## Rolling Period Analysis

One problem with the runs test as a measure of performance consistency is that the runs test accounts for whether or not there was value added, but not for the size of the value added. We can use the risk-adjusted return measures to look for both evidence of and magnitude of consistent performance. First we calculate the major risk, return, and relative risk-adjusted return statistics for the period in the spreadsheet below. We can see that Funds B and C dominate Fund A in all respects, with higher return, value added, and Information ratios as well as lower standard deviation and downside deviations.

 Fund A FundS Fund C Index Annualized 5-Vear Return: 10.68 15.47 18 25 14.42 Standard Deviation: 18X13 13.27 12.S2 12.46 Downside Deviation, T=0: 8 05 4.08 5.04 5.31 Value Added: 0.67 0.79 2.61 Tracking Error: 10.46 12.76 6.70 Information Ratio: 006 0.06 0,39

So we are left to choose from Funds B and C. These funds both outperformed the benchmark, with similar standard deviations. Fund C has a higher return, lower standard deviation, and higher Information ratio. But the choice is not totally clear-cut, because Fund B exhibited a lower downside deviation, if this was an important criterion in our manager selection. Before making a decision, we might be interested in evaluating the consistency of the managers with regard to these statistics. With a one-year evaluation horizon in mind, and with five years of data, we could calculate five yearly calendar cumulative returns, standard deviations, and derived risk adjusted returns. Exhibit14.4 shows the yearly risk and return statistics for each of the managers.

By looking at the statistics over finer periods, we can get a sense of the variability in the relative ranking of the funds over time. Exhibit

14.5 shows that Fund C has had a higher Information ratio for three out of the last five years, and that Fund B has had better relative risk adjusted performance for the most recent year.

We can also see that the relative ranking of the second and third portfolios has changed several times, in fact each portfolio has been the best performing or second-best performing fund for three out of the five periods. If we had more funds to select from, we could perform this analysis using rank and order statistics to see how consistently each manager ranked in the first quartile of managers, and so on.

When we measure the performance of a manager, we frequently do so over long periods of time, but not a long enough period of time to determine whether or not he has added statistically sufficient value over the benchmark. We can add to the number of observations and construct a more robust consistency analysis by performing a rolling period analysis. In a rolling period analysis, we calculate statistics for a fixed window of time, where with each new observation we add a new period and drop an earlier period. For example, instead of calculating annual Information ratios based on the five calendar years for which we have data, we can in fact calculate 17 yearly Information ratios with 20 quarters of data. We start with the first quarter where we have four prior quarters of data, and calculate an Information ratio for each quarter subsequent to that, adding the most recent and dropping the oldest quarter each time. Exhibit 14.6 shows how we can analyze performance on a rolling basis.

 Fund A Fund 6 FundC Index Fund A Fund B FundC Fund A FundB FundC (1 58) 21.42 9.05 0 23 age Return Annual Value Added Downside Deviation, 1 U Year 2000 Year 1999 073 1160 Year2000t (1318) 9.82 (10 87) (3.47) 688 Year 2000 9.34 0.48 2.36 10.38 3.70 Year 1999t 5.35 Year 1999 7.19 5.26 5.44 Year 1998 S.86 24.09 21.20 12.83 Year 1993. (7.17) 11.26 8.37 Year 1996 10.21 2.21 8.60 Year 1397 24.26 23.68 30.13 24.77 Year 1937: (0.51) (083) 5.38 Year 1397 0.00 2.03 0.00 Year 1336 20.46 7.04 15.OS 19.43 Year 1336: 1.03 Annualized Standar d D euiat ion Tracking (11 59) (1 35) Error Year 1936 0.00 0.38 0.00 Information Ratio Year 2000 Year 1999 17 30 19 3S 4.34 6.59 Year 2000t 19.58 6.12 7.31 Year 2ÛÛ0 (067) 1 61 (1.49) 0.69 (0.29) 1.07 10.39 13j84 Year 1999: 7.71 11.87 6 25 Year 1999 Year 1938 Year 1997 21.65 10.24 10.39 10.98 13.61 13.11 Year 1933: 6.52 18.63 3.10 Year 1398 (1.10) 0.60 2.70 (0.25) (0.07) 0.88 12.04 10.52 Year 1997: 2.00 13.09 6 «3 Year 1997 Year 1996 4.26 5 56 107 617 Year 1996: 2.16 4.99 4.75 Year 1996 0 48 (2.32) (0.28)
 Fund A Fund B Fund C Rank Year 2000: (0.67) 1.81 (-1 49) S-A-C Year 1993: 0.69 (0.29) 1.07 C-A-B Year 1998: (1.10) 0.6G 2,70 C-B-A Year 1997: CO 25) (0.07) 0.88 C-B-A Year 199S: 0.48 (2.32) (0.28) A-C-B

EXHIBIT 14.6 Rolling Period Analysis

 Rolling Annual Information Ratio Fund A Fund B Fund C Rank 2000 04 (0.67) 1.61 (1.49) B-A-C 2000 03 1.03 0.16 (0.59) A-8-C 2000 02 0.8S (0.34) 0.94 C-A-B 2000 01 2.26 0.36 0.88 A-C-B 1999 04 o.ss CO-29) 1.07 C-A-B 199\$ 03 (1.00) (0,18) 0.00 C-B-A 199\$ 02 (1.79) 0 63 0.55 B-C-A 1993 01 (2 91) 015 2.45 C-B-A 1998 04 (110) 0.60 2.70 C-B-A 1998 03 (1.14) 1.65 251 C-B-A 1998 02 (1.00) 0.36 0.81 C-B-A 1998 Q1 0.26 0.23 1.07 C-A-B 1997 04 (0 25) (0.07) 0.88 C-B-A 1997 03 (0 36) (7-20) 007 C-A-B 1997 02 0.80 (4-42) 0.74 A-C-B 1997 01 0.72 (3,31) 0.31 A-C-B 1998 04 0.48 (2.32) (0.23) A-C-B

Here we have calculated an Information ratio for each quarter starting in the fourth quarter of 1996. This quarter is the first for which we have four prior quarters of data including the current quarter and the three prior, which are not shown. Then, in the first quarter of 1997, we calculated a new Information ratio including the returns from the first quarter of 1997 and dropping the returns from the first quarter of 1996. We did this for each period up to the current period in order to derive the rolling time series of Information ratios. We can use this data in several ways. First, we can look at the rank consistency of the three funds over time. We can see that there was a definite pattern at the start and middle of the period under analysis, with Funds A and C dominating early on, and then a switch where Funds C and B dominated for a period. More recently there has been no clear pattern, as the ranking has switched several times. Exhibit 14.7 shows how we can get a better sense of the relative performance of the manager using a line graph.

In addition to giving us some information as to the relative performance of the managers, the rolling analysis gives us some information that we can use to put the performance into perspective. For example, the yearly Information ratio for Fund B in the third quarter 1997 (14 periods ago) is an outlier. We might discuss the situation with the manager and decide to exclude that period from the analysis when we better understand the reasons for the performance during that period. Or alternatively, we could decide that that period is more representative of the manager's expected performance over a particular market cycle and overweight that period in our analysis of performance. It also might be interesting to look more closely at the recent performance of Fund C.

Even though this fund had the best 5-year track record, it has not been as attractive in the most recent periods.

We can gain some additional insight into performance by creating a chart that graphs the movement over time of the 1-year tracking error versus the rolling 1-year value added. These are the two variables that make up the Information ratio. This kind of a chart is sometimes called a "snail trail" (see Exhibit 14.8).

Including the time period over which the observation occurred adds value over simply plotting the risk and return statistics. For example, here we can see that the rolling value added has been falling and the tracking error increasing over the past few periods.

EXHIBIT 14.7 Rolling Information Ratio Graph

Rolling 1-Vear Information Ratio

EXHIBIT 14.8 Information Ratio Snail Trail

EXHIBIT 14.8 Information Ratio Snail Trail

We have presented here one of many possible methods to use the return, risk, and risk-adjusted return statistics to analyze and rank the performance of portfolios. You can adapt the tools presented here, including consistency tests and rolling period analysis, to perform a more comprehensive analysis of performance.

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