Comparing Strategies With Different Cash-Flow Timing

Why Cash-Flow Timing Matters

Two strategies can deliver the same total dollars but produce different wealth at a future date because those dollars arrive at different times. A receipt available earlier can earn investment returns for longer. To compare strategies fairly on an accumulation basis, value every cash flow at the same horizon and use the same reinvestment assumptions for both.

This comparison answers a specific question: given the stated cash flows and reinvestment opportunities, how much accumulated wealth does each strategy produce at a chosen date? It does not, by itself, establish that the strategy with greater terminal wealth is preferable in every setting. The strategies may differ in risk, liquidity, or other features not represented by the cash-flow calculation.

Definition: The terminal wealth of a strategy at time $T$ is the algebraic sum of its cash flows accumulated to time $T$. A cash flow received before $T$ is reinvested according to the specified assumptions; a cash flow at $T$ is included without further accumulation. Outflows are entered as negative amounts.

Accumulating Cash Flows to a Common Horizon

Suppose a strategy has cash flows $C_0,C_1,\ldots,C_T$, where $C_t$ is the signed amount at time $t$, and time is measured in years. If a constant effective annual reinvestment rate $i$ applies between every pair of consecutive annual dates, then a cash flow at time $t$ accumulates for $T-t$ years. Its value at time $T$ is $C_t(1+i)^{T-t}$.

W_T=\sum_{t=0}^{T} C_t(1+i)^{T-t}

Here $W_T$ is terminal wealth. The exponent is the number of years from the cash-flow date to the comparison date, not the cash-flow date itself. In particular, the flow at $T$ has exponent zero, so it is included at its original amount.

If reinvestment rates vary from year to year, write $j_k$ for the effective rate during the interval from time $k-1$ to time $k$. A cash flow at time $t$ then accumulates through the rates for the intervals after $t$ and up to $T$:

W_T=\sum_{t=0}^{T} C_t\prod_{k=t+1}^{T}(1+j_k)

An empty product, as occurs when $t=T$, equals 1. For a comparison to be meaningful, use the same assumed rate for each calendar interval in every strategy. If an interval has a different rate, that change applies consistently to all funds invested during that interval.

Why the Accumulation Formula Works

The formula follows from compound accumulation and the additivity of account balances. A single amount $C_t$, invested at time $t$, earns the applicable rate during each subsequent interval. After one interval it is multiplied by $1+j_{t+1}$; after the next, by $1+j_{t+2}$, and so on. Its value at $T$ is therefore the product of those factors times $C_t$.

For multiple cash flows, each amount can be tracked separately because the accumulation factor is linear: multiplying a sum of amounts by a common factor gives the sum of their individually accumulated values. With different starting dates, each amount has its own product of applicable factors. Adding these accumulated amounts gives the terminal account value. A negative cash flow follows the same algebra: it reduces terminal wealth by the amount that the outflow would otherwise have accumulated to.

C_t\longmapsto C_t\prod_{k=t+1}^{T}(1+j_k) \quad\Longrightarrow\quad W_T=\sum_{t=0}^{T}C_t\prod_{k=t+1}^{T}(1+j_k)

This method presumes the stated cash flows occur at the stated dates and can be reinvested at the given rates, with no taxes, fees, or other adjustments unless those are explicitly included. It is a deterministic calculation: there is no mortality contingency or probability distribution in the examples below.

Worked Example: Two Equal Receipts Available at Different Dates

A hypothetical program offers one of two receipt schedules. Strategy C credits two separate receipts of $600 immediately, at time 0. Strategy D credits one receipt of $1,200 at time 1. Compare their terminal wealth at time 4. Assume a constant effective annual reinvestment rate of 6% for every year. The cash flows are deterministic; no mortality assumption or probability distribution applies.

Method. Accumulate each receipt from its own date to time 4. The two receipts under Strategy C each have four years to accumulate. The receipt under Strategy D has three years.

Strategy C. The accumulation factor for four years is $1.06^4=1.26247696$. Thus each $600 receipt accumulates to $600(1.26247696)=757.486176$. Adding both amounts gives $757.486176+757.486176=1,514.972352$, or $1,514.97 to the nearest cent.

Strategy D. The accumulation factor for three years is $1.06^3=1.191016$. Its terminal wealth is $1,200(1.191016)=1,429.2192$, or $1,429.22 to the nearest cent.

The terminal-wealth difference is $1,514.972352-1,429.2192=85.753152$, or $85.75 to the nearest cent. Under these assumptions, Strategy C produces more wealth at time 4 because its funds are available earlier.

Trap avoided. A deposit made at time 0 does not have zero value at the later comparison date. In particular, $600(1.06)^4+600(1.06)^4=1,514.972352$, which rounds to $1,514.97, not zero. The amount at time 0 is the starting balance, not the terminal wealth.

Comparing Net Terminal Wealth

Strategies may include both an initial outlay and later receipts. In that case, include every cash flow with its sign. A common initial outlay must also be accumulated to the comparison horizon; omitting it would compare receipts rather than the net wealth produced by each strategy.

Worked Example: Equal Initial Outlays and Differently Timed Receipts

Two hypothetical projects each require an outlay of $1,000 at time 0 and are compared at time 3. Project A pays $300 at time 1, $400 at time 2, and $500 at time 3. Project B pays $100 at time 1, $500 at time 2, and $600 at time 3. Assume a constant effective annual reinvestment rate of 5%. All cash flows are deterministic, with no mortality contingency or probability distribution.

Method. Accumulate the initial outlay and every receipt to time 3. The common outlay is negative, so it reduces each project's terminal wealth.

For Project A, the initial outlay accumulates to $-1,000(1.05)^3=-1,000(1.157625)=-1,157.625$. The receipts accumulate as follows: $300(1.05)^2=300(1.1025)=330.75$, $400(1.05)=420$, and the time-3 receipt remains $500$. The total accumulated receipts are $330.75+420+500=1,250.75$. Therefore, $W_3^{(A)}=-1,157.625+1,250.75=93.125$, or $93.13 to the nearest cent.

For Project B, the initial outlay is again $-1,157.625$ at time 3. Its receipts accumulate to $100(1.05)^2=110.25$, $500(1.05)=525$, and $600$ at time 3. Total accumulated receipts are $110.25+525+600=1,235.25$. Therefore, $W_3^{(B)}=-1,157.625+1,235.25=77.625$, or $77.63 to the nearest cent.

Project A's terminal wealth exceeds Project B's by $93.125-77.625=15.50$, or $15.50. The equal initial outlays cancel in the difference, but they were still included in each project's terminal-wealth calculation.

Trap avoided. Comparing only the total receipts at their payment dates ignores timing. Accumulating the receipts to time 3 shows that Project A's earlier allocation of receipts produces $15.50 more terminal wealth under the stated reinvestment rate.

When Reinvestment Rates Vary

A constant annual rate is not required. When the assumed rate changes over time, each cash flow is accumulated only through the intervals after it arrives. It would be incorrect to apply a rate from an interval that occurred before the cash flow was received.

Worked Example: Applying Interval-Specific Rates Consistently

Compare two deterministic receipt schedules at time 4. Strategy P pays $700 at time 1 and $500 at time 3. Strategy Q pays $400 at time 2 and $800 at time 4. The effective reinvestment rates are 4% from time 0 to 1, 5% from time 1 to 2, 3% from time 2 to 3, and 6% from time 3 to 4. No mortality assumption or probability distribution applies.

Method. For each receipt, multiply only the accumulation factors for the intervals between its payment date and time 4.

For Strategy P, the $700 received at time 1 earns the rates for intervals 1 to 2, 2 to 3, and 3 to 4: $700(1.05)(1.03)(1.06)$. Since $1.05(1.03)=1.0815$ and $1.0815(1.06)=1.14639$, this amount becomes $700(1.14639)=802.473$. The $500 received at time 3 earns only the final interval's rate: $500(1.06)=530$. Thus $W_4^{(P)}=802.473+530=1,332.473$, or $1,332.47 to the nearest cent.

For Strategy Q, the $400 received at time 2 earns the 3% and 6% rates: $400(1.03)(1.06)=400(1.0918)=436.72$. The $800 received at time 4 earns no further interest, so its value at the horizon is $800. Therefore, $W_4^{(Q)}=436.72+800=1,236.72$.

The difference is $1,332.473-1,236.72=95.753$, or $95.75 to the nearest cent. Strategy P has greater terminal wealth under these interval-specific assumptions.

Trap avoided. The $700 received at time 1 cannot earn the 4% rate for the interval from time 0 to 1; it was not available then. Including that rate would overstate its accumulated value.

Interpreting the Difference and Avoiding Common Errors

For two strategies $A$ and $B$, the terminal-wealth difference is $W_T^{(A)}-W_T^{(B)}$. A positive result means strategy $A$ produces more accumulated wealth at time $T$, given the chosen cash flows and reinvestment assumptions. It is not a statement that the difference would persist under other rates or horizons.

Several checks help keep a comparison reliable:

Use one horizon. Do not compare one strategy's value at time 3 with another's at time 4.
Count periods from the cash-flow date. A flow at time $t$ accumulates for $T-t$ annual intervals when rates are constant.
Use the relevant intervals. With changing rates, begin accumulating after the cash flow occurs.
Include all signed flows. Outlays are negative; receipts are positive. Do not silently omit a common investment cost.
Apply assumptions consistently. Both strategies use the same rate for any given calendar interval unless the problem explicitly specifies otherwise.
Interpret the result narrowly. Terminal wealth compares accumulated cash flows; it does not capture unmodeled risk, liquidity, taxes, or fees.

Key takeaway: To compare differently timed cash flows on an accumulation basis, move every signed cash flow to the same horizon using only the reinvestment factors that apply after its date. Then compare the resulting terminal wealth amounts.

Check Your Understanding

For each question, accumulate all cash flows to the stated horizon and use the assumptions given. Round dollar answers to the nearest cent.

Strategy A pays $500 at time 1 and $500 at time 2. Strategy B pays $1,000 at time 2. At time 3, which has greater terminal wealth if the effective annual reinvestment rate is 4%, and by how much? (Answer: Strategy A, by $20.80; its terminal wealth is $1,060.80 and Strategy B's is $1,040.00.)
A deterministic receipt of $900 is available under Strategy A at time 0 or under Strategy B at time 2. Compare terminal wealth at time 3 using a constant effective annual rate of 5%. (Answer: Strategy A, by $96.86; terminal wealths are $1,041.86 and $945.00.)
At time 4, compare Strategy A, which pays $200 at time 1 and $300 at time 3, with Strategy B, which pays $500 at time 2. Use a constant effective annual reinvestment rate of 3%. (Answer: Strategy B, by $2.90; terminal wealths are $530.45 and $527.55.)
Rates for the intervals from time 0 to 1, 1 to 2, and 2 to 3 are 2%, 4%, and 5%, respectively. Strategy A receives $1,000 at time 0; Strategy B receives $1,000 at time 1. Compare their terminal wealths at time 3. (Answer: Strategy A, by $21.84; terminal wealths are $1,113.84 and $1,092.00.)
Two hypothetical projects each have an outlay of $800 at time 0. Project A receives $300 at time 1 and $600 at time 2; Project B receives $100 at time 1 and $800 at time 2. Compare net terminal wealth at time 2 using a constant effective annual rate of 6%. (Answer: Project A, by $12.00; net terminal wealths are $19.12 and $7.12.)

What You'll Learn