# Risk Neutrality

# Risk Neutrality

*Risk neutrality* is an economic term that describes individuals’ indifference between various levels of risk. When confronted with a choice among different investment opportunities, risk-neutral decision makers only take into account the expected value of the alternative and not the associated level of risk. For example, a risk-neutral investor will be indifferent between receiving $100 for sure, or playing a lottery that gives her a 50 percent chance of winning $200 and a 50 percent chance of getting nothing. Both alternatives have the same expected value; the lottery, however, is riskier.

In economic theory it is generally accepted that most individuals are not risk-neutral. People tend to prefer safer choices to riskier ones, meaning they are risk-averse. In the above example, a risk-averse person would prefer getting the expected value of the lottery for sure rather than choose the gamble. In other words, a risk-averse person would require a premium above the expected value in order to play this lottery. Risk-neutral individuals would neither pay nor require a payment for the risk incurred.

In terms of utility theory, a risk-neutral individual’s utility of expected wealth from a lottery is always equal to his or her expected utility of wealth provided by the same lottery. This implies a linear utility function, or relationship between the wealth and the utility of wealth. A simple example of such utility function is *U* (*W* ) = *W*, where *W* is wealth and *U* (*W* ) is utility of wealth. In this case, a one-dollar increase in wealth will always result in the same increase in utility for any starting wealth level.

In finance, modern portfolio theory states that an investor needs to be compensated for the risk that she takes by receiving extra expected return. Such compensation is referred to as risk premium. The fact that historical returns for riskier asset classes, such as equities, consistently outperform returns for less risky assets (such as Treasury bills) confirms the existence of a considerable risk premium. For example, John Campbell, Andrew Lo, and A. Craig MacKinley reported in 1997 that in the period from 1889 to 1994 U.S. stock market returns were on average 4.18 percent higher than the returns on commercial paper. Risk neutrality, on the contrary, implies that investors do not require any risk premium since they are indifferent to the level of risk (p. 308). Therefore, in a world consisting of risk-neutral investors the difference between stock and commercial paper returns would be expected to be zero.

While risk neutrality is uncommon in the real world, the concept of risk neutrality plays an important role in option pricing. Risk-neutral valuation of options was first introduced by John Cox and Stephen Ross in 1976, and further developed by Cox, Ross, and Mark Rubinstein in 1979. This method is based on an observation that prices of derivatives are the same in the real as in the risk-neutral world. The risk preferences of individuals and the shape of their utility function do not affect the option prices as long as the agents prefer more wealth to less (the so-called nonsatiation condition). The fact that option prices are not affected by the risk preferences and utility functions of individuals allows us to price the options by calculating their expected values in the risk-neutral world and to use the risk-free discount rate to find their present value. The risk-neutral valuation principle is widely used in practice to calculate values of different financial derivatives.

**SEE ALSO** *Risk; Risk Takers; Risk-Return Tradeoff*

## BIBLIOGRAPHY

Campbell, John Y., Andrew W. Lo, and A. Craig MacKinlay. 1997. *The Econometrics of Financial Markets*. Princeton, NJ: Princeton University Press.

Cox, John C. and Stephen A. Ross. 1976. The Valuation of Options for Alternative Stochastic Processes. *Journal of Financial Economics* 3: 145–166.

Cox, John C., Stephen A. Ross, and Mark Rubinstein. 1979. Option Pricing: A Simplified Approach. *Journal of Financial Economics* 7: 229–263.

*Yulia Veld-Merkoulova*

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