Deriving the Risk-Free Rate from State-Contingent Claims

Summary

The discussion explains why the risk-free rate can be expressed as the reciprocal of the summed prices of state-contingent claims. Each claim pays one unit of currency in one possible state and nothing in the others. Buying a claim for every state therefore creates a portfolio that pays one unit in the next period regardless of which state occurs.

That portfolio is economically equivalent to a risk-free security. If its current price is below one, the ratio of its guaranteed future payoff to its price gives the gross risk-free return; subtracting one gives the net rate. The answer illustrates the idea with a portfolio price of 0.9 and a gross return of 1.11. This intuition assumes the claims span all relevant states and that their payoffs and prices are defined consistently. The brief exchange does not address complications such as trading frictions or incomplete markets.

Key ideas

• A state-contingent claim pays in a particular state of nature.
• Holding one claim for every possible state guarantees a unit payoff next period.
• The price of this complete set of claims is the price of a risk-free payoff.
• The gross risk-free return is the guaranteed payoff divided by the portfolio’s current price.

Tags

# Why is the Risk Free Rate 1 over Contingent Claim Prices?


# Why is the Risk Free Rate 1 over Contingent Claim Prices?












Reading Asset Pricing by John Cochrane (2005), in his second chapter he defines the risk free rate as:

Rf = 1 / sum [pc(s)]

Where pc(s) are state contingent claims, where s is the state of nature realised from a possible set S.

This is quite an elementary question I'm sure, but I just cannot grasp why this is the case. If anyone could enlighten me intuitively that would be most appreciated!

## Answer by nbbo2 (score 2)

https://quant.stackexchange.com/a/25710

By buying all the state contingent claims you ensure that you will receive 1 USD in the next period (since one of the states will occur), and that is the definition of a risk free security: something that is guaranteed to pay off 1. The price at which it sells today is lower than 1, and that discounting defines the Risk Free rate. If the risk free basket of claims sells for 0.9 then the interest rate is 1/0.9 = 1.11 or 11% in net terms.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.