10. Present Values & Discounting

Present Values & Discounting

• Author: S. Levkoff, PhD, CAP®

Agenda

• Key Topics:

  • Present & Future Values

  • Net Present Value (NPV)

  • Internal Rate of Return (IRR)

  • Evaluating Investments

  • Psychology of Discounting and Patience

  • Classical Theory of Asset Pricing

Future Values

• Definition: Future Value (FV) of a principal amount left in an account for a year at an interest rate (r) can be calculated as:

  • FV = Principal (1 + r)

Examples of Future Values

• Single Year Compounding:

  • FV after 1 year = Principal × (1 + r)

  • Example Formulation:

    1. After 1 year: FV = Principal × (1 + r)

    2. After 2 years: FV = Principal × (1 + r)²

• Using Simple Interest:

  • FV after 2 years with Simple Interest is calculated as:

    • FV = Principal + (Principal × r × Number of years)

Compounding Interest

• Definition: Compound interest allows earning interest on previously earned interest.

  • Formula for two years:

    • FV = Principal × (1 + r)² (compounded)

From Future to Present Values

• To convert Future Value back to Present Value (PV):

  • PV = FV / (1 + r)^t

  • Understanding the Relationship:

    • Moving Values Forward: Multiply by (1 + r)

    • Moving Values Backwards: Divide by (1 + r)

Net Present Values

• Definition: NPV measures the value of cash flow streams at different times brought to present value.

• Formula:

  • NPV = C₀ + C₁/(1+r) + C₂/(1+r)² + ... + Cₜ/(1+r)ⁿ

  • Converts various future payoffs to present values for analysis.

Lumberjack Example

• Situation: Decision on harvesting trees at different time periods affects cash flow.

  • Cash Flow Streams:

    • Early Harvest: {-100, 200, 0}

    • Late Harvest: {-100, 0, 300}

• Comparing NPVs for decision making:

  • NPV for Early Harvest and Late Harvest indicates better investment choices.

(Internal) Rate of Return (IRR)

• Definition: IRR is the discount rate making NPV = 0.

• IRR remains unchanged by variations in discount rates, allowing consistent assessment of investment viability.

Discount Rate vs. Discount Factor

• Discount Rate (r): The subjective interest rate reflecting opportunity cost.

• Discount Factor (DF):

  • DF = 1/(1 + r)

  • Used to convert future values back to present values (weights future benefits).

The Discount Factor and Patience

• The value of DF can be an indicator of an individual's patience.

• Closer DF is to 1, the more weight future payoffs receive (more patient).

• Closer DF is to 0, the less weight future payoffs receive (less patient).

Classical Theory of Asset Prices

• Financial assets provide future cash flow rights.

• Market equilibrium occurs when the price equals NPV, ensuring no arbitrage opportunities exist.

• Conditions for the classical theory: competitive markets and informed agents.