Chapter 3: Financial Decision Making and the Law of One Price: Study Notes

Financial Decision Making and the Law of One Price

Introduction

This chapter explores the principles of financial decision making, focusing on the Law of One Price, net present value (NPV), arbitrage, and risk-free interest rates. The problems and examples provided illustrate how these concepts are applied in real-world financial scenarios, including investment decisions, pricing securities, and evaluating cash flows.

Law of One Price

Definition and Application

• Law of One Price: States that identical goods or securities must sell for the same price in efficient markets, preventing arbitrage opportunities.

• Arbitrage: The practice of exploiting price differences in different markets for the same asset to earn risk-free profits.

• Example: If a security pays $100 in one year and the risk-free rate is 5%, its price today should be $95.24 ($100 / 1.05).

Net Present Value (NPV)

Definition and Calculation

• Net Present Value (NPV): The difference between the present value of cash inflows and outflows over a period of time.

• Formula:

$\text{NPV} = \sum \frac{\text{Cash Flow}_t}{(1 + r)^t}$

• Application: Used to evaluate investment opportunities. A positive NPV indicates a profitable investment.

• Example: Investing $114 million today to receive $120 million in one year at a 4% risk-free rate:

$\text{NPV} = \frac{114 \text{ million}}{1.04} = 109.62 \text{ million}$

Time Value of Money

Present and Future Value

• Present Value (PV): The current worth of a future sum of money given a specific rate of return.

• Formula:

$\text{PV} = \frac{\text{Future Value}}{(1 + r)^n}$

• Future Value (FV): The value of a current asset at a future date based on an assumed rate of growth.

• Formula:

$\text{FV} = \text{PV} \times (1 + r)^n$

• Example: Receiving $200 in one year is equivalent to $190.48 today at a 5% interest rate.

Arbitrage and Security Pricing

Arbitrage Opportunities

• Arbitrage: Occurs when a security is mispriced relative to its cash flows and the risk-free rate.

• Example: If an ETF trades for $120 but its underlying securities are worth $134, buy the ETF and sell the underlying securities for a risk-free profit.

Security Pricing with Risk-Free Rate

• Risk-Free Rate: The theoretical rate of return of an investment with zero risk, often based on government bonds.

• Pricing Formula:

$\text{Price} = \frac{\text{Cash Flow in One Year}}{1 + r}$

• Example: A security pays $150 in one year, risk-free rate is 7.14%:

$\text{Price} = \frac{150}{1.0714} = 140$

Investment Decision Making

Evaluating Projects

• Project Selection: Choose projects with the highest NPV to maximize shareholder value.

• Example Table:

Project	Cash Flow Today ($)	Cash Flow in One Year ($)
A	-20,000	35,000
B	-20,000	30,000
C	-40,000	60,000

• NPV Calculation:

$\text{NPV}_A = \frac{35,000}{1.10} - 20,000 = 11,818.18$

$\text{NPV}_B = \frac{30,000}{1.10} - 20,000 = 7,272.73$

$\text{NPV}_C = \frac{60,000}{1.10} - 40,000 = 14,545.45$

• Decision: Project C has the highest NPV and should be chosen.

Risk and Return

Risk Premium and Expected Return

• Risk Premium: The extra return required by investors for taking on additional risk.

• Expected Return: The weighted average of possible returns, considering probabilities.

• Formula:

$\text{Expected Return} = \sum p_i r_i$

• Example Table:

Security	Market Price Today	Cash Flow (Weak Economy)	Cash Flow (Strong Economy)
A	231	600	600
B	346	0	800

• Portfolio Payoff: Buying one share of each security yields $600 in both states.

• Expected Return Calculation:

$\text{Expected Return} = \frac{600 - (231 + 346)}{(231 + 346)} = 4.0\%$

International Financial Decisions

Currency and Interest Rate Arbitrage

• American Depositary Receipt (ADR): A negotiable certificate representing shares in a foreign company, traded in U.S. dollars.

• Currency Arbitrage: Exploiting differences in exchange rates and interest rates between countries.

• Example: If the exchange rate and interest rates differ between the U.S. and Japan, investors may borrow in one country and invest in another for arbitrage profits.

Summary Table: Key Formulas

Concept	Formula (LaTeX)
Net Present Value (NPV)	$\text{NPV} = \sum \frac{\text{Cash Flow}_t}{(1 + r)^t}$
Present Value (PV)	$\text{PV} = \frac{\text{Future Value}}{(1 + r)^n}$
Future Value (FV)	$\text{FV} = \text{PV} \times (1 + r)^n$
Expected Return	$\text{Expected Return} = \sum p_i r_i$

Conclusion

Understanding the Law of One Price, NPV, arbitrage, and risk-return relationships is essential for making sound financial decisions. These principles guide the evaluation of investments, pricing of securities, and identification of profitable opportunities in financial markets.