Corporate Finance: Net Present Value and the Separation Theorem

Course Introduction to Corporate Finance

• Instructor: Jessica Wachter, Professor of Financial Management and Professor of Finance at the Wharton School, University of Pennsylvania.
• Course Scope: This course covers the fundamental principles governing how corporations make financial decisions to maximize value.
• Primary Topics Covered:
  • Net Present Value (NPV) Rule: The foundational principle for evaluating investment opportunities.
  • Present Value Decisions: Determining the current worth of future cash flows.
  • Valuation of Bonds and Stocks: Applying present value concepts to financial securities.
  • Internal Rate of Return (IRR): An alternative metric used in capital budgeting and project evaluation.

Present Value Concepts and Calculations

• Future Value (FV):

  • Represents the amount an investment will grow to over a period of time at a specific interest rate.
  • Example Calculation: If you have an initial principal of $100 and an interest rate of $10\%$, the value after one year is calculated as:
    • $FV = 100 + 100 \times (0.10) = 110$
  • General Formula: $FV = PV(1 + r)$

• Present Value (PV):

  • Determines what a future sum of money is worth today, given a specific rate of return (discount rate).
  • Calculated by rearranging the FV formula: $PV = \frac{FV}{1 + r}$
  • Example Calculation: If you need $100$ in one year and the interest rate is $10\%$, the amount you must deposit today is:
    • $100 = PV(1 + r)$
    • $PV = \frac{100}{1 + 0.10}$
    • $PV = \frac{100}{1.10} = 90.91$

• General Form for Single Cash Flow:

  • Let $C$ represent the cash flow (CF) expected in one year.
  • $PV = \frac{C}{1 + r}$

Net Present Value (NPV) Rule

• Definition of NPV:

  • Net Present Value is the difference between the present value of cash inflows and the present value of cash outflows over a period of time.
  • Formula: $NPV = C_0 + \frac{C_1}{1 + r}$
    • $C_0$: The initial investment or cost (typically expressed as a negative number).
    • $C_1$: The expected payoff or cash flow in the next period.
    • $r$: The discount rate or market interest rate.

• Numerical Case Study (Investment Project):

  • Initial Cost: $0.5 \, M$ ($500,000$).
  • Next Year Payoff: $0.54 \, M$ ($540,000$).
  • Interest Rate ($r$): Assume $5 \%$.
  • Calculation:
    • $NPV = -0.5 + \frac{0.54}{1 + 0.05}$
    • $NPV = -0.5 + \frac{0.54}{1.05}$
    • $NPV = 0.0143 \, M$ ($14,300$).

• The NPV Decision Rule:

  • Accept projects if $NPV > 0$ (Positive NPV adds value to the firm).
  • Reject projects if $NPV < 0$ (Negative NPV destroys value).
  • Core Result: Following the NPV rule maximizes the total value of the corporation.

Investment Scenarios and Analysis

• Scenario 1: Suzy’s Consumption Choice

  • Initial Capital: $1 \, M$.
  • Bank interest rate ($r$): $20 \%$.
  • Trade-off: Suzy can either consume $1 \, M$ now or wait until "old age" to consume more.
  • If she saves the money, she will have $1.2 \, M$ later ($1 \, M \times 1.20 = 1.2 \, M$).
  • Graphical Representation: The trade-off between current and future consumption is shown as a line where the slope is $-(1 + r)$.

• Comparing Business Projects (Restaurant vs. Vineyard):

  • Assumptions: Initial cost for both is $0.7 \, M$. The market interest rate ($r$) is $30 \%$.     

  • The Restaurant Project:

    • Cost ($C_0$): $-0.7$
    • Payoff ($C_1$): $0.8$
    • Threshold check: To break even, the payoff must be at least $C_0 \times (1 + r)$.
    • $0.7 \times (1.30) = 0.91$
    • Since $0.8 < 0.91$, the project has an $NPV < 0$.
    • Action: Reject.

  • The Vineyard Project:

    • Cost ($C_0$): $-0.7$
    • Payoff ($C_1$): $0.91$ or higher (Slide 15 shows a value of $1.06$ generated for the firm).
    • Analysis: If the payoff is high enough such that $NPV > 0$, the project should be accepted.
    • Firm Value Calculation: Based on the investment, the value of the firm becomes the sum of the remaining cash and the present value of the project payoff.
    • $PV = 0.3 + \frac{0.91}{1.30} = 0.3 + 0.7 = 1.0$
    • Slide 15 indicates a scenario where firm value reaches $1.06$.

• Summary Logic of the NPV Rule:

  • Accept if $NPV = C_0 + \frac{C_1}{1 + r} > 0$.
  • This is mathematically equivalent to accepting if the future payoff ($C_1$) is greater than the future value of the initial cost ($-C_0(1 + r)$).

The Separation Theorem

• The Separation Theorem provides a crucial framework for corporate decision-making because it decouples individual preferences from financial management.
• Two Separate Decisions:
  1. What projects to choose? This decision should be made solely based on the NPV rule to maximize the value of the firm.
  2. When to consume? This is a personal decision for shareholders/individuals based on their own utility and timing preferences.
• Significance: Because of the existence of capital markets, managers can focus strictly on maximizing firm value through positive NPV projects, and individual shareholders can then use the market (borrowing or lending) to adjust their consumption timing to their liking.