Analyzing Returns on a 10-Asset Portfolio

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I present closing prices on 10 companies in monthly, weekly, and daily frequencies. These are given on mthly_p, wkly_p, and dly_p, respectively, for:

I also give the benchmark, or market index, value on the S&P 500.

By closing, I mean the last price recorded for a specific time (for example, end of day closing price). I include a fourth sheet, mthly_r, on which I compute monthly returns using the monthly closing prices. Notice that I compute the monthly return as the natural logarithm of the ratio of the current over the previous month's closing price, . Recall that with continuous compounding, . Therefore, , which is equal to . You can do it this way or, if you prefer, you can calculate month-over-month percentage changes, that is, . Monthly data range from March 1992 to December 2006. I have highlighted the first five years of monthly prices and that is where we begin our analysis.

In rows 180 and 181, I compute the average monthly returns and the volatilities (standard deviations). Then, I annualize them in rows 183 and 184. To annualize a monthly average, multiply it by 12. The logic is that if you earn 1 percent per month, then you earn 12 percent per year using an arithmetic averaging. Alternatively, if you geometrically link a 1 percent monthly return over the year, it grows to , which is equal to return for the year. I do the simpler arithmetic average. To annualize a variance, multiply it, too, by 12. To annualize a standard deviation, multiply it by . The logic here is that if returns are independent (that's a strong assumption, by the way), then over the year we have . The variance of this sum is the sum of their individual variances plus their covariances. But if returns are independent, then the covariances are zero. If returns are identically distributed, then for all i and thus, the annual variance is simply 12 times the monthly variance. The annualized standard deviation is therefore the square root.

If you check the formulas, you will see that I named the returns matrix (this is the matrix of highlighted monthly returns) R_97, and I also name the monthly mean returns μ_m. The variance-covariance matrix V is therefore:

Note that I leave the benchmark out. Our objective is to try to find a minimum variance portfolio of the 10 stocks. We intend to compare the performance of this portfolio to the benchmark but we do not hold the benchmark in that portfolio.

Upon inspection, we can see that there appear to be some attractive diversification opportunities here—Ford, for example, is negatively correlated with HD and INTC as well as TWX. It will be interesting to see how our portfolio loads on these securities. To that end, let's solve the minimum variance portfolio first. Recall that this portfolio finds the vector w that minimizes risk for a fully invested portfolio . From the development of the Lagrangian in Chapter 6, I add a row of ones and column of minus ones (recall our first order conditions—we don't have to repeat the math here) to V (the matrix we referred to as A). These are highlighted in gray. I now name the extended covariance matrix A and I add in the vector b as we did in Chapter 6. These results are presented in the spreadsheet (click on the Clipboard at the top left of the spreadsheet to review these names). The optimal portfolio is the solution .