Risk Preference, Utility Function Convexity, and Jensen’s Inequality

Risk Preference, Utility Function Convexity, and Jensen’s Inequality

Risk Preference and the Utility Function

In expected utility theory, an individual’s preferences over uncertain outcomes are represented by a non-decreasing utility function $u(x)$, where $x$ typically represents wealth. An important concept here is marginal utility, which is the additional utility obtained from a small increase in $x$. Formally, marginal utility is defined as the derivative $u’ (x)$. It measures how much utility increases when wealth increases by one unit. For instance, if $u’(x)$ is high, an extra dollar provides a significant increase in utility; if it is low, the extra dollar contributes only a little additional utility.

  • Risk-averse individuals prefer a certain outcome to a risky one with the same expected value. Their utility functions are concave, which also implies decreasing marginal utility. This means that as wealth increases, the additional satisfaction gained from each extra unit of wealth decreases.
  • Risk-neutral individuals are indifferent between certain and uncertain outcomes with the same expected value. Their utility functions are linear, implying constant marginal utility.
  • Risk-seeking individuals prefer risky outcomes over certain ones with the same expected value. Their utility functions are convex, meaning that the marginal utility is increasing.

For a risk-averse person, the concavity of the utility function implies decreasing marginal utility. In other words, for any increment $\Delta > 0$:

\[u(x+\Delta) - u(x) < u(x) - u(x-\Delta)\]

This inequality captures the intuition that the additional satisfaction (or utility) from gaining an extra $\Delta$ dollars is less than the loss in satisfaction from losing $\Delta$ dollars.

Relation with Jensen’s Inequality

For a concave function $u$ (which represents risk aversion) and a random variable $X$ representing uncertain wealth, Jensen’s inequality states:

\[u\big(\mathbb{E}[X]\big) \geq \mathbb{E}[u(X)]\]

Interpretation:

  • The utility of the expected wealth, $u\big(\mathbb{E}[X]\big)$, is greater than or equal to the expected utility of the wealth $\mathbb{E}[u(X)]$.
  • This inequality explains risk aversion: even though the risky prospect $X$ and the certain amount $\mathbb{E}[X]$ have the same expected monetary value, the risk-averse individual prefers the certainty of $\mathbb{E}[X]$.

Risk Premium

The certainty equivalent $C$ of a random wealth $X$ is defined as the amount of wealth that, if received with certainty, provides the same level of utility as the uncertain prospect:

\[u(C) = \mathbb{E}[u(X)]\]

The risk premium $\pi$ is the extra expected return that a risk-averse individual requires to be indifferent between a risky prospect and a certain outcome. It is defined as the difference between the expected value of the risky prospect and its certainty equivalent:

\[\pi = \mathbb{E}[X] - C\]

Expanded Discussion on the Risk Premium

For a risk-averse investor, accepting a risky asset means facing uncertainty about the outcome. To compensate for this uncertainty, the investor demands a risk premium—an additional expected return over the certainty equivalent.

Key points regarding the risk premium include:

  1. Compensation for Risk:
    • The risk premium represents the compensation required for bearing risk. A higher risk premium indicates that the investor requires more additional expected return to be willing to accept the uncertainty associated with the risky asset.
  2. Relationship with Utility Curvature:
    • The magnitude of the risk premium is related to the curvature (concavity) of the utility function. More risk-averse individuals (with a higher degree of concavity) will have a larger risk premium.
    • The Arrow-Pratt measure of absolute risk aversion, defined as $-\frac {u’‘(x)}{u’(x)}$, quantitatively links the curvature of the utility function to the risk premium. A larger value indicates greater risk aversion and typically a higher risk premium.
  3. Economic Interpretation:
    • In a financial context, the risk premium can be viewed as the excess return that an investor expects to earn from holding a risky asset rather than a risk-free asset. This concept is central to asset pricing models and helps explain phenomena such as the equity premium.
  4. Certainty Equivalence:
    • The certainty equivalent $C$ is always less than the expected value $\mathbb{E}[X]$ for a risk-averse individual. The difference $\pi = \mathbb{E}[X] - C$ quantifies the investor’s dislike for risk.
    • If the investor were risk-neutral, the certainty equivalent would equal the expected value (i.e., $C = \mathbb{E}[X]$) and the risk premium would be zero.