Imagine the City Council is evaluating two major urban development initiatives, each with different cash flow streams and objectives, to improve its infrastructure and quality of life. The council uses an annual effective discount rate of 6.5% for all its financial evaluations.

Project A: Green Corridor Development

This project aims to convert an abandoned railway line into a linear park and green space.

• It requires an initial investment of $120,000 today.
• Annual maintenance costs are projected to be $8,000 at the end of Year 1, Year 2, and Year 3.
• The project is expected to generate significant community benefits (e.g., increased property values, tourism, health benefits) which are monetized as equivalent cash inflows of $40,000 at the end of Year 1, $55,000 at the end of Year 2, and $70,000 at the end of Year 3.

Project B: Smart City Infrastructure Upgrade

This initiative focuses on deploying advanced digital infrastructure across a district to enhance connectivity and services.

• It requires an initial investment of $150,000 today.
• Expected revenues from enhanced services are $30,000 at the end of Year 1 and $50,000 at the end of Year 2.
• A major system upgrade costing $15,000 will be required at the end of Year 2.
• To ensure the financial viability and long-term sustainability of the project, a crucial government grant 'G' is anticipated to be received at the end of Year 3.

If the Net Present Value (NPV) of the Green Corridor Development project is equal to the Net Present Value (NPV) of the Smart City Infrastructure Upgrade project, calculate the value of the government grant 'G'.

• $81,335.77
• $93,784.49
• $106,494.61
• $111,186.27
• $113,292.15

Step 1: Define Parameters and Discount Factors

The annual effective discount rate is [math]i = 6.5\% = 0.065[/math]. We will calculate the present value factor [math]v^t = (1+i)^{-t}[/math] for each year [math]t[/math] up to 3 years.

Present Value Factors

Year (t)	Discount Factor ([math]v^t[/math])	Value
1	[math](1.065)^{-1}[/math]	[math]\approx 0.938967136[/math]
2	[math](1.065)^{-2}[/math]	[math]\approx 0.881659283[/math]
3	[math](1.065)^{-3}[/math]	[math]\approx 0.827849092[/math]

Step 2: Calculate Net Cash Flows for Project A (Green Corridor Development)

We determine the net cash flow ([math]CF_t[/math]) for Project A at each time point.

• Initial Investment ([math]t=0[/math]): Outflow of $120,000.
• Year 1 ([math]t=1[/math]): Inflow of $40,000 (community benefits) and outflow of $8,000 (maintenance costs).
• Year 2 ([math]t=2[/math]): Inflow of $55,000 (community benefits) and outflow of $8,000 (maintenance costs).
• Year 3 ([math]t=3[/math]): Inflow of $70,000 (community benefits) and outflow of $8,000 (maintenance costs).

Cash Flows for Project A

Year (t)	Description	Inflow ($)	Outflow ($)	Net Cash Flow ([math]CF_t[/math]) ($)
0	Initial Investment	0	120,000	-120,000
1	Community Benefits / Maintenance	40,000	8,000	32,000
2	Community Benefits / Maintenance	55,000	8,000	47,000
3	Community Benefits / Maintenance	70,000	8,000	62,000

Step 3: Calculate Net Present Value (NPV) for Project A

The Net Present Value ([math]NPV_A[/math]) for Project A is the sum of the present values of all its net cash flows.

[[math]]NPV_A = CF_0 + CF_1 \cdot v + CF_2 \cdot v^2 + CF_3 \cdot v^3[[/math]]

[[math]]NPV_A = -120,000 + 32,000(1.065)^{-1} + 47,000(1.065)^{-2} + 62,000(1.065)^{-3}[[/math]]

Using the discount factors from Step 1:

[[math]]NPV_A = -120,000 + 32,000(0.938967136) + 47,000(0.881659283) + 62,000(0.827849092)[[/math]]

[[math]]NPV_A = -120,000 + 30046.94835 + 41437.98620 + 51326.64371[[/math]]

[[math]]NPV_A = 2811.57826[[/math]]

Step 4: Calculate Net Cash Flows for Project B (Smart City Infrastructure Upgrade)

We determine the net cash flow ([math]CF_t[/math]) for Project B at each time point, including the unknown grant 'G'.

• Initial Investment ([math]t=0[/math]): Outflow of $150,000.
• Year 1 ([math]t=1[/math]): Inflow of $30,000 (revenues).
• Year 2 ([math]t=2[/math]): Inflow of $50,000 (revenues) and outflow of $15,000 (system upgrade cost).
• Year 3 ([math]t=3[/math]): Inflow of 'G' (government grant).

Cash Flows for Project B

Year (t)	Description	Inflow ($)	Outflow ($)	Net Cash Flow ([math]CF_t[/math]) ($)
0	Initial Investment	0	150,000	-150,000
1	Revenues	30,000	0	30,000
2	Revenues / System Upgrade	50,000	15,000	35,000
3	Government Grant	G	0	G

Step 5: Set up Net Present Value (NPV) for Project B in terms of G

The Net Present Value ([math]NPV_B[/math]) for Project B is the sum of the present values of its net cash flows, including the unknown grant 'G'.

[[math]]NPV_B = CF_0 + CF_1 \cdot v + CF_2 \cdot v^2 + CF_3 \cdot v^3[[/math]]

[[math]]NPV_B = -150,000 + 30,000(1.065)^{-1} + 35,000(1.065)^{-2} + G(1.065)^{-3}[[/math]]

Using the discount factors from Step 1:

[[math]]NPV_B = -150,000 + 30,000(0.938967136) + 35,000(0.881659283) + G(0.827849092)[[/math]]

[[math]]NPV_B = -150,000 + 28169.01408 + 30858.07490 + 0.827849092G[[/math]]

[[math]]NPV_B = -90972.91102 + 0.827849092G[[/math]]

Step 6: Equate NPVs and Solve for G

The problem states that the Net Present Value of Project A is equal to the Net Present Value of Project B ([math]NPV_A = NPV_B[/math]).

[[math]]2811.57826 = -90972.91102 + 0.827849092G[[/math]]

Now, we solve for G:

[[math]]0.827849092G = 2811.57826 + 90972.91102[[/math]]

[[math]]0.827849092G = 93784.48928[[/math]]

[[math]]G = \frac{93784.48928}{0.827849092}[[/math]]

[[math]]G = 113292.14819[[/math]]

Rounding to two decimal places, the value of the government grant G is approximately $113,292.15.

Key Insights

• Net Present Value (NPV) is a crucial metric for evaluating project profitability, representing the present value of expected cash inflows minus the present value of expected cash outflows.
• Accurate identification and classification of all cash flows (inflows, outflows, and initial investments) at their correct time periods are fundamental for NPV calculation.
• The choice of the discount rate significantly impacts the NPV. Consistent application of the specified annual effective discount rate across all projects is essential for comparable evaluations.
• When comparing projects, equating their NPVs allows for solving for unknown variables (like a grant or a future payment) that make the projects equally financially attractive under the given discount rate.
• Precision in discount factor calculations (carrying sufficient decimal places) and subsequent arithmetic is important to avoid rounding errors, especially when solving for an unknown variable.

Answer was verified by AI

Answer was verified numerically