Investment Theory and Risk Management, + Website

Changing the Investment Horizon Returns Frequency

Investors have different investment horizons for a variety of reasons—liquidity needs and risk management, to name a few. I have included both weekly and daily closing prices for these 10 stocks and the S&P benchmark so that we could study the frequency implications to portfolio construction. Sticking with the five-year period ending in February 1997, I have constructed on wkly_r.xlsx in Chapter 7 Examples.xlsx, a time series of weekly returns. These are highlighted and their corresponding weekly means and standard deviations are given in rows 261 and 262. The covariance matrix is denoted by V_w and I have constructed two new portfolios that are the weekly analogs to the portfolios found on mthly_r.xlsx.

The first question that comes to mind is how changing the frequency alters the sample statistics like means and variances, and therefore the portfolio. If you compound the weekly mean returns in row 261, for example, to get monthly returns, and compare these to those found on mthly_r.xlsx, you will see that they are very close. (1 + rw)4 – 1 should be close to the monthly mean return rm. Or, if you prefer, multiply rw by 4 to get the arithmetic monthly equivalent. Likewise, σw√4 should be close to σm. The major point of interest, however, is in comparing optimal portfolios. I solve for both the minimum variance portfolio as well as a targeted return portfolio, in which the weekly targeted return compounds to the monthly targeted ...