimal places, terminal wealth is $9,834.54 and the realized annual effective return is 4.22%. The check is that $8{,}000(1.042156\ldots)^5$ reproduces approximately $9,834.543. The common mistake is to compound $9,000 for 5 years: it is received at year 2, so only 3 years of reinvestment are available.

Worked Example: Sale Proceeds and a Replacement Investment

An investor pays $2,000 at time 0 for an asset. It pays $100 at time 1, then is sold at time 1.5 for $2,100. The $100 payment is deposited in an account earning an annual effective rate of 3% until the horizon at time 3.5. At the sale date, the investor uses the $2,100 sale proceeds to purchase a replacement note. That note pays $100 at time 2.5 and $2,200 at time 3.5. The $100 replacement-note payment is reinvested for one year at an annual effective rate of 4%. Find terminal wealth and the realized annual effective return over the full 3.5-year horizon.

Method. Accumulate each cash flow to the horizon for the time it is actually held. The $2,100 purchase of the replacement note is not accumulated separately; the note’s later payments represent what that purchase produces.

The original asset’s time-1 payment is reinvested for $3.5-1=2.5$ years:

100(1.03)^{2.5} =100(1.076697\ldots) =\$107.6697\ldots

The replacement note’s time-2.5 payment is reinvested for $3.5-2.5=1$ year:

100(1.04)^1=\$104

The $2,200 payment at the horizon needs no further accumulation. Therefore,

W_{3.5}=107.6697\ldots+104+2{,}200 =\$2{,}411.6697\ldots

The realized annual effective return is

R=\left(\frac{2{,}411.6697\ldots}{2{,}000}\right)^{1/3.5}-1 =0.0549\ldots

Rounded to two decimal places, terminal wealth is $2,411.67 and the realized annual effective return is 5.49%. The check is that a 5.49% annual effective rate over 3.5 years gives terminal wealth close to $2,411.67, with the small difference due to rounding. A common mistake is to accumulate the $2,100 sale proceeds to the horizon and also add the replacement note’s payments. That counts the same proceeds twice.

Worked Example: Interim Payments Before Early Maturity

An investor pays $5,000 for an asset that pays $200 at the end of each of years 1 and 2, then matures at the end of year 3 for $5,200. The investor’s horizon is the end of year 5. Each payment is deposited immediately into an account earning an annual effective rate of 3% until the horizon. Find terminal wealth and the realized annual effective return.

Method. Treat each payment as a separate deposit. The year-1 payment accumulates for 4 years, the year-2 payment for 3 years, and the maturity payment for 2 years.

\begin{aligned} W_5 &=200(1.03)^4+200(1.03)^3+5{,}200(1.03)^2\\ &=200(1.12550881)+200(1.092727)+5{,}200(1.0609)\\ &=225.101762+218.5454+5{,}516.68\\ &=\$5{,}960.327162 \end{aligned}

The realized annual effective return is calculated over all 5 years, not just the 2 years after maturity:

R=\left(\frac{5{,}960.327162}{5{,}000}\right)^{1/5}-1 =0.0358\ldots

Rounded to two decimal places, terminal wealth is $5,960.33 and the realized annual effective return is 3.58%. As a check, accumulating each of the three payments at 3% for its stated number of years gives the displayed terminal wealth. The common mistake is to give the year-1 and year-2 payments the same reinvestment period as the year-3 maturity proceeds.

Common Traps and a Practical Check

The calculation has two distinct stages: constructing terminal wealth from the actual timeline, and annualizing that wealth from the original investment date. Keeping these stages separate makes the timing easier to audit.

Using the wrong accumulation period. A payment at $t_j$ accumulates for $T-t_j$ years, not $T$ years. A sale payment at time $\tau$ accumulates for $T-\tau$ years if it is deposited.
Using the original asset’s yield as the reinvestment rate. The asset’s yield describes its own cash flows under a yield calculation. Proceeds after sale or maturity earn the rate of the account or replacement investment actually specified.
Annualizing from the wrong starting date. The denominator in the terminal-wealth ratio is the original outlay, and the exponent is the full horizon $T$. Using only the post-sale period changes the question being answered.
Double-counting sale proceeds. If the proceeds purchase a replacement asset, track the replacement asset’s later cash flows. Do not also accumulate the same purchase amount as if it remained in a separate account.

A useful final check is to write a timeline with the original outlay, every payment date, the maturity or sale date, the reinvestment treatment, and the horizon. Confirm that every amount appears once in terminal wealth and that its accumulation period starts when that amount is received. These realized-return calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the time remaining to the horizon for every payment before calculating terminal wealth.

An investor pays $4,000 for an asset that matures at time 1 for $4,600. The horizon is time 3, and the proceeds earn an annual effective rate of 5%. Find terminal wealth. (Answer: $5,071.50)
An investment of $1,000 matures after 2 years for $1,100. The investor’s horizon is 4 years, and maturity proceeds earn 2% annually effective. Find terminal wealth. (Answer: $1,144.44)
An asset purchased for $1,000 pays $250 at time 1 and is sold for $1,000 at time 2. Both amounts are deposited at 3% annually effective until the horizon at time 4. Find terminal wealth. (Answer: $1,334.08)
An investor pays $1,500 at time 0 and sells the asset at time 1.5 for $1,650. The sale proceeds earn 4% annually effective until time 3.5. Find terminal wealth. (Answer: $1,784.64)
An investor pays $1,000 for an asset that pays $30 at time 1, $30 at time 2, and $1,030 at time 3. Each payment is reinvested at 2% annually effective until time 5. Find terminal wealth. (Answer: $1,135.92)

What You'll Learn