Each red dot represents the mean and return or risk and with consistently positive standard deviation of a portfolio. The blue expected returns relative to the market. line is the efficient frontier. portfolios or the efficient frontier have maximum re Backtesting turn for a given level of risk or, alterna- Now that we have identified a set of portfolios level of that are both efficient and stable, we can per eturn. Clearly, a rational investor will se- form an ex post analysis, examining turnover, lect a portfolio on the efficient frontier. drawdown, and realized average return to see The portopt function in Financial Tool how these portfolios actually performed. box lets us determine directly which Turnover portfolios of assets lie along the efficient Turnover refers to the change in portfolio frontier given the means and covariances holdings over time due to trading A port- of individual asset returns folio with annual turnover of 25% will re- Finding a Stable Region place a quarter of its assets over a one-year Figure 3. Average furnover for portolio sequences in the Because theefficient frontiershifts over time period. Since trading is costly, low turnover stable region is around 25% a once-efficient portfolio may be not be is a desirable feature of a portfolio strategy the efficient frontier in subsequent time pe- Figure 3 shows the annual turnover for the riods. In addition, it is not clear which port- portfolio sequences on our efficient fron folio to select on the efficient frontier tiers, with the blue line re. One solution is to study the time evolution of sults of the analysis after removing market efficient frontiers and identify a sequence of returns. Note that in the stable region, with portfolios that remain relatively stable from the first eight portfolio sequences, the an- I one efficient frontier to the next. We can use nual turnover remains at 259 IATLAB to visualize this stable region. Drawdown Figure 2 shows efficient frontiers plotted as a function of time. matlab has calculated Evaluating the maximum drawdown of a efficient frontiers with 40 portfolios on each por portfolio is a good way to measure ex post risk maximum drawdown refers to the amount a frontier at one-month intervals and plotted the results. Figure 2 underscores the value portfolio declines in value relative to its peak of taking the market out of the data: We can value. It represents the worst possible per mance over any time period. Figure 4. Drawdown in the stable region is the same identify sequences of portfolios-those in the deep blue region-with little variation of In Figure 4, the green line represents the maximum drawdown of the DJIa over our back-test period-roughly 20%. The flat part of the blue One of the simplest performance measure- line represents the maximum ments is to determine a portfolios average drawdown for the portfolio mean and standard deviation of returns. sequences through the stable We have already determined that the port region and closely mirrors the folio sequences in the stable region have maximum drawdown of the reasonable levels of risk compared to the DJIA. Since our goal is to as- benchmark. But do those same portfolios semble portfolios with risk and deliver superior returns? return characteristics similar We plot the average of ex post returns versus the Dow Jones Average, risk of a portfolio or index In Figure 5,the this is a good result-it shows red star represents the return and risk of the that the risk of these portfolio DJIA benchmark over our backtest period. Each blue circle corresponds to a portfolio Figure 2. Efficient frontiers at one-month intervals with marker retums removed our benchmark. quence. The circles closest to ReprintedfromTheMathworksNews&notesIOctober2006iwww.mathworks.corReprinted from T heMathWorksNews&Notes | October 2006 | www.mathworks.com Each red dot represents the mean and standard deviation of a portfolio. The blue line is the efficient frontier. Portfolios on the efficient frontier have maximum re￾turn for a given level of risk or, alterna￾tively, minimum risk for a given level of return. Clearly, a rational investor will se￾lect a portfolio on the efficient frontier. The portopt function in Financial Tool￾box lets us determine directly which portfolios of assets lie along the efficient frontier given the means and covariances of individual asset returns. Finding a Stable Region Because the efficient frontier shifts over time, a once-efficient portfolio may be not be on the efficient frontier in subsequent time pe￾riods. In addition, it is not clear which port￾folio to select on the efficient frontier. One solution is to study the time evo­lution of efficient frontiers and identify a sequence of portfolios that remain relatively stable from one efficient frontier to the next. We can use MATLAB to visualize this stable region. Figure 2 shows efficient frontiers plotted as a function of time. MATLAB has calculated efficient frontiers with 40 portfolios on each frontier at one-month intervals and plotted the results. Figure 2 underscores the value of taking the market out of the data: We can identify sequences of portfolios—those in the deep blue region­—with little variation of return or risk and with consistently positive expected returns relative to the market. Backtesting Now that we have identified a set of portfolios that are both efficient and stable, we can per￾form an ex post analysis, examining turnover, drawdown, and realized average return to see how these portfolios actually performed. Turnover Turnover refers to the change in portfolio holdings over time due to trading. A port￾folio with annual turnover of 25% will re￾place a quarter of its assets over a one-year period. Since trading is costly, low turnover is a desirable feature of a portfolio strategy. Figure 3 shows the annual turnover for the portfolio sequences on our efficient fron￾tiers, with the blue line representing the re￾sults of the analysis after removing market returns. Note that in the stable region, with the first eight portfolio sequences, the an￾nual turnover remains at 25% or less. Drawdown Evaluating the maximum drawdown of a portfolio is a good way to measure ex post risk. Maximum drawdown refers to the amount a portfolio declines in value relative to its peak value. It represents the worst possible perfor￾mance over any time period. In Figure 4, the green line represents the maximum drawdown of the DJIA over our back-test period—roughly 20%. The flat part of the blue line represents the maximum drawdown for the portfolio sequences through the stable region and closely mirrors the maximum drawdown of the DJIA. Since our goal is to as￾semble portfolios with risk and return characteristics similar to the Dow Jones Average, this is a good result—it shows that the risk of these portfolio sequences is comparable to our benchmark. Figure 2. Efficient frontiers at one-month intervals with market returns removed. Figure 3. Average turnover for portfolio sequences in the stable region is around 25%. Figure 4. Drawdown in the stable region is the same as the DJIA. Average Return One of the simplest performance measure￾ments is to determine a portfolio’s average mean and standard deviation of returns. We have already determined that the port￾folio sequences in the stable region have reasonable levels of risk compared to the benchmark. But do those same portfolios deliver superior returns? We plot the average of ex post returns versus risk of a portfolio or index. In Figure 5, the red star represents the return and risk of the DJIA benchmark over our backtest period. Each blue circle corresponds to a portfolio sequence. The circles closest to