how to calculate downside deviation? Alternative EOC 12

I’m completely confused by the calculation of downside deviation from the CFAI material.

According to CFAI material page 23 ,

downside deviation = square root of sum min(rt - r*, 0)^2 divided by n-1

however the answer for EOC 12 is using a different formula: square root of sum min(rt - r*, 0)^2 divided by n-1, then multiply sq rt of n. Why there is such a difference? Also the hurdle rate is not given in the question, but the answer assumes it’s 5%. Shall we do the same on the exam?

hurdle rate is a footnote at bottom of the page

Im pretty shot for the day but any chance n = 250? if so, its just the conversion to annual downside dev

Yes, the hurdel rate is a footnote for the example,but it wasn’t given in EOC 12, however the answer used 5%, which is confusing.

Also any idea why the calculation is multipled by sq rt of 12? This part is not in the formula in the material.

Congrats on the charterholder and thank you for still coming back here to answer our questions, cpk. I have seen your name in the forums before.

In the quesion, n=12 months

That question data applied to both 11 and 12. so 5% is applicable for 12 also.

Now why multiply by sqrt(12)

annual variance = monthly variance * 12

(here think downside deviation is like std deviation).

so annual std deviation = sqrt(monthly variance * 12) = monthly std deviation * sqrt(12)

Got it. Thanks a lot!

Can someone explain how did they come up with portfolio return and indexu return?

I can’t come up with same numbers 0.6133 or –0.449. I added monthly returns or chain linked but still different number.

How did CFAI come up with those number?


geometric gave me the answer

(1.03 * 1.04 * 0.98 * 0.98 * 0.99 * … )^(1/12) - 1 = 0.006133 = 0.6133%

Look at the big grey box in the middle of the chapter where all these same statistics are calculated for another set of numbers. That is a pretty big grey box for this chapter.

yes that gives me the answer.

However why do we need to raise to (1/12)th of power? and then multiply with 12?

its monthly returns. shouldn’t annualized return just be compounding of the returns? (especially when those periods are less than a year?)

^1/12 gives you the average monthly geometric return

* 12 gives you the annualized return from there on.

geometric 12 month return = 1.0761346

the (1.0761346(^1/12) - 1) * 12 gives you 7.36%

They do the same in the grey box as well. and call the 7.36% here (7.5% in the example as the geometric return annualized).

ok got it. thanks! its geometric avg return not the chain linked return I guess…

I always thought that if the returns are less than a year, you just chain link them and not geometric avg.

Thanks CPK

This is probably insignficant, but I can’t help but comment because this is confusing and misleading. Calculating an annualized return for a 12 month period by taking the geometric mean and multiplying by 12 is both theoretically incorrect and against industry practice. Annuallized return for a 12 month period is equivalent to the cumulative return over the 12 month period, there’s not a lot of room for argument here.

Sachin: You’re thoughts on chain linking returns <= a year are correct.

In any case, the correct solution to annualizing the geometric mean would be to use (1+r)^12-1 instead of r*12 which is the cause of the discrepancy.


you are misunderstanding

they are saying r1 = (1+r)^(1/12) - 1

and then doing r1 * 12 for average geometric return.

I’m not misunderstanding, and it’s a small detail but it’s incorrect. Smoothing has nothing to do with it, this is purely about inconsistent use of compounding.

The explanation is great. However, how do we know whether smooting is required or not. I can’t see any hints for using ^1/12 then - 1 then *12. I kept compounding the return then -1 at the beginning.

CAN SUCH Q come?

send an email to CFAI and ask them.

let us know what they say