Markowitz paradigm

"The investor does (or should) consider expected return a desirable thing and variance of return an undesirable thing" (Harry Markowitz - 1952)

Markowitz formulated the investor choices' as a problem of maximizing their utilities instead of simply maximizing their wealth. In investment, the same problem appears: investors must find their optimal choice (understand here "allocation") between the risks and the returns.

The expected return of a portfolio of assets is equal to the weighted expected returns of each asset in the portfolio.

1. Suppose an equal weighted portfolio between assets A, B and C. The expected returns of each asset are respectively 5%, 7% and 3%. What is the expected return of the portfolio?

2. Let's assume you have a column vector ? of dimension 3x1 with each element being the weight of each asset, and another column vector ? of same dimension. Compute ?^? ?. Compare the result with 1 and deduce the expected value of a portfolio in matrix form.

3. Write python function "portfolio_return" that takes as input two arrays (an array of weights and an array of expected returns) and that returns the expected return of the portfolio.

The variance is a mathematical quantity defining the spread around the mean. In our case, it represents how much your investment could deviate from your expected value (mean). In other words and as implicitly stated by Markowitz, the variance reflects the risks you are taking having a given portfolio.

We call the variance of a portfolio the sum of the product of weights jQuery2240001496395712750287_1620464397151 of each pair of assets by their

covariance jQuery22407101482584775022_1620464527491,?. Mathematically, we write the following:

?? ??_p= (?_i?_j?_i,j )

? ?=?. ?=?

4. Suppose an equal-weighted portfolio of two assets A and B, with variance 0.0225 and 0.04 respectively and covariance 0.0045. What is the variance of the portfolio?

5. Suppose we have a third asset C with variance 0.0025 and covariance with A of -0.0015 and covariance with B of 0.0075. What is the variance of the portfolio?

6. Compute the following matrix product:

?^??? with ? being the covariance matrix of the three assets above. Compare your result with 5 and conclude.

7. a python function that takes as input all the necessary parameters and returns the variance of a portfolio

Note: saying that the variance measures accurately the risk of our portfolio implicitly gives credibility to the notion of mean. In fact, the variance is just the symmetric spread around the mean... so you are assuming that you are in a Gaussian world.

Goiing back to the concept of utility, Markowitz expressed mathematically the problem as follows:

???????= ? (1/2)?²

Where:

- ? is the expected return of the investment

- ?² is the variance associated to this investment

- is a parameter reflecting the risk aversion of the investor

Let's assume that an investor has the above utility function and is looking to maximize it.

8.Explain the rational of the utility function above.

9.What would is the impact of having a very high risk aversion parameter? A very low risk aversion parameter?

10.What would be the utility function of an investor neutral to risk?

11. If the above utility were to be applied to a portfolio of assets, what would be the vector dimensions of each parameter in the formula? Re-write the utility formula using the matrix notations used in question 3 and 4.

12python function that takes the ap propriate parameters and return the utility.