Exercise
A municipal government is evaluating two potential public works projects, a Community Park Revitalization (CPR) and a Downtown Infrastructure Upgrade (DIU), over a 4-year planning horizon. They need to compare the present value of the net benefits (benefits minus costs) of these projects to allocate funds efficiently. The annual effective discount rate for public projects is [math]5\%[/math].
Project CPR (Community Park Revitalization):
- Initial cost (outflow at time [math]t=0[/math]): $[math]200,000[/math]
- Projected net annual benefits (inflows at year-end):
- End of Year 1: $[math]60,000[/math]
- End of Year 2: $[math]75,000[/math]
- End of Year 3: $[math]80,000[/math]
- End of Year 4: $[math]65,000[/math]
Project DIU (Downtown Infrastructure Upgrade):
- Initial cost (outflow at time [math]t=0[/math]): $[math]180,000[/math]
- Projected net annual benefits (inflows at year-end):
- End of Year 1: $[math]50,000[/math]
- End of Year 2: $[math]60,000[/math]
- End of Year 3: $[math]X[/math] (unknown benefit)
- End of Year 4: $[math]70,000[/math]
The municipal council has decided that Project DIU will only be approved if its Net Present Value (NPV) is equal to or greater than the NPV of Project CPR.
Calculate the minimum value of [math]X[/math] (the net benefit at the end of Year 3 for Project DIU) required for Project DIU to be approved.
- $68,122.78
- $75,095.39
- $78,852.10
- $80,000.00
- $82,798.12
The annual effective discount rate is [math]i = 5\% = 0.05[/math]. We will calculate the present value factor [math]v^t = (1+i)^{-t}[/math] for each year [math]t[/math] up to 4.
| Year (t) | PV Factor ([math]v^t = (1.05)^{-t}[/math]) | Approximate Value |
|---|---|---|
| 1 | [math](1.05)^{-1}[/math] | 0.952381 |
| 2 | [math](1.05)^{-2}[/math] | 0.907029 |
| 3 | [math](1.05)^{-3}[/math] | 0.863838 |
| 4 | [math](1.05)^{-4}[/math] | 0.822702 |
The NPV for Project CPR is the sum of the present values of its cash flows (benefits minus initial cost).
| Year (t) | Cash Flow ([math]CF_t[/math]) | PV Factor ([math]v^t[/math]) | Present Value ([math]CF_t \times v^t[/math]) |
|---|---|---|---|
| 0 | -$200,000 | 1 | -$200,000.00 |
| 1 | $60,000 | [math](1.05)^{-1}[/math] | $57,142.86 |
| 2 | $75,000 | [math](1.05)^{-2}[/math] | $68,027.21 |
| 3 | $80,000 | [math](1.05)^{-3}[/math] | $69,107.04 |
| 4 | $65,000 | [math](1.05)^{-4}[/math] | $53,475.63 |
The Net Present Value of Project CPR is:
The NPV for Project DIU includes an unknown net benefit [math]X[/math] at the end of Year 3.
| Year (t) | Cash Flow ([math]CF_t[/math]) | PV Factor ([math]v^t[/math]) | Present Value ([math]CF_t \times v^t[/math]) |
|---|---|---|---|
| 0 | -$180,000 | 1 | -$180,000.00 |
| 1 | $50,000 | [math](1.05)^{-1}[/math] | $47,619.05 |
| 2 | $60,000 | [math](1.05)^{-2}[/math] | $54,421.77 |
| 4 | $70,000 | [math](1.05)^{-4}[/math] | $57,589.14 |
The sum of the present values of the known cash flows for Project DIU (excluding [math]X[/math]) is:
Project DIU will be approved if its NPV is equal to or greater than the NPV of Project CPR. For the minimum value of [math]X[/math], we set [math]NPV_{DIU} = NPV_{CPR}[/math].
- Net Present Value (NPV) is a fundamental tool for capital budgeting, allowing comparison of projects by converting all future cash flows to their present-day equivalents.
- The chosen discount rate is critical, as it reflects the time value of money and the project's risk, directly impacting the calculated NPV.
- To solve for an unknown cash flow within an NPV framework, set the project's NPV (including the unknown) equal to a target NPV (e.g., a benchmark or another project's NPV) and algebraically isolate the unknown variable.
- Each cash flow must be correctly discounted to time [math]t=0[/math] using its specific discount factor [math](1+i)^{-t}[/math] to ensure accuracy in NPV calculations.