Financial Economics — Risk Sharing, Arbitrage and State Prices

Financial Economics — What Prices Do Is Share Risk

If corporate finance asks about one firm’s investment and capital structure, financial economics asks who bears risk across the whole economy. Share prices and interest rates are, before they are calculator outputs for “what is fair”, relative prices that decide who gives up consumption in which state.

NPV and WACC calculations are covered in the corporate finance course. This course looks at the equilibrium and no-arbitrage constraints behind those formulas.

1. State-contingent claims have the same structure as weather insurance

Suppose next year the economy is either in a boom or a recession. A security that pays 1 won in the boom and 0 won in the recession is called an Arrow–Debreu security. Real shares, bonds and options are bundles of such securities.

If the boom security costs 0.40 won and the recession security 0.50 won, a risk-free 1 won costs 0.90 won. The risk-free interest rate is about 11.1%.

The no-arbitrage price of an asset paying 120 won in the boom and 80 won in the recession is $120×0.40 + 80×0.50 = 88$ won. If it trades at 90 won in the market, selling it and buying the replicating portfolio leaves an arbitrage profit.

2. In complete markets, every risk can be traded

A market is complete when there are as many linearly independent assets as independent states. In a complete market, any desired consumption plan can be built from a combination of securities, and identical state-by-state cash flows must have the same price.

In an incomplete market this fails. Risks with no hedging instrument — labour income risk, house prices, disasters — are not fully reflected in prices and stay with individuals. “A market has opened” and “risk has been transferred” are different statements.

3. No arbitrage is an accounting constraint, not a preference

An arbitrage opportunity is a portfolio with zero net investment today that yields a non-negative payoff in every state and a positive payoff in some state. No assumption about preferences or risk aversion is needed. If two assets pay the same cash flows in every state but have different prices, buying the cheap one and selling the dear one is enough.

Preferences enter after arbitrage has gone. The price of the remaining risk — which states get higher state prices — depends on marginal utility and risk sharing.

Check your understanding

1. If the boom and recession state prices are 0.45 and 0.50, what is the risk-free interest rate?
2. If an asset paying 150 won in the boom and 50 won in the recession trades at 80 won, which way does the arbitrage run?

Building a risk-free 1 won from the boom and recession securities costs 0.95 won, so $r_f ≈ 5.3\%$. The replication price is $150×0.45 + 50×0.50 = 92.5$ won, so the market price of 80 won is too low. Buy the asset and sell the replicating portfolio.

References

• John Cochrane, Asset Pricing, ch. 1–3
• MIT OpenCourseWare, 15.401 Finance Theory
• Kenneth Arrow, “The Role of Securities in the Optimal Allocation of Risk-Bearing”