1.4 Advanced Risk and Risk-Adjusted Return Measures
1.4.2 Maximum Relative Drawdown
be given with stating the observed historical time period and the considered time subperiods. Especially, when calculating theMADDfor two different portfolios, it is mandatory to apply theMADDcalculation to the same historical time period and the same subintervals. Otherwise the two calculatedMADDs are not comparable.
1.4.2.1 Interpretation
We have dealt with MADD by looking at the drawdown of a portfolio alone.
However, if we analyze an active portfolio managed against a benchmark, this absolute drawdown measure is irrelevant. What has to be analyzed is its relative counterpart MRDD. This measure provides information about the active asset manager’s consecutive negative alphas and allows to analyze the asset manager’s ability to regain positive alphas.
Example 14
Figure1.31shows the monthly excess returns of a portfolio versus its benchmark between July 2012 and December 2013. The data used are identical to Exam- ple 13 forMADD displayed in Table1.22. However, we have now added the benchmark using data provided in Example 12, see Table1.23. For the observed time periodŒ0; T the value ofT is1:5years withN D 18representing the18 subintervals (here: months).
In Fig.1.31we can identify four drawdown periods with consecutive negative alphas. Such a relative drawdown period ends when it is followed by a subperiod (in our case a month) with a positive alpha. This especially means that a drawdown period with consecutive negative alphas can also be one single subperiod only. It also shows that the choice of N, i.e., the choice of the subperiods (daily, weekly, monthly, etc.) plays a critical role in the MRDD calculation. Drawdown periods in our example are:
• Drawdown period 1: November and December 2012, 2 months.
• Drawdown period 2: February 2013, 1 month.
• Drawdown period 3: May and June 2013, 2 months.
• Drawdown period 4: August–November 2013, 4 months.
The stated length in the list above is the respectivedrawdown duration. After having determined the drawdown periods with the consecutive negative alphas we have to calculate the magnitude of the negative returns for each of the four drawdown periods. This is shown graphically in Fig.1.32.
Using the idea of Eq. (1.4) from page8we get:
• Drawdown period 1: cumulative relative return of1:36%.
• Drawdown period 2: cumulative relative return of0:28%.
• Drawdown period 3: cumulative relative return of0:69%.
• Drawdown period 4: cumulative relative return of1:93%.
Therefore, the worst relative drawdown occurred in period 4 withMRDD D 1:93% spanning the 4 months from August to November 2013. The calcula- tions of the relative drawdowns are shown in Table1.23on the next page. The necessary formulas for the cellsD2toH 2are:
Table1.23Example14:CalculationofthemaximumrelativedrawdownMRDD ABCDEFGH MonthlyMonthlyRelativeCumulativeportfolioCumulativebenchmarkAlphaofthe portfoliobenchmarkMonthlydrawdownreturninthereturninthecumulatedreturnsin 1Monthperformanceperformancealpha(yes/no)?drawdownperioddrawdownperiodthedrawdownperiod 207/20126.10%6.01%0.09%No 308/20125.50%5.45%0.05%No 409/20124.70%4.63%0.07%No 510/20125.00%6.99%1.99%No 611/20125.10%4.16%0.94%Yes5.10%4.16%0.94% 712/20126.70%7.07%0.37%Yes1.26%2.62%1.36% 801/20136.03%5.97%0.06%No 902/20133.23%2.95%0.28%Yes3.23%2.95%0.28% 1003/20135.12%4.66%0.46%No 1104/20135.21%4.91%0.30%No 1205/20134.10%4.01%0.09%Yes4.10%4.01%0.09% 1306/20134.50%3.87%0.63%Yes8.42%7.72%0.69% 1407/20131.75%2.95%4.70%No 1508/20133.71%4.52%0.81%Yes3.71%4.52%0.81% 1609/20134.20%3.93%0.27%Yes0.65%0.41%1.06% 1710/20134.26%4.99%0.73%Yes3.59%5.42%1.84% 1811/20134.00%3.84%0.16%Yes0.56%1.37%1.93% 1912/20135.10%4.99%0.11%No 20MRDD=-1.93% Source:Own,forillustrativepurposesonly
Fig. 1.32 Graph of the cumulative relative returns in Example 14 with drawdown periods and recovery periods.Source: Own, for illustrative purposes only
• Monthly portfolio alpha˛monthlyin cellD6: 0:94% D B6C 6
• Occurrence of a relative drawdown (i.e., negative monthly alpha) in cellE6:
yes D IF.D6 < 0;”yes”, ”no”)
• Cumulative portfolio returnrPfcumin the drawdown period in cellF 6: 5:10% D IF.D6 < 0; .1CB6/.1CF 5/1;””)
• Cumulative benchmark returnrBmcumin the drawdown period in cellG6: 4:16% D IF.D6 < 0; .1CC 6/.1CG5/1;””)
• Cumulative alpha˛cumin the drawdown period in cellH 6: 0:94% D IF.D6 < 0; F 6G6;””)
The maximum relative drawdownMRDD in cell H 20is then obtained by MIN.H 2WH19/ D 1:93%.
Therefore, the worst drawdown period was the last one spanning August–
November 2013 with a duration of 4 months and a cumulative alpha of1:93%.
Figure1.33shows the same situation again, now displaying the cumulative alpha on the vertical axis. This allows to better spot the time point when the loss of a drawdown period is recovered. This recovery period is an important measure since it shows the capability of an asset manager to cope with difficult investment periods (based on historical experience).
Fig. 1.33 Drawdown diagram for Example 14.Source: Own, for illustrative purposes only
To finalize this section, let us look at another graphical representation of the relative drawdown. Figure1.32shows the relative drawdowns and the length of the recovery periods, but the magnitude of the drawdown is not obvious to see.
Therefore, drawdowns are often represented like in Fig.1.33which graphically represents column H of Table1.23. Here, the magnitude of each single relative drawdown and theMRDDcan be seen directly (but not the recovery period).
End of Example 14
1.4.2.2 Conclusion
The maximum relative drawdown is a key asymmetrical risk measure in active portfolio management which shows the maximum consecutive negative alpha a portfolio had in the past. In combination with the measured drawdown and recovery period an investor can evaluate the portfolio manager’s ability to regain losses versus the benchmark using historical data. As in the case of the absolute drawdown, the length of the historical time period and the choice of the subperiods is key. The shorter the subperiods (for example, days vs. months) and the analyzed historical time period, the less severe theMRDDwill be.
The historical time period should cover at least one market crash in order to deliver a meaningfulMRDD. For example, doing anMRDDanalysis for an active equity portfolio in the beginning of 2008 based on data for 2004–2007 would be irrelevant, since no stock market crash occurred during this time. Here, extending the time period to include the internet bubble 1999–2003 would be necessary, assuming the equity portfolio already existed then.
MRDDis a powerful instrument to compare the drawdown risk and the recovery potential of actively managed funds. The historical time periods and subperiods used for this comparison have to be identical for all analyzed portfolios. Otherwise, an MRDDcomparison is meaningless.