Question

1. Suppose now you can allocate your money in the n risky stocks and in the risk-free asset (so there n + 1 assets). What is the optimization problem that you need to solve? What is the constraint for this problem?

2. What is the shape of the frontier when the investor has access to the risk-free asset? Why are two portfolios on the capital allocation line perfectly correlated?

3. Usually, when there are n risky stocks and one risky free asset we draw the tangent to the efficient frontier and the tangency point is called the optimal portfolio. How do we find the composition of this portfolio?

4. Consider the investing possibilities as in (3). Give the expression that computes the weight of the kth asset of the tangency portfolio.

Answer 1

Question 1) A Rational investor always wishes to maximise his return from a portfolio for taking a certain level of risk. Hence, he allocates his funds between a risky asset like stocks and risk free assets like treasury bills. Risk loving investors are only concerned with maximising their returns from a portfolio, risk neutral investors are somewhat conerned for the level of risk taken for earning a particular return, while a risk averse investor requires a higher return for a given level of risk taken. Since investors differ in terms of their risk-return tradeoffs, it is imperative to assign an investment utility score to each investor which reflects the risk-return relationship.

Assuming that I am a risk averse rational investor who wishes to maximise returns but at the same time take a reasonable level of risk

Example of an utility function which should be maximised:

Utility score= expected return- (0.005 (variance)*risk aversion coefficient)

risk aversion coefficient is a number proportionate to the amount of risk aversion of the investor and is usually set to integer values less than 6, and 0.005 is a normalizing factor to reduce the size of the variance which is square od standard deviation, i.e, a measure of volatality.

Constraint for this problem would be the amount of funds at my disposal that can be invested.

Note: Alternatively, we could also maximise 'Return from the portfolio' and consider 'Risk from the portfolio' a constraint. Since we have taken both risk and return into account with the Utility function, hence we have treated the 'amount of funds' at disposal as the constraint.