Why the Market Price of Risk Can Be Positive or Negative

Summary

This note explains that the market price of risk does not have to be positive. Its sign reflects how investors value exposure to a risk, in particular the relationship between that risk and the investor’s marginal utility of consumption. Investors may accept a negative average return for exposure that pays off in states when their need for wealth is high.

The answer contrasts stochastic volatility and correlation risk, described as negatively priced, with equity market risk, described as positively priced. Equity exposure tends to be more valuable in bearish states, when investors especially value returns, so investors require average compensation to bear it. The discussion is conceptual and offers no empirical estimates or formal model; the signs are presented as economic intuition rather than a universal rule for every market, investor, or specification of risk.

Key ideas

• The price of risk can be positive or negative depending on how the risk relates to investors’ marginal utility.
• A risk that pays off when investors especially value wealth may command a negative average return.
• The answer characterizes stochastic volatility and correlation risk as negatively priced.
• It characterizes equity market risk as positively priced because exposure is less valuable in bearish states.

Tags

# The positivity of the market price of risk


# The positivity of the market price of risk












Does the market price of risk, be it of stochastic volatility, interest rate or equity return, have to be positive? What is the rationale if it does?

## Answer by Igor Pozdeev (score 6, accepted)

https://quant.stackexchange.com/a/40999

No, it can be negative. The price of risk is what you agree to receive on average in exchange for positive returns when the risk measure is high, and determined by the covariance of the risk measure with your marginal utility of consumption. That said, stochastic volatility risk is negatively priced: you happily agree to a negative return on average in exchange for positive compensation in times of higher variance, because it is these times when your marginal utility is high. Correlation risk is negatively priced. Equity market risk is positively priced: when the stock market is bullish, you tend not to need positive returns as much as you need them when the market is bearish, so you require a positive compensation on average to be exposed to this risk.

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