CHAPTER 21Exercises

EXERCISE 1 PORTFOLIO CHOICE AND THE NOTION OF VALUE AT RISK (VAR)1

The objective of this exercise is to clarify important points about VaR for market risk when choosing a portfolio. The same message applies for CVaR, but the formulas differ, as we will see in Exercise 5.

Calculating VaR

Suppose an individual has an expected utility function such that:

equation

where:

  • images is the expected portfolio return;
  • images is the portfolio variance;
  • r is the risk aversion coefficient.

Answer the following questions by assuming that there are only three risky assets in the market and that the investor's portfolio may contain these three assets.

Question 1

What conditions must U and r verify to define a utility function of a risk-averse investor?

  1. Write the problem of maximization of the investor's utility in matrix form, given that the sum of the weights associated with the three risky assets is equal to 1.
  2. Find the first-order conditions of this portfolio choice problem.
  3. Give the analytical expression for the weight of an optimal portfolio.
  4. Suppose that the expected returns of the three risky assets are respectively:

The matrix of the correlation coefficients is given by:

The variances ...

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