Discount-Rate Sensitivity of NPV

Why Discount-Rate Sensitivity Matters

A project’s NPV depends on both its dated cash flows and the rate used to discount them. If the rate changes, the cash flows do not change, but their present values do. Comparing NPVs at two specified rates shows the size and direction of that effect. Breaking the change down by date shows which cash flows drive it.

This analysis is useful when assessing how dependent a valuation is on its discount-rate assumption. It does not, by itself, determine which rate is appropriate. The calculations below assume that the cash flows are fixed, occur at specified annual dates, and are discounted using a constant effective annual rate. The rate and timing units must match: here, a cash flow at time $t$ is $t$ years from time 0.

Definition: For fixed cash flows $C_0,C_1,\ldots,C_n$, where $C_t$ is the net cash flow at time $t$, the NPV at effective annual rate $i>-1$ is $$ \operatorname{NPV}(i)=\sum_{t=0}^{n}\frac{C_t}{(1+i)^t}. $$ A negative cash flow is an outflow and a positive cash flow is an inflow. The time-0 cash flow is not discounted. NPV sensitivity describes how this value changes when the discount rate changes while the cash flows and their dates remain fixed.

Recalculating NPV at Two Rates

For a finite change in the rate, the direct method is to calculate the NPV separately at each rate and subtract. To see which dated cash flows account for the difference, compare each cash flow’s present value at the two rates. This decomposition is exact; it does not rely on an approximation.

Worked Example: Separating the Change by Cash-Flow Date

A hypothetical project has a net outflow of $1,000 at time 0, followed by inflows of $600 at the end of each of years 1 and 2. Calculate the NPV at 5% and 10%, then identify how much each future cash flow contributes to the NPV change. Both rates are effective annual rates.

Method. Discount each cash flow at each rate, then subtract the 5% NPV from the 10% NPV. The time-0 outflow is unchanged because its discount factor is 1.

At 5%, the present values of the two inflows are

\frac{600}{1.05}=571.428571,\qquad \frac{600}{1.05^2}=\frac{600}{1.1025}=544.217687.

Thus,

\operatorname{NPV}(0.05)=-1000+571.428571+544.217687 =115.646258\approx \$115.65.

At 10%, the present values are

\frac{600}{1.10}=545.454545,\qquad \frac{600}{1.10^2}=\frac{600}{1.21}=495.867769.

Therefore,

\operatorname{NPV}(0.10)=-1000+545.454545+495.867769 =41.322314\approx \$41.32.

The total change from 5% to 10% is

41.322314-115.646258=-74.323944\approx -\$74.32.

The year-1 inflow contributes $545.454545-571.428571=-\$25.974026$, and the year-2 inflow contributes $495.867769-544.217687=-\$48.349918$. Their sum is $-\$74.323944$, matching the change in total NPV. Rounded to cents, the NPV falls by $74.32.

Check and trap avoided. The component changes add to the total change, and the initial outflow contributes zero to the change. A common error is to report the decrease as a positive change. With change defined as the 10% NPV minus the 5% NPV, the answer is negative.

Why the Rate Changes Present Values

The sensitivity formula follows directly by differentiating the NPV expression with respect to $i$. The cash flows and their dates are held fixed, and $i>-1$, so each term is differentiable. The time-0 term $C_0$ is constant; for each future term, the power rule gives the following derivative.

\frac{d}{di}\left[C_t(1+i)^{-t}\right] =-tC_t(1+i)^{-t-1} =-\frac{tC_t}{(1+i)^{t+1}}.

Adding these derivatives gives the local rate sensitivity of the whole project:

\frac{d\,\operatorname{NPV}(i)}{di} =-\sum_{t=1}^{n}\frac{tC_t}{(1+i)^{t+1}}.

The derivative measures the approximate NPV change for a small change in the rate near $i$. Since rates are represented as decimals, a change of one percentage point is a rate change of $0.01$. For a small change $\Delta i$, the local approximation is

\Delta\operatorname{NPV}\approx \frac{d\,\operatorname{NPV}(i)}{di}\Delta i.

This is a local approximation, not a replacement for recalculating NPVs when the rate change is substantial. The exact change from $i_1$ to $i_2$ is the sum, over all dated cash flows, of the change in each cash flow’s discounted value.

Interpretation: A positive future cash flow has a negative derivative contribution, so increasing the rate lowers its present value. A negative future cash flow has a positive derivative contribution, so increasing the rate makes its present value less negative. The time-0 cash flow has no rate sensitivity. For a cash flow $C_t$, the magnitude of its local derivative contribution is $t|C_t|/(1+i)^{t+1}$.

Cash-Flow Signs Can Offset Sensitivity

If all future cash flows are positive, increasing the discount rate lowers each one’s present value, so the NPV cannot increase. If future cash flows have mixed signs, however, the contributions can offset each other. A higher rate reduces the present value of a future outflow in absolute magnitude as well as reducing the present value of an inflow. Therefore, the direction and size of the total change must be calculated rather than inferred from one cash flow alone.

Worked Example: A Future Outflow Offsets Part of the Change

A hypothetical project has cash flows of $-\$1,000$ at time 0, $+\$1,500$ at year 1, and $-\$600$ at year 2. Calculate the NPVs at 5% and 10%, and explain how the future outflow affects the change. Rates are effective annually.

Method. Calculate each future cash flow’s present value at both rates, retaining its sign. In particular, do not treat the year-2 outflow as though it were a positive receipt.

At 5%, the present values are

\frac{1500}{1.05}=1428.571429,\qquad -\frac{600}{1.05^2}=-\frac{600}{1.1025}=-544.217687.

So the NPV is

\operatorname{NPV}(0.05)=-1000+1428.571429-544.217687 =-115.646258\approx -\$115.65.

At 10%, the present values are

\frac{1500}{1.10}=1363.636364,\qquad -\frac{600}{1.10^2}=-\frac{600}{1.21}=-495.867769.

Thus,

\operatorname{NPV}(0.10)=-1000+1363.636364-495.867769 =-132.231405\approx -\$132.23.

The change is

-132.231405-(-115.646258)=-16.585147\approx -\$16.59.

The year-1 inflow contributes $1363.636364-1428.571429=-\$64.935065$. The year-2 outflow contributes $-495.867769-(-544.217687)=+\$48.349918$: it becomes less negative at the higher rate. Together the contributions are $-64.935065+48.349918=-\$16.585147$, equal to the total NPV change.

Check and trap avoided. The outflow’s present value is negative at both rates, but its change is positive. Treating the outflow as a positive amount when calculating the NPV, or assuming every future cash flow contributes a negative change, gives the wrong result.

Later Cash Flows Are Not Automatically More Sensitive

The derivative contribution contains a factor of $t$, but it also contains the discounting factor $(1+i)^{-(t+1)}$. Consequently, the date alone does not determine the magnitude of local sensitivity. For equal positive cash flows at dates $t=1$ and $t=2$, the respective derivative magnitudes, divided by the common cash-flow amount $C$, are

\frac{1}{(1+i)^2}\quad\text{and}\quad \frac{2}{(1+i)^3}.

At $i=1$, these are both $1/4$. At other rates their relative sizes can differ. More generally, compare $t/(1+i)^{t+1}$ for the dates and rate in question, taking account of the cash-flow amounts as well. This comparison concerns local sensitivity at a specified rate; the exact effect of a finite rate change is found by recalculating the discounted values.

Worked Example: Locating the Largest Contribution to a Finite Change

A hypothetical project has an outflow of $2,000 at time 0 and inflows of $800 at year 1, $900 at year 3, and $700 at year 4. Compare its NPV at 0% and 10%, then identify the largest individual contribution to the change. Rates are effective annually.

At 0%, no future cash flow is discounted:

\operatorname{NPV}(0)=-2000+800+900+700=\$400.00.

At 10%, the relevant powers are $1.10$, $1.10^3=1.331$, and $1.10^4=1.4641$. The discounted inflows are

\frac{800}{1.10}=727.272727,\qquad \frac{900}{1.331}=676.183321,\qquad \frac{700}{1.4641}=478.109419.

Therefore,

\operatorname{NPV}(0.10) =-2000+727.272727+676.183321+478.109419 =-118.434533\approx -\$118.43.

The change from 0% to 10% is

-118.434533-400=-518.434533\approx -\$518.43.

The year-1 inflow contributes $727.272727-800=-\$72.727273$. The year-3 inflow contributes $676.183321-900=-\$223.816679$, and the year-4 inflow contributes $478.109419-700=-\$221.890581$. These add to $-\$518.434533$, as required. The largest decrease comes from the year-3 inflow, not the latest inflow. The result also shows why it is safer to compare actual discounted-value changes than to rank cash flows by date alone.

Check and trap avoided. The component changes sum to the total change, and rounding to cents gives a decrease of $518.43. Assuming that the year-4 cash flow must have the largest contribution just because it is latest would overlook the cash-flow amount and the actual discount factors.

A Reliable Sensitivity Workflow

For a project with several dated cash flows, use the same sequence each time. Keep the rate units, dates, and cash-flow signs consistent throughout. The component check is especially useful when a project has both future inflows and future outflows.

List dated cash flows with signs.
Record each amount as positive for an inflow or negative for an outflow, and retain the stated date for discounting.

Calculate NPV at each specified rate.
Discount every future cash flow at each rate, leaving the time-0 amount unchanged.

Subtract in the requested order.
For a change from $i_1$ to $i_2$, calculate $\operatorname{NPV}(i_2)-\operatorname{NPV}(i_1)$. A negative result is a decrease.

Attribute and verify the change.
Subtract each cash flow’s first present value from its second. The component changes must sum to the total NPV change.

Interpret signs and dates carefully.
Positive future flows normally lose present value when the rate rises; negative future flows become less negative. Do not infer that a later date must dominate.

Common Mistakes

Reversing the subtraction. State whether the comparison is new rate minus old rate. A negative difference means NPV fell under that convention.
Dropping the sign of an outflow. Keep negative future cash flows negative when finding their present values and changes.
Including the initial cash flow in the rate effect. A time-0 cash flow is not discounted, so its value is the same at every rate.
Ranking sensitivity only by date. Compare the cash-flow amount, date, and discount factor together; later does not always mean a larger effect.
Using a local derivative as an exact finite change. The derivative approximates the effect of a small rate change near the chosen rate. For a specified finite change, recalculate the NPVs or each discounted cash flow.

Key takeaway: NPV sensitivity is determined by how each fixed, dated cash flow changes in present value as the discount rate changes. Recalculate at the specified rates, preserve signs, and verify that the individual contributions add to the total change.

Check Your Understanding

For each problem, treat the stated rates as effective annual rates and the cash flows as occurring at the listed annual dates.

A project has cash flows of $-\$500$ at time 0 and $+\$300$ at each of years 1 and 2. Find the NPV at 0% and 8%, and the change from 0% to 8%. (Answer: NPV at 0% = $100.00; NPV at 8% = $34.98; change = $-\$65.02$.)
A project has cash flows of $-\$1,000$ at time 0, $+\$600$ at year 1, and $+\$600$ at year 3. Find its NPVs at 5% and 10%, and the change from 5% to 10%. (Answer: NPV at 5% = $89.73; NPV at 10% = $-\$3.76$; change = $-\$93.49$.)
A project has cash flows of $-\$1,000$ at time 0, $+\$900$ at year 1, and $-\$400$ at year 2. Find its NPVs at 0% and 10%, and explain the direction of the change. (Answer: NPV at 0% = $-\$500.00$; NPV at 10% = $-\$512.40$; change = $-\$12.40$, because the positive inflow’s decrease exceeds the offset from the outflow becoming less negative.)
At $i=5\%$, compare the magnitudes of the local derivative contributions from a $+\$500$ cash flow at year 1 and a $+\$500$ cash flow at year 3. (Answer: Year 1: $500/(1.05)^2\approx \$453.51$ per unit rate; year 3: $1500/(1.05)^4\approx \$1,234.05$ per unit rate. The year-3 contribution has the larger magnitude at this rate.)
A project consists only of a $-\$2,500$ cash flow at time 0. What is its NPV at 4% and 9%, and how much does NPV change between those rates? (Answer: NPV is $-\$2,500.00$ at each rate; change = $0.00 because a time-0 cash flow is not discounted.)

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