Factor Models in Risk Attribution

Active returns come with active risk. A good asset manager will follow the benchmark, and even beat it. The best-case scenario is to have portfolio (_P_) returns that are consistently above the benchmark (_B_). That's a low-risk win. But the more variance in that difference, the greater the __active risk__, measured commonly as the __tracking error (TE)__ (or even "tracking risk"), which is the sample standard deviation of those active returns. $$\displaystyle TE = s(R_P - R_B) $$

Less tracking error is better, of course. For passive funds designed to follow a benchmark, less than 0.10% is good. Diversified active managers might go with 2%-6%, with only the truly aggressive higher than that. What would make you willing to take on more tracking error in a manager?

No. The higher the benchmark returns, the better that looks, and maybe you don't even need the manager. The manager's value is in getting a higher active return.

Absolutely!

The ratio of mean active return to tracking error is the __information ratio (IR)__. $$\displaystyle IR = \frac{R_P - R_B}{TE} = \frac{R_P - R_B}{\sigma(R_P-R_B)} $$ If you wanted to give an active manager a limit as to how much risk was acceptable for a certain active return, would this be a minimum or maximum IR?

Yes!

No. A minimum, actually.

The information ratio is active return over active risk, so the more the better.

Now the decomposition. Just like the fundamental factor model that breaks down active returns, these models can break down active risk. Recall that active returns were the factor contributions, with whatever is left being security selection. It's the same here. The __active factor risk__ is measured in variance, and the __active risk squared__ is the squared tracking error. What's left is the __active specific risk__. How might you use this to restate the value of active risk squared?

Exactly!

No. It's the sum of these.

$$ \mbox{Active Risk Squared = Active Factor Risk + Active Specific Risk}$$

| Portfolio | Industry factor | Style Factor | Total Factor | Active Specific | |---|---|---|---|---| | P1 | 15 | 12 | 27 | 9 | | P2 | 10 | 8 | 18 | 18 | All of these are variances, so 15 really means 15 percent squared, or 0.0015, depending on how you like to see these. With this in mind, the active risk error on the first portfolio is the square root of the squared active risk, or 6%. $$\displaystyle TE = \sqrt{0.0027 + 0.0009} = 0.06 = 6 \% $$ How would you characterize the second portfolio?

No. Actually both portfolios have a 6% tracking error.

No. The active factor risk variances are smaller for the second portfolio.

That's right. This is just a matter of seeing that 18 is larger than 9 in the table. From there, you can find that the second portfolio has the same tracking error as the first portfolio. $$\displaystyle TE = \sqrt{0.0018 + 0.0018} = 0.06 = 6 \% $$ Then this can be enhanced a bit, just like before. In the first portfolio, the active factor risk of 27 is 75% of the total active risk squared of 36, and active specific risk is 25%. In the second portfolio, it was 50/50. So this gives you some specific idea of where risk is coming from with these managers.

To summarize: [[summary]]

For example, you might have two managers' portfolios where the risks can be separated with your fundamental factor model like this:

Higher active returns

Higher benchmark returns

Minimum

Maximum

It's the sum of active factor risk and active specific risk

It's the difference between active factor risk and active specific risk

More active risk

More active factor risk

More active specific risk

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