Sorry but I have Stalla…if you just put the details I need then I can help, if not someone else can chime in.
Yes, alpha=portfolio-benchmark. But tracking risk (tracking error, std. of alpha) is not calcuated by Std. portofolio-std. benchmark.
derswap07, I read the * on page 54. It says the “Tracking error is the std. of the difference between the fund return and market index return”, not “Tracking error is the difference between the std. of the fund return and the std. of the market index return” Is this the hung up?? I hope I am helping.
Yeah, alpha is (Rport - Rbm), but tracking error is not SD(port) - SD(bm). I think that Var(Portfolio) - Var(bm) = Var(alpha), where Var(X) means “the variance of X.” This is because Alpha is by definition uncorrelated to the benchmark over long periods of time.
bchadwick Wrote: ------------------------------------------------------- > I think that Var(Portfolio) - Var(bm) = > Var(alpha), where Var(X) means “the variance of > X.” This is because Alpha is by definition > uncorrelated to the benchmark over long periods of > time. not ture, Rp is correlated with Rm though
OK, I’m not saying I’m right here, but this is what I was thinking: Rp is correlated with Rbm (usually), and can be decomposed into a portion that is correlated with the BM and a portion that isn’t. The portion that is correlated with the benchmark has some factor Beta representing how a Beta% change in BM translates to a 1% change in the portfolio, and you can get this beta from a regression. If alpha and beta components are uncorrelated, wouldn’t the portfolio variance be the following? : Var(portfolio) = Beta^2*Var(BM) + Var(non-correlated components) The non-correlated component is what you call alpha, and so flipping things around algebraically, you’d get : Var(alpha) = Var(portfolio) - Var(BM)*Beta^2 And then you can take the square root of Var(alpha) to get SD(alpha). I’ve never tried to do it this way - I usually look at the SD of the intercept that my statistical software outputs, but I’m trying to think it through here…
Thanks guys, I think we are getting too deep into this. I will post this in S’s office hours tomorrow rather than wasting everybody’s time & let you know the answer. Thanks for trying.
bigwilly Wrote: ------------------------------------------------------- > I think these terms get very confused as they are > used interchangibly but: > > Tracking Error = Alpha = Return of Portfolio - > Benchmark You may be tempted to think “error” is the difference but Tracking error is NOT alpha. I think in this sense Tracking error is like “standard error” which is the SD of sample distribution (if this confuses you further) > Tracking Error Risk = Active Risk = Tracking Risk > = Std Deviation of Tracking Error or Alpha Again, I won’t say “SD of tracking error”. I’d say “SD of alpha” - sticky
Tracking Error is Alpha as they are both the Portfolio Return - Benchmark Return, but like I said Tracking Error can also mean Active Risk, depending on the source…
bigwilly Wrote: ------------------------------------------------------- > Tracking Error is Alpha as they are both the > Portfolio Return - Benchmark Return, but like I > said Tracking Error can also mean Active Risk, > depending on the source… I don’t think you can say tracking error is the alpha. The definition of tracking error has been very clear — Schweser, CFAI text and even wiki: http://en.wikipedia.org/wiki/Tracking_error I agree this may sound a bit different from the general way we define an “error”, but definition is definition. - sticky
Well there are a few related concepts that just need to be clarified: 1) “periodic alpha” is the difference between a portfolio return and the portfolio benchmark over a holding period and is assumed to be the result of manager decisions to hold securities in different quantities than replicating the benchmark. Transaction costs also create (negative) alpha, even if you are indexing. Usually “periodic” is understood so it just gets called alpha. 2) “expected alpha” is the average periodic alpha for a representative sample of holding periods. It is technically “expected alpha” when it is forward looking, and may be “realized alpha” when backward looking. It is different than periodic alpha in that it is an average or future expectation, rather than simply the difference between a portfolio and its benchmark. 3) “tracking error” is the standard deviation of periodic alpha over a number of periods. Tracking error gives you a sense of a) how much variability you will get in your returns due to manager decisions (but not the variability of returns that has to do with the performance of the benchmark itself), and b) a possible statistical test as to whether a manager can outperform consistently. If mean(alpha)/(tracking error/Sqrt(#periods)) > 1.96 or whatever the appropriate value of the t statistic is, then it is likely that the manager is actually adding value. 4) The information ratio is average(alpha)/sd(alpha) and tells you how effectively a manger is converting extra risk into extra return.
haha, even Investopedia is unsure: “Tracking errors are reported as a “standard deviation percentage” difference. This measure reports the difference between the return you received and that of the benchmark you were trying to imitate.” Is it just me or is first half the tracking error/risk as defined as std deviation of Rp-Rb and the second half defined as Rp-Rb???
See I’m not crazy!!! It’s as if there is no standard definition of TE!!! Also from another source: Tracking Error -------------------------------------------------------------------------------- What It Is: Tracking error is the difference between a portfolio’s return and the benchmark or index it was meant to mimic or beat. Tracking error is sometimes called active risk. There are two ways to measure tracking error. The first is to simply subtract the benchmark’s cumulative returns from the portfolio’s returns. However, the more common way is to calculate the standard deviation of the difference in the portfolio and benchmark returns over time.
And now Stalla has Tracking Error = Tracking risk as Std Deviation of the portfolio’s active return (Rp-R) so I guess I’ll stick with that definition.
I think that we all know it…it is the damn material throwing all those terms casuing the confusion.
I’m pretty sure investopedia’s first definition is wrong. I don’t think anyone else ever measures tracking error as “[subtracting] the benchmark’s cumulative returns from the portfolio’s returns.” That is simply alpha, or active return. Does CFAI text come down to a specific definition? That’s the key for the exam. I plan on using “standard deviation of active returns” unless I see otherwise in CFAI assigned reading.
Thanks Sticky, That’s exactly my thoughts.