Consider A Project With Free Cash Flows In One Year Of 14,41

Consider A Project With Free Cash Flows In One Year Of 144100or 1

Consider a project with free cash flows in one year of $144,100 or $186,700, with each outcome being equally likely. The initial investment required for the project is $92,300, and the project's cost of capital is 24%. The risk-free interest rate is 7%.

a. What is the NPV of this project?

b. Suppose that to raise the funds for the initial investment, the project is sold to investors as an all-equity firm. The equity holders will receive the cash flows of the project in one year. How much money can be raised in this way—that is, what is the initial market value of the unlevered equity?

c. Suppose the initial $92,300 is instead raised by borrowing at the risk-free interest rate. What are the cash flows of the levered equity, what is its initial value, and what is the initial equity according to Modigliani and Miller (MM)?

Paper For Above instruction

The valuation of investment projects and the implications of leveraging are foundational concepts in corporate finance, reflected in the understanding of net present value (NPV), capital structure, and firm valuation under different financing scenarios. This essay examines a hypothetical project with probabilistic cash flows, applies relevant financial formulas, and explores how financing choices influence firm value, using the principles established by Modigliani and Miller.

Part A: Calculating the NPV of the Project

The project presents two possible cash flows in one year: $144,100 and $186,700, each with an equal probability of 50%. The initial investment is $92,300, and the project’s cost of capital is 24%. To compute the NPV, we first determine the expected cash flow, then discount it at the required rate, and finally subtract the initial investment.

The expected cash flow (ECF) is calculated as:

ECF = 0.5 × $144,100 + 0.5 × $186,700 = $72,050 + $93,350 = $165,400.

Next, discounting this expected cash flow at 24%:

Present Value = ECF / (1 + r) = $165,400 / (1 + 0.24) ≈ $165,400 / 1.24 ≈ $133,387.

NPV is then:

NPV = Present Value of expected cash flows - Initial investment = $133,387 - $92,300 ≈ $41,087.

Therefore, the project’s NPV is approximately $41,087. A positive NPV indicates that the project adds value and should be considered favorable from an investment standpoint.

Part B: Valuation as an Unlevered Equity Firm

In selling the project as an all-equity firm, the initial market value of the unlevered equity equals the current value of the expected cash flows discounted at the appropriate rate, which in this context is the project’s cost of capital (24%). This approach assumes that the cash flows are attributable solely to equity holders, without debt considerations.

Using the same expected cash flow calculation and discount rate as part A, the initial equity value equals approximately $133,387. Rounding to the nearest dollar, this is $133,387. This valuation captures the current worth of the project's cash flows to equity investors in a no-debt structure.

Part C: Financing the Investment via Risk-Free Borrowing

If the initial $92,300 is raised by borrowing at the risk-free rate of 7%, the leverage impacts both the cash flows and valuation according to the Modigliani-Miller theorem (without taxes). The borrowed amount accrues interest, and the cash flows to equity are affected by the repayment obligations and the financial structure.

First, we determine the future obligation of debt:

Debt at time 0: $92,300.

Interest at 7%: $92,300 × 0.07 ≈ $6,461.

Total debt to be repaid in one year: $92,300 + $6,461 ≈ $98,761.

Cash flows to equity depend on the project’s outcomes:

- If the economy is strong, the project cash flow is $186,700.

- If weak, cash flow is $144,100.

In each case, debt repayment occurs first, and residual cash flows are allocated to equity holders:

- For a strong economy:

Equity cash flow = $186,700 - $98,761 ≈ $87,939.

- For a weak economy:

Equity cash flow = $144,100 - $98,761 ≈ $45,339.

These are probabilistic outcomes with equal likelihood. The expected cash flow to equity is:

(0.5 × $87,939) + (0.5 × $45,339) = $43,969.50 + $22,669.50 = $66,639.

The initial value of the levered equity is the present value of these expected cash flows, discounted at the risk-free rate of 7%:

PV = $66,639 / (1 + 0.07) ≈ $66,639 / 1.07 ≈ $62,268.

Note that the initial equity value (per MM without taxes) should correspond to the equity portion, which is the value of the firm minus the debt. Since the total firm value is approximately $133,387, and debt is $92,300, the initial equity value aligns with $41,087, matching the prior valuation in a no-leverage scenario. This demonstrates the effect of leveraging on equity valuation under MM assumptions.

Conclusion

The valuation of projects with probabilistic cash flows reveals significant insights into investment decision-making. The positive NPV ($41,087) indicates the project’s profitability. The valuation as an unlevered firm produces a similar figure (~$133,387), aligning with the expected discounted cash flows. Introducing leverage at the risk-free rate alters cash flows to equity and the firm's valuation, consistent with MM propositions that capital structure changes do not affect total firm value in perfect markets but do influence the distribution of cash flows and individual equity value.

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