No More Worries!

Our orders are delivered strictly on time without delay

Paper Formatting

Double or single-spaced
1-inch margin
12 Font Arial or Times New Roman
300 words per page

No Lateness!

Our orders are delivered strictly on time without delay

Our Guarantees

Free Unlimited revisions
Guaranteed Privacy
Money Return guarantee
Plagiarism Free Writing

Net Present Value (NPV)

Elaborate on why the net present value (NPV) of a relatively long-term project is more sensitive to changes in the cost of capital than is the NPV of a short-term project.
Provide two examples of NPV calculations that support your position.

Full Answer Section

Later Cash Flows are Discounted More Heavily: Cash flows that occur further in the future are discounted more heavily. In a long-term project, a larger proportion of the total cash flows typically occur in the later years. These later cash flows are the most susceptible to changes in the discount rate because they are divided by a larger $(1 + r)^{t}$ term. A small increase in $r$ drastically reduces the present value of these distant cash flows, leading to a much larger change in the overall NPV. Conversely, a short-term project has most of its cash flows occurring relatively soon, where the discounting effect is less pronounced.
Time Value of Money Amplification: The core concept of NPV is the time value of money, which states that a dollar today is worth more than a dollar in the future. The cost of capital represents the opportunity cost of investing money in the project. When this cost changes, the "value erosion" of future cash flows is either accelerated (higher $r$ ) or slowed down (lower $r$ ). This effect is much more pronounced over extended periods.

Examples of NPV Calculations

Let's consider two projects, one short-term (2 years) and one long-term (10 years), with an initial investment of $$1, 000$ for both.

Scenario 1: Initial Cost of Capital = 10%

Example 1: Short-Term Project (2 Years)

Initial Investment ( $CF_{0}$ ): $- $1, 000$
Year 1 Cash Flow ( $CF_{1}$ ): $$600$
Year 2 Cash Flow ( $CF_{2}$ ): $$600$

$NPV_{Short} = ( 1 + 0.10 ) ^{0} -$1 , 000 + ( 1 + 0.10 ) ^{1} $600 + ( 1 + 0.10 ) ^{2} $600$ $NPV_{Short} = - $1, 000 + 1.10 $600 + 1.21 $600$ $NPV_{Short} = - $1, 000 + $545.45 + $495.87$ $NPV_{Short} = $41.32$

Example 2: Long-Term Project (10 Years)

Initial Investment ( $CF_{0}$ ): $- $1, 000$
Annual Cash Flow ( $CF_{1 - 10}$ ): $$170$ (for 10 years)

$NPV_{Long} = - $1, 000 + t = 1 \sum 10 ( 1 + 0.10 ) ^{t} $170$ Using a financial calculator or spreadsheet for the sum of present values of an annuity: Present Value of Annuity Factor for 10 years at 10% $\approx 6.14457$ $NPV_{Long} = - $1, 000 + ($170 \times 6.14457)$ $NPV_{Long} = - $1, 000 + $1, 044.58$ $NPV_{Long} = $44.58$

Scenario 2: Cost of Capital Increases by 2% to 12%

Now, let's see the effect of changing the cost of capital from 10% to 12%.

Example 1: Short-Term Project (2 Years) with 12% Cost of Capital

$NPV_{Short} = ( 1 + 0.12 ) ^{0} -$1 , 000 + ( 1 + 0.12 ) ^{1} $600 + ( 1 + 0.12 ) ^{2} $600$ $NPV_{Short} = - $1, 000 + 1.12 $600 + 1.2544 $600$ $NPV_{Short} = - $1, 000 + $535.71 + $478.32$ $NPV_{Short} = $14.03$

Change in NPV for Short-Term Project: $$41.32 - $14.03 = $27.29$
Percentage Change: $($27.29/$41.32) \times 100% \approx 66.05%$

Example 2: Long-Term Project (10 Years) with 12% Cost of Capital

$NPV_{Long} = - $1, 000 + t = 1 \sum 10 ( 1 + 0.12 ) ^{t} $170$ Present Value of Annuity Factor for 10 years at 12% $\approx 5.65022$ $NPV_{Long} = - $1, 000 + ($170 \times 5.65022)$ $NPV_{Long} = - $1, 000 + $960.54$ $NPV_{Long} = - $39.46$

Change in NPV for Long-Term Project: $$44.58 - (- $39.46) = $84.04$
Percentage Change: $($84.04/$44.58) \times 100% \approx 188.52%$

Conclusion

As shown by the examples:

For the short-term project, a 2% increase in the cost of capital resulted in a decrease in NPV from $$41.32$ to $$14.03$ , a change of $$27.29 (or approximately 66.05%).
For the long-term project, the same 2% increase in the cost of capital resulted in a decrease in NPV from $$44.58$ to $- $39.46$ , a change of $$84.04 (or approximately 188.52%).

The long-term project's NPV changed by a significantly larger absolute amount and percentage, demonstrating its higher sensitivity to changes in the cost of capital. This is because the impact of the discount rate is compounded over a much greater number of periods for long-term projects, particularly affecting those cash flows furthest in the future.

Sample Answer

The Net Present Value (NPV) of a relatively long-term project is more sensitive to changes in the cost of capital than is the NPV of a short-term project primarily due to the compounding effect of discounting over a longer period.

Explanation

The NPV formula is:

$NPV = t = 0 \sum n ( 1 + r ) ^{t} CF ^{t}$

Where:

$CF_{t}$ = Cash flow in period $t$
$r$ = Discount rate (cost of capital)
$t$ = Time period
$n$ = Total number of periods (project life)

The term $(1 + r)^{t}$ in the denominator is the discount factor. As the time period $t$ increases, the exponent becomes larger.

Exaggerated Impact of the Discount Rate: When the discount rate ( $r$ ) changes, its impact is compounded over each period. For a short-term project, the number of periods ( $t$ ) is small, so the exponent $t$ is small. A change in $r$ will have a relatively minor effect on $(1 + r)^{t}$ . However, for a long-term project, $t$ is large, meaning even a small change in $r$ is magnified exponentially over many periods. This significantly alters the present value of distant cash flows.