Question

1. If I were to provide you with 50 years of monthly returns on the S&P500 and on the Barclays Aggregate Bond Index, how would you go about calculating the efficient frontier of investment opportunities?  By that I mean, determine
   1. What variables you need to calculate from the return data?
   2. What optimization problem would you need to solve, i.e., what are you going to maximize or minimize?
   3. What restrictions would you need to place on the optimization problem?
   4. How would you determine the starting point of the efficient frontier?

If I asked you to determine the optimal portfolio for a client, what information would you need?  Once that information is provided, what optimization problem would you solve to identify the optimal portfolio (what are you going to maximize or minimize)?  What constraints would you place on this problem?

Answer 1

a) The main variables needed for an efficient frontier are Risk (standard deviation of returns) and return of the portfolio. The efficient frontier is a set of optimal portfolios which have highest returns for a given level of risk, or lowest risk for a given level of returns. With Returns on Y axis and Risk on X axis.

b) You will Maximize returns for a given level of risk. Or minimize Risk for a given level of return. So if you take the graph, along the x-axis - Standard Deviation (Risk) you will plot all portfolios, the ones with the highest returns will form the efficient frontier. Another way to go about this is, you can plot the various portfolios which provide the lowest risk for a given level of returns. Combining these two, you should get your efficient frontier.

c) Constrains or restrictions would be -
1) Transactions costs are tricky to account for.
2) Optimization is hard.
3) Returns have to be normally distributed for mean-variance optimization.
4) May not be able to account for short sales as retail investor, thus, there should be a restriction of portfolio weights should be positive.

d) The starting point of the efficient frontier (left side of the graph) will be the portfolio with the lowest variance (risk). This is called the minimum variance portfolio.