On Some Practical Issues when Using AlgoQuant to Compute the Markowitz Efficient Frontier

Markowitz suggests in his Nobel Prize-winning paper Markowitz(1952) that when one selects a portfolio, he/she should consider both the return and the risk of the portfolio. Most of us, if not all, are risk-averse. Risk-averse means that if there are two  portfolios with the same return, but different risks (in this article by risk we mean the standard deviation of the portfolio), we would choose the one with the smaller risk without any hesitation. Therefore given a set of risky assets and an expected return, we are interested in finding their best combination, i.e. the weights which will minimize the risk of the portfolio. And if we find the minimum risk of the portfolio for any return, we can draw a curve on risk-return plane. This curve is the famous efficient frontier. Assuming there are  risky assets, and their return vector and covariance matrix are  and  respectively, then the points on the efficient frontier are computed by solving the following problem: where is the pre-defined expected return, and is the weight vector. The above problem can be solved using Lagrange multipliers. And we denote this problem as “Problem 1”. In AlgoQuant, we use another approach to compute the efficient frontier. The problem we solve is based on the utility function: , ,  and  are the same parameters in Problem 1. The newly added parameter , is risk-averse coefficient. And this problem is denoted as “Problem 2”. The larger the , the less risk the investor is willing to take. Although most of us are risk-averse, the degrees of risk-averse are different among individuals. As a result, a coefficient that describes the risk-averse degree is introduced. Note that in some papers, risk...