Why the Reinvestment Assumption Matters
An investment may produce cash flows before the end of the period over which its performance is assessed. Those receipts do not automatically become part of the terminal value: their value at the horizon depends on what happens to them in the meantime. If a receipt is held without earning interest, its value is unchanged. If it is reinvested, it accumulates according to the reinvestment rate and the time remaining.
A zero-rate case is a useful baseline. A stated benchmark case asks what the terminal value would be if interim receipts earned a specified rate. A case in which receipts earn the investment's yield has a special mathematical relationship to the original investment, but only when the cash-flow and yield assumptions are consistent. These are different assumptions, not interchangeable descriptions of one guaranteed outcome.
This formula assumes that the stated rate applies over each period between a receipt and the horizon, with the same effective rate each period. It also assumes receipts can be reinvested immediately and that there are no taxes, transaction costs, or other changes to the cash flows. With different rates in different periods, the relevant accumulation factor must reflect each period's rate; a single constant \(j\) would not describe that case.
The timing convention is essential. A receipt at time 1 in a three-period investment is reinvested for two periods, not three. A receipt at time 3 is already at the horizon and receives no accumulation. The cash-flow timeline, rather than the label of a payment, determines the exponent.
Three Reinvestment Cases
The three cases below answer different questions. The zero-rate case isolates the sum of the receipts. The benchmark case uses an externally selected comparison rate. The investment-yield case asks what happens if every interim receipt earns the same rate used to price the investment's original cash flows.
| Case | Rate applied to interim receipts | Interpretation |
|---|---|---|
| Zero-rate | \(j=0\) | Receipts are retained at their nominal amounts until the horizon. |
| Benchmark | A stated rate \(b\) | A comparison scenario based on the chosen benchmark, not necessarily the investment's yield. |
| Investment yield | The investment's yield \(i\) | A consistent yield-based accumulation of all cash flows, subject to the yield equation. |
When comparing cases, keep the underlying cash-flow dates and horizon fixed. Changing the reinvestment rate should change only the accumulation applied to each interim receipt. It should not change the original payment amounts or silently move them to different dates.
Worked Example: Zero-Rate and Benchmark Accumulation
A hypothetical three-year investment produces receipts of $200 at the end of year 1, $300 at the end of year 2, and $500 at the end of year 3. Find its terminal value at the end of year 3 if interim receipts earn (a) zero interest or (b) a 5% effective annual benchmark rate. Assume each receipt is available for reinvestment immediately after it is paid.
Method. Accumulate each receipt from its payment date to year 3. The year-1 receipt has two years to accumulate; the year-2 receipt has one; the year-3 receipt has none.
(a) Zero rate. With no interest, each receipt remains at its original amount:
(b) 5% benchmark. Use \(j=0.05\) and the correct number of remaining years for each receipt:
The benchmark terminal value exceeds the zero-rate value by \(1{,}035.50-1{,}000=35.50\). Check: only the first two receipts earn interest, and their accumulated increments are $20.50 and $15.00; these sum to $35.50. The common mistake avoided here is giving the final receipt an extra year of interest.
Why Reinvestment at the Yield Has a Special Result
Let \(P\) be the price paid at time 0 for an investment with receipts \(C_1,\ldots,C_n\). Suppose the investment yield is \(i\), so that the cash flows satisfy the yield equation
If each receipt is reinvested at that same effective rate \(i\) until time \(n\), its terminal value is \(C_t(1+i)^{n-t}\). Multiplying the yield equation by \((1+i)^n\) gives
This is the justification for the yield-based accumulation identity: when the yield equation holds for the stated price and cash flows, accumulating all receipts at the yield produces \(P(1+i)^n\) at time \(n\). Each present value contribution is simply moved forward by the same total number of periods. The identity is not a claim that the investment will actually earn that terminal amount in every market condition. It is a consequence of the yield equation and the stipulated reinvestment assumption.
The result applies when the timing convention in the yield equation matches the accumulation calculation, the price and all included cash flows are consistent, and the same effective rate per period is used throughout. It cannot be applied merely because a rate is called a yield. If receipts are reinvested at a different rate, or the yield equation omits a cash flow that is included in terminal value, the equality need not hold.
Worked Example: Checking the Yield-Based Terminal Value
An investment costs $1,400 at time 0 and pays $220 at the end of year 1, $242 at the end of year 2, and $1,331 at the end of year 3. Verify that its yield is 10% effective annually, then calculate terminal value under zero-rate, 6% benchmark, and yield-rate reinvestment.
Method. First verify the yield equation by discounting the receipts at 10%. Then accumulate each receipt to year 3 under each requested assumption.
Yield check. At 10%, the discount factors for years 1, 2, and 3 are \(1/1.1\), \(1/1.21\), and \(1/1.331\):
Thus 10% satisfies the yield equation for the stated price and cash flows.
Zero-rate case. Add the receipts without accumulating interest:
6% benchmark case. Accumulate the first receipt for two years and the second for one year:
Yield-rate case. Accumulate at 10%:
The identity gives the same result: \(1{,}400(1.1)^3=1{,}400(1.331)=1{,}863.40\). Check: the yield-rate result agrees both with direct accumulation and with the price accumulated for three years. The common mistake avoided is applying \(1.1^3\) to every receipt; each receipt is accumulated only for the time remaining after it is paid.
Comparing a Benchmark With the Yield
A benchmark rate may be lower or higher than the investment yield. For positive receipts, increasing the reinvestment rate increases terminal value whenever a receipt has a positive accumulation period. This follows because \(C_t(1+j)^{n-t}\) increases with \(j\) when \(C_t>0\) and \(n-t>0\). A receipt paid at the horizon is unchanged across rates. Therefore, for positive receipts at or before the horizon, a benchmark below the yield produces a terminal value below the yield-rate case, unless all receipts are paid at the horizon.
That comparison does not establish which rate will be available in practice. A benchmark is an assumption selected for analysis. Actual reinvestment rates may vary over time, and the zero-rate case is a limiting comparison, not necessarily a realistic description of an account or market. If a cash flow is a payment rather than a receipt, it is signed negative in a net terminal-value calculation, so conclusions about how terminal value moves with the rate require care.
Worked Example: Three Assumptions for One Cash-Flow Stream
A hypothetical investment costs $1,000 and pays $210 at the end of year 1, $330.75 at the end of year 2, and $578.8125 at the end of year 3. Its yield is 5% effective annually. Find the terminal value at year 3 under zero-rate, 2% benchmark, and yield-rate reinvestment, then compare the outcomes.
Method. Check the stated yield, then apply each rate to the exact time remaining for each receipt.
Yield check. Discounting each receipt at 5% gives:
The yield statement is consistent with the $1,000 price.
Zero-rate case.
2% benchmark case. The year-1 receipt accumulates for two years, and the year-2 receipt for one:
Yield-rate case. By the identity, the terminal value is \(1{,}000(1.05)^3\). Direct accumulation confirms it:
The benchmark value exceeds the zero-rate value by \(1{,}134.6615-1{,}119.5625=15.099\). The yield-rate value exceeds the benchmark value by \(1{,}157.625-1{,}134.6615=22.9635\). Check: since \(0\%<2\%<5\%\) and the interim receipts are positive, the terminal values are ordered \(1{,}119.5625<1{,}134.6615<1{,}157.625\). Rounded to the nearest cent, they are $1,119.56, $1,134.66, and $1,157.63. The common mistake avoided is treating the 2% benchmark as if it were the investment's 5% yield.
Common Mistake: Using the Wrong Number of Periods
For a horizon at time \(n\), the exponent on a receipt paid at time \(t\) is \(n-t\). A frequent incorrect calculation uses \(n\) for every receipt, as if all payments were made at time 0. Another error is to give a receipt one period of interest even though it arrives exactly at the horizon.
In the first worked example, an incorrect 5% calculation might use \(200(1.05)+300+500=1{,}010\), giving the first receipt only one period of growth and none to the second receipt. The correct timeline gives the first receipt two years and the second one year:
A reliable procedure is to mark time 0, every payment date, and the terminal date before writing any accumulation factor. Then count the periods from each payment date to the horizon. Round only after all cash flows have been accumulated and added; premature rounding can slightly alter the total.
Check Your Understanding
For each problem, identify the horizon and accumulate every receipt for only the periods remaining until that horizon.
- A project pays $120 at the end of year 1, $250 at the end of year 2, and $400 at the end of year 4. Find the year-4 terminal value under zero-rate and 3% effective annual reinvestment. (Answer: zero-rate $770.00; 3% benchmark $796.35.)
- An account receives $500 at the end of year 1 and $200 at the end of year 3. Find the year-3 terminal value under zero-rate and 4% effective annual reinvestment. (Answer: zero-rate $700.00; 4% benchmark $740.80.)
- An investment costs $800 and pays $105, $220.50, and $578.8125 at the ends of years 1, 2, and 3. Verify its 5% yield, then find its year-3 terminal value at zero, 3%, and 5% reinvestment. (Answer: 5% yield verified; zero-rate $904.31; 3% benchmark $917.32; 5% yield-rate $926.10.)
- A fund pays $300 at the end of year 1, $500 at the end of year 2, and $600 at the end of year 3. Find its year-3 terminal value at zero and at a 6% effective annual benchmark rate. (Answer: zero-rate $1,400.00; 6% benchmark $1,467.08.)
- An investment costs $1,000 and pays $250, $400, and $300 at the ends of years 1, 2, and 3. Find the year-3 terminal value at a 5% effective annual benchmark rate, and state whether it is above or below the initial cost. (Answer: $995.63, which is $4.38 below the initial cost.)