Financial Decision-Making and the Law of One Price

The one decision rule

Every financial decision in this course can be reduced to:

Does the present value of the benefits exceed the present value of the costs?

If yes, do it. If no, don’t. We’ll call the difference net present value (NPV):

$$\text{NPV} \;=\; \text{PV(benefits)} \;-\; \text{PV(costs)}.$$

Positive NPV creates value for shareholders. Negative NPV destroys it. Zero NPV is indifferent. That’s the entire decision framework.

Everything else in this course is machinery for computing PV correctly in specific situations: discounting future cash flows (Unit 2), valuing risky cash flows (Unit 4), choosing the right discount rate when financing is mixed (Unit 5).

Why we use market prices

To compute PV we need to convert future cash flows into today’s dollars. The right exchange rate to use is whatever rate competitive financial markets are quoting, not the rate you personally think captures the time value of money.

Why? Because if your personal rate differs from the market rate, someone can arbitrage you. Here’s the canonical example: I’m willing to sell a contract that pays $110 in one year for $95 today. The market rate for one-year risk-free borrowing is 5%. The market price of $110 in one year is $110 / 1.05 = $104.76. By selling for $95 I’m leaving $9.76 on the table, and someone will swoop in, borrow $95 from me, buy the same $110 contract in the market for $104.76, borrow the rest from the bank at 5%, and pocket the difference at no risk.

The Law of One Price formalizes this: two assets that produce identical cash flows must have identical prices. Otherwise risk-free profit exists, and someone will compete it away in seconds.

Arbitrage in practice

Pure textbook arbitrage is rare. But the threat of arbitrage is what disciplines prices. Three forms you’ll see in real markets:

1. Triangular arbitrage in foreign exchange. If 1 EUR = 1.10 USD and 1 USD = 110 JPY, then 1 EUR had better trade for about 121 JPY. Mismatches are exploited and closed in milliseconds.
2. ETF arbitrage. If a stock-index ETF trades at $99.50 but the underlying basket is worth $100.00, authorized participants simultaneously buy the ETF and sell the basket, profiting from the spread and pushing the prices back together.
3. Cash-and-carry. Buy a commodity today for $S$, store it, sell a futures contract at $F$. If $F > S \cdot (1+r) + \text{storage}$, you’ve locked in risk-free profit.

In each case the trader doesn’t need to forecast where prices are going: only that two prices for the same cash flows should match.

Pricing by replication

The Law of One Price gives us a powerful pricing tool: if we can build the cash flows of a target asset using other traded assets, the target asset must trade for the cost of that portfolio.

Example. A risky bond promises $100 in one year if the firm survives, $0 if it defaults. The market prices a one-year risk-free bond paying $100 at $95.24 (5% rate). And the market prices a one-year credit- default swap that pays $100 in default at $4 today.

Can we build the risky bond’s cash flows? Yes: long a risk-free bond ($95.24 in) + short a CDS ($4 out, since you collect the $4 premium and pay $100 in default) = a portfolio that pays $100 if no default and $0 if default. So the risky bond should cost $95.24 − $4 = $91.24.

If the risky bond is selling for $89, buy it and short the replicating portfolio. Risk-free profit of $2.24 per bond. If it’s selling for $93, do the reverse. Either way the price gets pulled toward $91.24.

What about risk?

The Law of One Price as stated assumes you can match cash flows state by state: when the firm defaults, when it doesn’t, in every scenario. For most cases you’ll see in this course, that’s fine; we build portfolios that match outcomes and price them by no-arbitrage.

When the cash flows can’t be exactly replicated (e.g. you can’t short a private firm’s equity), the principle still gives upper and lower bounds. Unit 4 will introduce a separate machinery (CAPM and factor models) for pricing the risk premium on assets whose cash flows can’t be perfectly hedged.

Why this is the organizing principle

Every corporate-finance decision (invest in a project, issue debt, buy back stock, acquire a competitor) can be framed as: “What’s the NPV of doing this, relative to the next-best alternative?” The Law of One Price is what makes the discount rate in that NPV calculation unambiguous: it’s whatever rate competitive markets are pricing similar-risk cash flows at.

Memorize the framework, then we’ll spend the rest of the course filling in the details.