# Tracking Error Calculation

I have quarterly returns for a fund up to 5 years and also the benchmark mark. meaning I calculation the excess return.

How can I calculate the Annualized Tracking Error and why?

How does the formula change for monthly returns.

Thanks

" Wiley's prep material was a huge part of my success." - Lindsey G., USA

For annualized tracking error I think you need to take your quarterly returns and multiply them to get annual return
annual = (1+q1)(1+q2)(1+q3)(1+q4)
do the same for benchmark unless it is already in annual terms

then tracking error is standard deviation of (portfolio return - benchmark return)

for monthly returns it’s same formula, standard deviation of (portfolio return - benchmark return), just that they are monthly returns not annual

to get monthly return take 4th root of your quarterly returns
i.e. (1+q)^(1/4) unless you have monthly return for portfolio and benchmark already
if you don’t then your tracking error will be same for first 3 months, for the next 3 months, etc.

——————————————————-
> For annualized tracking error I think you need to
> take your quarterly returns and multiply them to
> get annual return
> annual = (1+q1)(1+q2)(1+q3)(1+q4)
> do the same for benchmark unless it is already in
> annual terms
>
> then tracking error is standard deviation of
> (portfolio return - benchmark return)
>
> for monthly returns it’s same formula, standard
> deviation of (portfolio return - benchmark
> return), just that they are monthly returns not
> annual
>
> to get monthly return take 4th root of your
> quarterly returns
> i.e. (1+q)^(1/4) unless you have monthly return
> for portfolio and benchmark already
> if you don’t then your tracking error will be same
> for first 3 months, for the next 3 months, etc.

I don’t think that’s correct, cause if you annualize to get the returns, then you will only have 5 year of annual return and you get the STD of that will be different than getting the VAR from quarter to quarter for 5 years and then annualizing it.

Compute alpha vs the benchmark for each time period (quarter, or monthly, or whatever) as

Alpha = (Return_portfolio - Return_Benchmark)

Tracking error = Standard Deviation(Alphas)

The tracking error above is the tracking error for whatever period (quarterly, monthly, etc) the raw data is from.

To go from quarterly SD to Annual SD, multiply the SD(Alphas) by SQRT(4) b/c 4 is the number of quarters in the year. To go from Monthly to Annual, multiply by SqRt(12) instead.

You want a quote?  Haven’t I written enough already???

——————————————————-
> Compute alpha vs the benchmark for each time
> period (quarter, or monthly, or whatever) as
>
> Alpha = (Return_portfolio - Return_Benchmark)
>
> Tracking error = Standard Deviation(Alphas)
>
> The tracking error above is the tracking error for
> whatever period (quarterly, monthly, etc) the raw
> data is from.
>
> To go from quarterly SD to Annual SD, multiply the
> SD(Alphas) by SQRT(4) b/c 4 is the number of
> quarters in the year. To go from Monthly to
> Annual, multiply by SqRt(12) instead.

YES CHAD!!!!…. you are right.. but i can’t get my head around that. please explain.

someone explained because VAR(X+Y) = Var(x) + Var(y), but you can’t do the same with SD

VAR(x+y) = VAR(X) + VAR(Y) + 2*COV(X,Y)

but if X and Y are uncorrelated, VAR (x+y) = VAR(x) + VAR(y)

So, if you assume that the variance every month should be (on average) the same every month (and the change in alpha is uncorrelated to what the change in alpha was at any point previously), then

VAR(annual alpha) = VAR(monthly alpha)*12

and since

SD = SqRt(VAR)

therefore

SD(Annually) = SqRt( VAR(monthly) * 12 ) = SqRt( VAR(monthly) ) * SqRt(12) = SD(monthly)*SqRt(12)

Basically, you use SD for building confidence intervals and interpreting the dispersion of a random process, but you use variance for adding together processes (and then at the end take the square root to get SD)

You want a quote?  Haven’t I written enough already???

——————————————————-
> VAR(x+y) = VAR(X) + VAR(Y) + 2*COV(X,Y)
>
> but if X and Y are uncorrelated, VAR (x+y) =
> VAR(x) + VAR(y)
>
>
> So, if you assume that the variance every month
> should be (on average) the same every month (and
> the change in alpha is uncorrelated to what the
> change in alpha was at any point previously),
> then
>
> VAR(annual alpha) = VAR(monthly alpha)*12
>
> and since
>
> SD = SqRt(VAR)
>
> therefore
>
> SD(Annually) = SqRt( VAR(monthly) * 12 ) = SqRt(
> VAR(monthly) ) * SqRt(12) = SD(monthly)*SqRt(12)
>
>
> Basically, you use SD for building confidence
> intervals and interpreting the dispersion of a
> random process, but you use variance for adding
> together processes (and then at the end take the
> square root to get SD)

you are the man. thanks.