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## #1 2010-06-23 14:44:05

George,Y
Member
Registered: 2006-03-12
Posts: 1,332

### Computing Expectation of Negative returns

Hi boys and girls

I recently encounter an interesting task on studying fund rating methodologies. And I fund Lipper's Preservation Measure is

Sum(Min(0,ri))/T or Sum(Min(0,ri))/N*(T/N)

What it actually does is turn all the positive return r's to 0 and compute the average.

If we assume r normally distributed as N(u,s²)

The negative expectation can be modeled as  E- =∫r*pdf dr   over (-∞,0)

I came up with the answer

E-= u*N(-u/s)-(s/√2π)*exp(-u²/2s²)

But Michael Stutzer in his paper Mutual Fund Ratings: What is the Risk in Risk-Adjusted Fund Returns? derived an approximation as

Could you check this out and tell me why the difference? Thanks!

Last edited by George,Y (2010-06-23 14:52:16)

X'(y-Xβ)=0

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## #2 2010-06-24 03:50:30

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Computing Expectation of Negative returns

Here is the paper.  I can't find the computation you claim.

Also, be careful George.  You need to assume an infinite amount of real numbers in order to calculate that integral

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #3 2010-06-26 00:25:30

George,Y
Member
Registered: 2006-03-12
Posts: 1,332

### Re: Computing Expectation of Negative returns

at page 19

"You need to assume an infinite amount "
If infinite independent factors were real, normal curve would have explained everything. Unfortunately, there is always "fat tail" phenomenon, which indicates only finite factors in the real world.

X'(y-Xβ)=0

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## #4 2010-06-26 00:34:13

George,Y
Member
Registered: 2006-03-12
Posts: 1,332

page 39 appendix

X'(y-Xβ)=0

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