How NPV Changes With the Valuation Date

Why the Valuation Date Matters

A net present value is a value stated at a particular date. If the date changes, the cash flows have not changed, but their value must be expressed in dollars at the new date. Moving the valuation date forward accumulates value; moving it backward discounts value. Under one constant effective annual interest rate, the change has a simple relationship that applies to the entire signed cash-flow stream.

The key is to distinguish a cash flow’s date from the valuation date. A cash flow before the valuation date is accumulated to it; a cash flow after the valuation date is discounted back to it. The same rule applies to positive inflows and negative outflows.

Definition: Suppose cash flow $C_k$, positive for an inflow and negative for an outflow, occurs at time $t_k$, measured in years. Under a constant effective annual rate $i$, its value at focal date $\tau$ is $C_k(1+i)^{\tau-t_k}$. The value of the full cash-flow stream at $\tau$ is

V(\tau)=\sum_k C_k(1+i)^{\tau-t_k}.

Here $i>-1$, and the same rate and year-time convention are used for every cash flow and valuation date. A positive exponent accumulates a cash flow forward; a negative exponent discounts it backward.

This definition permits the focal date to fall before, between, or after the cash-flow dates. If dates are fractional years, the model uses the corresponding fractional powers of $1+i$. It is therefore important to use a consistent time unit: if $i$ is an effective annual rate, the dates and date differences in the exponents must be measured in years.

The Date-Shift Relationship

Let $\tau_1$ be the original valuation date and $\tau_2$ the new valuation date. With one constant rate, each cash flow changes value by the same multiplier when moving between these dates. This is true whether the cash flow occurs before both dates, between them, or after both.

Theorem: Under the constant-rate model in the definition, the value of the same cash-flow stream at two focal dates satisfies

V(\tau_2)=V(\tau_1)(1+i)^{\tau_2-\tau_1}.

If $\tau_2>\tau_1$, the value is accumulated forward. If $\tau_2<\tau_1$, it is discounted backward.

To justify the relationship, write the value at the new date by its definition, then separate the exponent into the change in focal date and the exponent used at the original date:

\begin{aligned} V(\tau_2) &=\sum_k C_k(1+i)^{\tau_2-t_k}\\ &=\sum_k C_k(1+i)^{(\tau_2-\tau_1)+(\tau_1-t_k)}\\ &=(1+i)^{\tau_2-\tau_1}\sum_k C_k(1+i)^{\tau_1-t_k}\\ &=(1+i)^{\tau_2-\tau_1}V(\tau_1). \end{aligned}

The factor can be taken outside the sum because it is the same for every cash flow. This argument depends on using the same constant rate for the full valuation. If rates vary over time, the value of each cash flow may be affected by a different accumulation factor, so this single-factor relationship generally does not apply.

Worked Examples

Worked Example: Moving the Valuation Date Forward

A project has cash flows of $-\$1{,}000$ at time 0, $+\$400$ at time 1, and $+\$800$ at time 3. The effective annual interest rate is 10%. Find the value at time 0 and at time 2. Verify the date shift by direct valuation at time 2.

Method. First calculate $V(0)$ from the cash flows. Then accumulate it two years to obtain $V(2)$, and check the result by valuing each cash flow directly at time 2. The common mistake this method avoids is treating every cash flow as if it must be discounted, even when it falls before the focal date.

At time 0, the time-0 outflow is already at the focal date. The later inflows are discounted:

\begin{aligned} V(0) &=-1{,}000+\frac{400}{1.10}+\frac{800}{1.10^3}\\ &=-1{,}000+363.636364+601.051841\\ &=-35.311796. \end{aligned}

Thus $V(0)=-\$35.31$, rounded to the nearest cent. Moving from time 0 to time 2 is a two-year accumulation:

$$ V(2)=V(0)(1.10)^2=(-35.311796)(1.21)=-42.727273. $$

For the direct check, the time-0 outflow accumulates two years, the time-1 inflow accumulates one year, and the time-3 inflow is discounted one year:

\begin{aligned} V(2) &=-1{,}000(1.10)^2+400(1.10)+\frac{800}{1.10}\\ &=-1{,}210+440+727.272727\\ &=-42.727273. \end{aligned}

Both methods give $V(2)=-\$42.73$, to the nearest cent. The negative value remains negative because the date-shift multiplier, $1.21$, is positive.

Worked Example: Shifting Between Dates That Surround Cash Flows

A hypothetical investment has a $+\$100$ receipt at time 0, a $-\$300$ payment at time 2, and a $+\$500$ receipt at time 4. The effective annual rate is 10%. Calculate its value at time 1, translate that value to time 3, and verify the time-3 value directly.

Method. Value each signed cash flow at time 1, apply the two-year shift factor, and then revalue each flow at time 3. This avoids the trap of accumulating or discounting based on whether a cash flow is an inflow or outflow; the direction depends on the dates, not the sign.

At time 1, the time-0 receipt accumulates one year, the time-2 payment is discounted one year, and the time-4 receipt is discounted three years:

\begin{aligned} V(1) &=100(1.10)-\frac{300}{1.10}+\frac{500}{1.10^3}\\ &=110-272.727273+375.657400\\ &=212.930128. \end{aligned}

Moving from time 1 to time 3 means accumulating two years:

$$ V(3)=212.930128(1.10)^2 =212.930128(1.21) =257.645455. $$

Directly at time 3, the time-0 receipt accumulates three years, the time-2 payment accumulates one year, and the time-4 receipt is discounted one year:

\begin{aligned} V(3) &=100(1.10)^3-300(1.10)+\frac{500}{1.10}\\ &=133.10-330+454.545455\\ &=257.645455. \end{aligned}

The results agree: $V(1)=\$212.93$ and $V(3)=\$257.65$, each rounded to the nearest cent. In particular, the negative payment was accumulated from time 2 to time 3, just as the positive receipt at time 0 was accumulated.

Worked Example: A Zero Value at Different Focal Dates

A cash-flow stream consists of $-\$100$ at time 0, $+\$55$ at time 1, and $+\$60.50$ at time 2. At an effective annual rate of 10%, determine its value at time 0 and time 3. Use both the date-shift rule and direct valuation at time 3.

Method. Calculate the original value, shift it to the new date, and check by direct accumulation. The common mistake avoided here is assuming a zero value at one date can become nonzero merely because the focal date changes.

At time 0, discount both later receipts:

\begin{aligned} V(0) &=-100+\frac{55}{1.10}+\frac{60.50}{1.10^2}\\ &=-100+50+50\\ &=0. \end{aligned}

The date-shift rule gives

$$ V(3)=V(0)(1.10)^3=0(1.331)=0. $$

To check directly, accumulate each cash flow to time 3:

\begin{aligned} V(3) &=-100(1.10)^3+55(1.10)^2+60.50(1.10)\\ &=-133.10+66.55+66.55\\ &=0. \end{aligned}

The value is zero at both dates. More generally, because $1+i$ is positive, multiplying by a date-shift factor preserves whether a value is positive, negative, or zero.

Interpretation and Common Traps

Changing the valuation date changes the amount being reported, not the underlying cash-flow stream. A value at a later date is not directly comparable in dollars to a value at an earlier date unless the timing difference is accounted for. The date-shift relationship provides that adjustment under its stated constant-rate assumption.

Reversing the date difference. For a move from $\tau_1$ to $\tau_2$, use the exponent $\tau_2-\tau_1$. A move forward has a positive exponent; a move backward has a negative exponent.
Ignoring cash flows before the focal date. These are accumulated forward to the focal date; they are not omitted just because they are in the past.
Changing the sign rule. Inflows remain positive and outflows remain negative. Their signs do not determine whether to discount or accumulate; timing does.
Using inconsistent time units. An effective annual rate requires time differences in years. A difference of six months is $0.5$ years in the fractional-year model.
Applying one factor when the rate assumption changes. The derivation requires a common rate. If the rate varies across the time span, do not assume the whole value can be translated by one power of $1+i$.

For instance, the wrong calculation for Example 1 would be to discount $V(0)$ to time 2 by dividing by $1.10^2$. That would give $-35.311796/1.21=-29.183303$, rather than the correct $-42.727273$. The correction is to recognize that time 2 is later than time 0, so the value must be accumulated: multiply by $1.10^2$.

Key takeaway: With one constant effective annual rate, moving a stream’s focal date from $\tau_1$ to $\tau_2$ multiplies its value by $(1+i)^{\tau_2-\tau_1}$. The sign and zero status are preserved, but the stated dollar value changes with the valuation date.

Check Your Understanding

For each problem, use the stated constant effective annual rate and consistent year-based dates.

A stream has cash flows of $-\$200$ at time 0, $+\$120$ at time 1, and $+\$100$ at time 2. At 5%, calculate the value at time 0 and use the date-shift rule to find the value at time 3. (Answer: $V(0)=\$4.99$; $V(3)=\$5.78$, to the nearest cent.)
A stream has $-\$500$ at time 1, $+\$200$ at time 3, and $+\$400$ at time 4. At 8%, calculate the value at time 2, then shift it to time 4. (Answer: $V(2)=-\$11.88$; $V(4)=-\$13.86$, to the nearest cent.)
At a 6% effective annual rate, a stream’s value at time 1 is $+\$250$. What is its value at time 4, and does its sign change? (Answer: $V(4)=+\$297.75$, to the nearest cent; its sign remains positive.)
A stream has cash flows of $-\$400$ at time 0, $+\$220$ at time 1, and $+\$242$ at time 2. At 10%, calculate its value at time 0 and verify the value at time 3 by direct accumulation. (Answer: $V(0)=\$0.00$; $V(3)=\$0.00$.)
At a constant effective annual rate of 4%, a stream’s value at time 2 is $-\$240$. Find its value at time 5 and state whether moving the valuation date changes its sign. (Answer: $V(5)=-\$269.97$, to the nearest cent; the sign remains negative.)

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