Sharpe ratio

[lkuo123]

Hi, David,

For Sharpe ratio, sometimes, I saw people use arithmetric return for the numerator, not geometric returns (for both portfolio returns and risk free returns). Do you know why? Which method is most commonly use? I though using geometric returns make more sense as the performance returns are geometric, why the extra effort (we need to get arithmetric returns in order to get Sharpe returns)?

Thank you for your help on this.

 

[curt]

I always use arithmetic return I think its most suitable way of calculating rate of return.

 

[David Harper CFA FRM]

Hi lkuo123,

I do agree with you, geometric returns make more sense to me. But I cannot make a decisive argument. I checked three of my library sources (e.g., including the FRM assigned Amenc) and (i) in regard to the Sharpe ratio itself, it don't see it explictly addressed. I mean, E(R) is not defined along this dimension and (ii) as an input Expected return, E(r), in my sources, consistent with my training, favors geometric especially when going to multi-period (long-term).

So, I don't know why arithmetic in the Sharpe per se. However, it might just be because the user prefers arithmetic returns generally in regard to expected return. It is debatable (one argument says arithmetic for ex ante, geo for realized ex post. I don't really buy it...) Ultimately, the issue is: if you accept skewed returns (e.g., our basic lognormal is skewed), either the mean (arith) or median (geo; which must be lower where there is skew by roughly 1/2 variance as "volatility erodes returns") is a valid mathematical idea. The specific appeal of geo, to me, is (i) it's naturally time additive (ii) it's more realistic (and conservative!) to where your wealth/portfolio ends up. I attached a paper from the CFA: they argue you should use geo for E(returns), FWIW.

David

 

[lkuo123]

David,

Thank you very much for looking into your souces. It helps me a lot.

I was very confused because so many users are very firm that using arithmetric returns is the correct way - it is everywhere, including online how to get Sharpe ratio by using arithmetric returns and so on. However, very little reasoning was provided on why not geometric returns.

Sincerely,
Lesley

 

[David Harper CFA FRM]

Lesley -

sure thing, I'm not sure how much it helps. If you have a good link (that defends arithmetic), I'd be interested. I'm familiar with arithmetic assertions but not with a convincing defense? ... I tried to approach the sharpe ratio via i. it's roots in MPT or even ii. ratio consistency, but nothing definitive. (I am actually surprised i've not encountered this nor do references seem to address it...I mean, IMO, Sharpe is used often enough in practice). David

 

[lkuo123]

Hi, David,

It appears to be general practice for a lot users to use arithmetic mean for Sharpe ratio.

http://www.spreadsheetml.com/finance/sharperatio_performanceindex.shtml

I saw other websites, all use arithmetic mean, not geometric. However, none of them gives any reasonings. It appears to be acceptable to everyone.

Lesley

 

[cwright]

David, I don't seem to be able to see the attached report from the CFA Institute you mention...

Hi lkuo123,

I do agree with you, geometric returns make more sense to me. But I cannot make a decisive argument. I checked three of my library sources (e.g., including the FRM assigned Amenc) and (i) in regard to the Sharpe ratio itself, it don't see it explictly addressed. I mean, E(R) is not defined along this dimension and (ii) as an input Expected return, E(r), in my sources, consistent with my training, favors geometric especially when going to multi-period (long-term).

So, I don't know why arithmetic in the Sharpe per se. However, it might just be because the user prefers arithmetic returns generally in regard to expected return. It is debatable (one argument says arithmetic for ex ante, geo for realized ex post. I don't really buy it...) Ultimately, the issue is: if you accept skewed returns (e.g., our basic lognormal is skewed), either the mean (arith) or median (geo; which must be lower where there is skew by roughly 1/2 variance as "volatility erodes returns") is a valid mathematical idea. The specific appeal of geo, to me, is (i) it's naturally time additive (ii) it's more realistic (and conservative!) to where your wealth/portfolio ends up. I attached a paper from the CFA: they argue you should use geo for E(returns), FWIW.

David

 

[cwright]

Never mind, I think I found it: "The Misuse of Expected Returns"

http://cfainstitute.org/learning/pr...4356.aspx?WPID=Topic_List_Tabbed&PageName=All

 

[David Harper CFA FRM]

Hi cwright, yes, that is exactly the paper to which i referred but forgot to attach,