Calculate and interpret tracking error

CFA level I / Quantitative Methods: Application / Common Probability Distributions / Discrete uniform and binomial distribution functions

Tracking error is the total return on a portfolio (gross of fees) minus the total return on its benchmark index. It measures the deviation of a portfolio's return from the returns on its benchmark index.

Example 3: Meeting a tracking error objective

John is managing an emerging market mutual fund. The objective of the fund is to tracking the emerging market index and keep the tracking error within a band of 100 basis points 90 percent of the times. John manages to keep the tracking error within that range four out of five times which is 80 percent success. But he was expected to have a success rate of 90 percent. To judge his performance, find the probability that, given an assumed success rate of 90 percent, performance could be as bad as or worse than that delivered. Assume that his performance is following a binomial distribution i.e. the success probability is expected to remain constant across all nodes.

Solution:

The probability of success is 0.90, and the probability of failure is 0.10.

We need to measure the F(4) i.e. the cumulative probability that the number of successes is less than or equal to 4.

F(4) = p(0) + p(1) + p(2) + p(3) + p(4)

where p(n) is the probability of exact n successes in a binomial distribution

p(0) = 5C0(0.9)0(0.1)5 = 0.00001
p(1) = 5C1(0.9)1(0.1)4 = 0.00045
p(2) = 5C2(0.9)2(0.1)3 = 0.00810
p(3) = 5C3(0.9)3(0.1)2 = 0.07290
p(4) = 5C4(0.9)4(0.1)1 = 0.32805

F(4) = 0.00001 + 0.00045 + 0.00810 + 0.07290 + 0.32805 = 0.40951 = 40.95 percent.

Please note that the F(4) can also be calculated as 1-p(5) = 1 - 5C5(0.9)5(0.1)0 = 1 - 0.59049 = 0.40951.

We can say that there is a 40.95 percent probability that John would show the same performance or the worse performance if he had the skill to meet the success rate of 90 percent.

We can evaluate different managers using this probability. The lower is the value of the probability, the worse is the manager.


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