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# Shortcut to calculating semi-variance (mean squared deviation)

#### afterworkguinness

##### Active Member
Hi all,
Is there some shortcut to calculating semi-variance (mean squared deviation / used in information ratio) ?

#### afterworkguinness

##### Active Member
*correction; I meant used in the Sortino ratio not IR

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi afterwork,

I am not aware of such a shortcut.
In our XLS (T1.b.4) http://www.bionicturtle.com/how-to/spreadsheet/2012.1.b.4.-sortino ,
I only calculate an "ex post" Sortino with the realized return (i.e., typical longhand based on actual return series). I suppose you could calculate an "ex ante" Sortino using E(r): [E(r) - MAR]/downside deviation, although I am not sure why it would be useful, thanks,

#### ShaktiRathore

##### Well-Known Member
Subscriber

mean squared deviation is nothing but volatility of downside returns in a set of portfolio returns. The volatility of returns that are falling below a MAR over a period.
Sortino Ratio = (R - MAR) / SDd
where R is the Mean annual return,
(or Minimum Acceptable Return: MAR)
and SDd is the downside risk: SDd2 = (1/n) Σ (r - MAR)2
where the sum (or average) includes only returns for which r < MAR

Consider portfolio returns foe n=5 days, as 2%,3%,5%,.5%,.6%
let MAR=1%
r - MAR==2-1%,3-1%,5-1%,.5-1%,.6-1%
r - MAR==0%,0%,0%,-.5%,-.4% ignoring positive excess returns and take them as 0 as we are interested in downside returns and their volatility

SDd2 =(1/n)Σ (r - MAR)2=(0^2+0^2+0^2+(-.5%)^2+(-.4%)^2)/5=(.25%+.16%)/5=.41%/5=.0802%
SDd2 =.0802%=>SDd=.2831%

Follow four simple steps,

So directly look for returns less than MAR -> and take their square deviations from MAR ->and sum up squares to get variance and divide by total portfolio retunrs(obs.) =>and finally sqrt to get result.

thanks

#### afterworkguinness

##### Active Member

mean squared deviation is nothing but volatility of downside returns in a set of portfolio returns. The volatility of returns that are falling below a MAR over a period.
Sortino Ratio = (R - MAR) / SDd
where R is the Mean annual return,
(or Minimum Acceptable Return: MAR)
and SDd is the downside risk: SDd2 = (1/n) Σ (r - MAR)2
where the sum (or average) includes only returns for which r < MAR
Consider portfolio returns foe n=5 days, as 2%,3%,5%,.5%,.6%
let MAR=1%
r - MAR==2-1%,3-1%,5-1%,.5-1%,.6-1%
r - MAR==0%,0%,0%,-.5%,-.4% ignoring positive excess returns and take them as 0 as we are interested in downside returns and their volatility

SDd2 =(1/n)Σ (r - MAR)2=(0^2+0^2+0^2+(-.5%)^2+(-.4%)^2)/5=(.25%+.16%)/5=.41%/5=.0802%
SDd2 =.0802%=>SDd=.2831%

Follow four simple steps,

So directly look for returns less than MAR -> and take their square deviations from MAR ->and sum up squares to get variance and divide by total portfolio retunrs(obs.) =>and finally sqrt to get result.

thanks

I was asking about a shortcut as this can be rather time consuming on the test.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
ShaktiRathore It was my understanding that the downside deviation (i.e., denominator in Sortino) does not include the zeros; i.e., when P<MAR P>MAR, these positive excess values are EXCLUDED, not treated as zero. Although I had understood this to be the GIPS-compliant method (at the time I sat for the CIPM but this was several years ago ....), it seems to be controversial (see comments to my youtube video) and I do agree there is a disadvantage. Just FYI. Same video below

afterworkguinness I don't see how the realized (ex post) sortino can be performed any shorter than Shakti has shows. Of course, ex ante only requires them to give you the key input variables. I do not see GARP ever testing the calculation of the ex post Sortino.

I will submit to GARP with respect to this AIM, I think it is an unrealistic AIM:
• Current AIM: "Compute and interpret tracking error, the information ratio, and the Sortino ratio."
• Recommend AIM: "Compute and interpret tracking error and information ratio. Interpret Sortino"
Same video as above, note comments about the treatment of zeros:

#### ShaktiRathore

##### Well-Known Member
Subscriber
I agree with David that Sortino ratio calculation can be quite time consuming and seeing the less time available during exam time it would consume a lot of time.
@afterworkguinness this is the best short cut that i could give , and yes i agree with david that short cut better than this if there please let me know as i want to know how it is done. otherwise just follow the four steps which i think will not take much of your time:
1. look for returns less than MAR (in less than 30s)
2. take their square deviations from MAR(2 mins)
3. sum up squares to get variance and divide by total portfolio retunrs(1 min)
4. finally sqrt to get result.(less than 30s)
so total time u spent is 30s+2 min+ 1 min+ 30s= 4 mins. approx
for exam total time is 4 hrs for 100 Qs. that makes 240/100= 2.4 mins per Q. so u need extra 4-2.4 mins=1.6 mins for calculation of sortino ratio. You can easily garner extra 1.6 mins from other direct Qs that are their in the exam.
I agree with david that interpretation of the ratio is more important than its calculation. But if AIms includes calculation also than you cannot ignore it as there is possibility of it appearing in the exam.
Nevertheless thanks and ATB for exam...

#### afterworkguinness

##### Active Member
Thanks for the replies !

#### coquin22

##### New Member
a real good example i think it wud be less than 4min as we just ignore any return more than threshold but not forger n is accounted for all returns..
i had this for CAIA L1 lastyear i needed 7-8 mins for the answer...