## Wednesday, April 6, 2011

### Baskets of Eggs: Risk, Return and Diversification

Interactive tools can make really dry material, like portfolio theory, come alive. When I first started teaching, without such tools, I used to dread this part of the course. Now its fun, and here is what I do.

I start with historical stock data on a few stocks. If you don’t have access to current data, you can download it from financial web sites, like Yahoo finance. Step 1 is to show the students what recent returns have looked like, for stocks, for the index, and for some simple (e.g. equally weighted) portfolios using the Efficient Portfolio module. The following picture will give you an idea of what we discuss:

The example uses:
two years of weekly data on 30 large cap US stocks.  You can type in or copy/paste portfolio weights.  The histogram above is for the equally weighted portfolio.   We discuss how returns are measured, that volatility is commonly used as a measure of risk, and talk about skewness and fat tails.  The students use their own data and analyze stocks they want, I just explain the concepts.  The green line is a normal distribution for comparison and the JB statistic is the Jarque-Bera test for normality.  The fat tails discussion lets us talk about Value at Risk.  The discussion is mainly conceptual; for advanced students, you can go into formulas.  The problem sets on this part depend on the level at which you are teaching, ranging from graphical conclusions only to having students reproduce the calculations on their own in a spreadsheet.  Even at the basic level, I require some Excel calculations for one or two stocks.  More on this below.

The next step is diversification.  The “Portfolios” button lets you calculate frontiers and lots of other things.  I have circled the parts I commonly use.

The yellow “curve” is the efficient frontier with short sales permitted (you can turn off short sales).  The black dot corresponds to the portfolio weights shown.  If you click on the frontier, it calculates the corresponding weights.  The “Plot Securities” button shows you where each stock lies, and it is easy to illustrate that the frontier does a lot better than any individual stocks.  Again, you can require verification of the calculations for a subset of stocks.

The Backtest button tests historical performance; you can use a constant mix (i.e. the portfolio weights are always the same) or you can set the target return and recompute the weights every period.  The covariance matrix is recalculated and the performance is measured one-period ahead, out of sample.  An interesting comparison here is to look at constant mix portfolios, some that correspond to high returns and some that correspond to low returns.  It provides intuition on the volatility of high risk and low risk portfolios.

Having illustrated the basic ideas and how they worked historically, Step 2 is to take the model for a test drive.  This is done with a real time trading simulation.  In this simulation (typically done with 30 stocks, to keep the problem manageable), the students have to calculate an optimal portfolio in Excel, using Solver.  This has the additional advantage of teaching them a transferable skill.  They have to decide what constraints to impose, and then decide how to rebalance.  This is all going forward in time.   The FTS Real Time System has a set of analytics designed exactly around this.  You can find details in the Portfolio Diversification project, so I won’t go into the details here, except to show you the analytics available:

The first two calculate optimal portfolios based on the single index model, so students can see in real time how these perform relative to equally weighted portfolios as well as their own portfolio.  They can set their own expected returns or use CAPM. The covariances and returns allows them to copy the covariance matrix; again, they can specify their own if they want.  The portfolio tracking lets them see how far they are from a specified portfolio and calculates the required rebalancing.  The backtesting capability is the same as described above.

At the end of all of this exercise, the report produced is not on how much money you made but on the decisions you made: the portfolio you selected, the constraints you imposed, the rebalancing decisions, and of course the performance.  Some instructors have students manage two portfolios, a low risk and a high risk one.  The exercise is typically run for about as month.  The amount of work required by the students can be tailored; for example, if you ask them to buy and hold, there is very little work except at the beginning and end.  If you require a daily rebalancing, that is more work.  But about two weeks of class time with a month long exercise bring all this material to life in an intuitive, step-by-step fashion.

In future posts, I’ll describe similar exercises, one on options and one on equity valuation.