Utility Formula

By introducing the concept of risk and efficient frontier, Markowitz inaugurated a sort of modern securities portfolio management.

In a portfolio of securities whose returns and volatility are known, and in which the price movement of one does not completely influence that of the other, it is possible to calculate the weight of each security in order to achieve maximum performance while taking the minimum risk. From an objective and rational perspective, investors should always choose the optimal portfolio with the minimum risk relative to the chosen level of returns.

But what is the level of risk the investor is able to endure? The answer is subjective. This is how the utility formula and risk aversion coefficient introduce to portfolio management.

• Risk Aversion Coefficient

A quantitative and practical method will be used to measure the risk aversion of an investor. The risk aversion coefficient can be considered as what an investor is willing to sacrifice expected return in order to lower variance by one unit. A higher risk aversion coefficient means investor are more willingly to pass up the opportunity for a large gain in favor of safety. Since it is a trade-off between risk and return, the risk aversion coefficient has to be positive for the portfolio optimization.

• Mean-Variance(quadratic) Utility Formula:

\(U = \mathop{\mathbb{E}[R]} – 0.5 \ast A\ast {\sigma}^2 \)

In this formula, U represents the utility or score to give this investment in a given portfolio by comparing it to a risk-free investment.
\(\mathop{\mathbb{E}[R]} \) is the expected return of the portfolio and \({\sigma}^2 \) is the square of volatility.
The part of the equation to the right of the “minus” sign indicates the risk of the strategy itself, taking into account the investor’s risk aversion. The formula as a whole therefore gives us the difference between the total expected return of a portfolio and the risk involved.