1.3 Traditional Risk and Risk-Adjusted Return Measures .1 Volatility.1Volatility
1.3.2 Tracking Error
Tracking error (TE) is one of the most commonly used relative risk measures in active management.18 Its definition is in line with the definition of volatility.
However,TE does not use an absolute return time series but a relative return time series, i.e., a time series of alphas.
Tracking error is the typical risk measure used when managing a portfolio versus an index as a benchmark. Therefore,TEis a relative risk measure, similarly constructed like its absolute counterpart volatility. Like volatility, tracking error is a symmetrical risk measure since it looks at both deviations above and below the average of the subperiod alphas.
Definition: Tracking Error
Like in the previous sections, we again split time interval Œ0; T into N equidistant subintervals which are usually days or months. Let us denote the subperiod percentage returns of the portfolio as always with
rPf1 ; rPf2; : : : ; rPfN; the subperiod percentage returns of the benchmark with
rBm1 ; rBm2 ; : : : ; rBmN ; and the alpha for thek-th subperiod with˛k; 1kN.
Then, the tracking error of the portfolio versus its benchmark is the standard deviation of the subperiod percentage alpha returns˛1; ˛2; : : : ; ˛N.
Depending on the chosen subperiods, we get the respective tracking error:
daily alphas as input lead to a daily tracking error. Monthly alphas lead to a monthly tracking error. Mathematically, the monthly tracking error can be calculated as19
TEmonthly D p
Var.˛1; ˛2; : : : ; ˛N/ D Stdev.˛1; ˛2; : : : ; ˛N/ (1.34) withNrepresenting the number of months inŒ0; T . Formula (1.34) can easily be modified for using daily returns as input. Then,N is the number of days in Œ0; T :
TEdaily D p
Var.˛1; ˛2; : : : ; ˛N/ D Stdev.˛1; ˛2; : : : ; ˛N/: (1.35) (continued)
18Lhabitant (2004, p. 59).
19Lhabitant (2004, p. 59).
In practice the tracking error can easily be calculated using the subperiod alphas within time period Œ0; T . By using monthly returns, this means mathematically:
˛monthly D vu ut 1
N 1 XN kD1
.˛k˛/2: (1.36)
1.3.2.1 Note
The nametracking errorstems from passive portfolio management where the goal is to achieve an alpha of zero. If, for example, for each month alpha is zero, then the alpha of the whole period is of course zero as well. If, however, one or more of the monthly alphas are different from zero it is most likely that over the whole time period alpha is different from zero as well. This deviation for the whole period can be measured with the standard deviation of each subperiod’s alpha, i.e., tracking error.20More specifically:
• Tracking error quantifies theerrormade by the passive portfolio manager when tracking the benchmark.
• A perfect passive portfolio manager generates an alpha of zero in each subperiod, i.e.,˛kD0for1kN. This automatically leads toTED0.
• The less perfect the passive portfolio manager, the higherTEand the more the portfolio manager will deviate from his overall alpha target of zero.
It is important for tracking error calculation that the type of input determines the interpretation of the output. If the alphas used as input to calculate tracking error are monthly alphas in % over the last 3 years, the calculated tracking error is a monthly tracking error and also in %.
As with volatility, a daily or monthly tracking error is not very meaningful. Only an annualized tracking error derived from a daily or monthly tracking error is used.
This annualized tracking error can then be compared to an annualized alpha over the same time intervalŒ0; T . To annualize a tracking error,T has to be at least1 year, i.e.,T 1.
In order to annualize a monthly tracking errorTEmonthly, we have to scale it by the square root of12, i.e.:
TEp:a: D p
12 TEmonthly: (1.37)
Similarly, if a daily tracking error is given, this has to be scaled with the square root of the number of business days per year which is roughly252, i.e.:
20Natenberg (1994, pp. 60–61).
TEp:a: D p
252 TEdaily: (1.38)
The reasons for the scaling in Eqs. (1.37) and (1.38) are (like in the case of annualizing a volatility) rooted in financial engineering and will not be discussed further here. As a standard deviation, the tracking error can never be negative. And, if we assume a normal distribution for subperiod alphas, as we did when interpreting volatility, the standard deviation concept allows a neat interpretation of the tracking error.
1.3.2.2 Interpretation
For interpretation purposes, we assume that the future annualized alpha ˛p:a is a normally distributed random variable. Let then TEp:a: be the corresponding annualized tracking error. Then the probability distribution for the annualized alpha
˛p:ais described by a symmetrical bell-shaped curve which peaks atEŒ˛p:a:. This expected value can be interpreted as the p.a. alpha target of the actively managed portfolio versus its benchmark, see Fig.1.11:
In Fig.1.11, the probability of the return being in the intervalŒ˛0; ˛00is the area under the curve between ˛p:a: D ˛0 and˛p:a: D ˛00 (the entire area under the graph is1). The percentages in the diagram show the area under the graph in the respective intervals, for example, the probability for the return to lie between
˛0DEŒ˛p:a:and˛00DEŒ˛p:a:CTEp:a:is34:13%.
Thus, the probability of the annualized returnEŒ˛p:a:to be
• at mostTEp:a:off from the expected annualized returnEŒ˛p:a:is68:3%,
• at most2TEp:a:off from the expected annualized returnEŒ˛p:a:is95:4%,
• at most3TEp:a:off from the expected annualized returnEŒ˛p:a:is99:7%.
Return density
Annualized alphaαp.a.
IE[αp.a.]
IE[αp.a.]−3·TEp.a. IE[αp.a.]−2·TEp.a. IE[αp.a.]−TEp.a. IE[αp.a.]+TEp.a. IE[αp.a.]+2·TEp.a. IE[αp.a.]+3·TEp.a.
34.13%
34.13%
13.59%
13.59%
2.14% 2.14%
0.13% 0.13%
Fig. 1.11 Graphical interpretation of the tracking error: normal distribution of the annualized alpha for a portfolio managed against a benchmark with alpha targetEŒ˛p:a:and tracking error TEp:a:.Source: Own
Table 1.9 Example 5: Tracking error calculation
A B C D
Monthly portfolio Monthly benchmark
1 Month performance performance Monthly alpha
2 07/2012 6.10 % 6.01 % 0.09 %
3 08/2012 5.50 % 5.45 % 0.05 %
4 09/2012 4.70 % 4.63 % 0.07 %
5 10/2012 5.00 % 6.99 % 1.99 %
6 11/2012 5.10 % 4.16 % 0.94 %
7 12/2012 6.70 % 7.07 % 0.37 %
8 01/2013 6.03 % 5.97 % 0.06 %
9 02/2013 3.23 % 2.95 % 0.28 %
10 03/2013 5.12 % 4.66 % 0.46 %
11 04/2013 5.21 % 4.91 % 0.30 %
12 05/2013 4.10 % 4.01 % 0.09 %
13 06/2013 4.50 % 3.87 % 0.63 %
14 07/2013 1.75 % 2.95 % 4.70 %
15 08/2013 3.71 % 4.52 % 0.81 %
16 09/2013 4.20 % 3.93 % 0.27 %
17 10/2013 4.26 % 4.99 % 0.73 %
18 11/2013 4.00 % 3.84 % 0.16 %
19 12/2013 5.10 % 4.99 % 0.11 %
20 TEmonthlyD 1.29 %
21 TEp:a:D 4.48 %
Source: Own, for illustrative purposes only
Example 5
Let us now take a look at the example of a fund and its benchmark. Table1.9 shows the monthly performance of both and the resulting alpha over a time period of18months, i.e.,T D1:5years andN D18months.
Using Eqs. (1.34) and (1.37) to calculate the monthly and annualized tracking error over time periodŒ0; T , respectively, we need the following functions in cellsD20andD21:
• Monthly tracking errorTEmonthlyin cellD20:
1:29% D STDEV:S.D2WD19/
• Annualized tracking errorTEp:a:in cellD21: 4:48% D SQRT.12/STDEV:S.D2WD19/
End of Example 5
1.3.2.3 Conclusion
Tracking error isthe risk measure in benchmark-oriented portfolio management.
As a relative risk measure, it corresponds in its definition and interpretation to volatility as an absolute risk measure in benchmark-agnostic portfolio management.
Both risk measures are symmetrical, and their interpretation which is based on the mathematical concept of confidence intervals is only valid, if the underlying data (either an absolute return or a relative return time series) are normally distributed.
This especially means that tracking error (as well as volatility) treats positive and negative deviations in the same way.