Question on Tracking Error

[BHeng9611]

Hi all,

Kindly refer to the attach taken from FRM part 1 chapter 5, I have 1 question to ask regarding tracking error.
1. According to the TE formula (TE = SD(Rp - Rb)), may I know why is it Sqrt(30%^2 + 15%^2 - 2*0.014) and not Sqrt(30%^2 - 15%^2 - 2*0.014)? I supposed benchmark refers to market?

Thank you in advance,
BX

 

[gsarm1987]

Hi all,

Kindly refer to the attach taken from FRM part 1 chapter 5, I have 1 question to ask regarding tracking error.
1. According to the TE formula (TE = SD(Rp - Rb)), may I know why is it Sqrt(30%^2 + 15%^2 - 2*0.014) and not Sqrt(30%^2 - 15%^2 - 2*0.014)? I supposed benchmark refers to market?

Thank you in advance,
BX

Ex-post, answer for tracking error should be array should be: R(P-B)1, R(P-B)2, R(P-B)3......R(P-B)N , we take Stdev of that (see blue number below). however for ex-ante, this does not consider the variances and covariance and is not the correct formula for tracking error. the calc in screen shot implies ex-ante situation i.e we know the variances of both arrays in advance and want to predict what the tracking error would be.
where we are long the portfolio and short the benchmark portfolio then we can also say that the tracking error is : SQRT(Stdev(P)^2 + Stdev(B)^2 - 2*Cov(P,B)). notice that the sign before two is negative, that denotes negative relation between P & B. Now when you ask why the 2nd term (variance of Benchmark) isnt negative. answer is simple. if you look closer at the formula, it actually says: SQRT(Variance(P) + Variance(B) -2 Cov(P,B)). variance is always positive because its a squared term and we just interpret here in the spreadsheet as Std^2. meaning same thing. just like 2*Corr*StdP*StdB mean 2Cov(P,B). Benchmark can be the market portfolio. can be S&P500 Nasdaq100, anything depending on the mandate, which ever the manager chooses to compete against.

Did a hypothetical example in excel, pasting here:

Serial	P	Short B	Combined
1​	2​	-6​	-4​
2​	3​	-2​	1​
3​	5​	-4​	1​
4​	1​	-5​	-4​
5​	3​	-8​	-5​
6​	4​	-6​	-2​
Stdev	1.29	1.86	2.41
Var	1.67	3.47	5.81
Cov	0.33
Tracking error	2.11

Blue is expost (realised)
Red is exante

I hope its clear?
@David Harper CFA FRM i hope im not overthinking on the spreadsheet re: ex-post and ex-ante

 

[David Harper CFA FRM]

HI @gsarm1987 Not overthinking it, to me! I agree 100% with what you wrote, and with your spreadsheet. It really is the difference between an ex post versus ex ante tracking error. In my opinion, TE is truly meant to be an ex post measure. The ex ante version is just if you aren't able to (i.e., if you don't have the data) compute on a historical vector of Δreturns. So if you only have σ(P), σ(B) and ρ(P,B), the ex ante version is merely utilize the variance property that variance(X - Y) = variance(X) + variance(Y) - 2*COV(X,Y).

Your XLS infers an ex ante TE by utilizing the historically-derived params, so that's a valid illustration of ex ante TE. I suppose I view your 2.41 as the realized (ex post), which varies with each sample (history is the one sample that happened!), that is estimated by the 2.11 ex ante value. Thank you!