Treynor vs Sharpe index

[verma.rahul]

Hi David

In "Portfolio theory and performance analysis" Noel Amenc explains that Treynor index is appropriate for portfolio that only constitutes a part of investors assets and Sharpe index for a portfolio that represents individual's total investment.

But according to me it should be other way round as Treynor index basically judge the *extra* returns with respect to beta (systematic risk) which means that portfolio should be well diversified and this can be insured only if one considers investor's total investments. And same goes Sharpe index i.e. it uses total risk (both systematic and unsystematic) and hence it can be used for a part of investor's total portfolio.

Also, can you please clarify how can one get a zero beta asset by short selling. As per the definition, a short seller sells an asset that is not owned by him/her but anyhow s/he is looking for its price to fall down which is an effect of market forces/Interest rate volatility i.e. systematic risk.
Also, does risk free asset have a systematic risk ??? otherwise why does merton model take a third portfolio into consideration ???

 

[David Harper CFA FRM]

Hi frm_aspirant,

I too struggle with this, I went to my library for relief but found none; I feel it maybe can be debated. But I *think* Amenc means the following, where (IMO) he later has better wording, 4.2.44: "The Sharpe ratio and the Treynor ratio are based on the same principle, but use a different denition of risk. The Sharpe ratio can be used for all portfolios [notice this approach is consistent with your view in regard to Sharpe!]. The use of the Treynor ratio must be limited to well-diversied portfolios."

if portfolio risk = systematic risk + specific (idiosyncratic) risk,
1. if your total portfolio has any specific risk, the total risk (Sharpe) should be used; i.e., since it is your total portfolio, specific risk can find no elimination in further diversification.
2. but if your partial portfolio has any specific risk, we assume it will be diversifed away (to zero) in your larger portfolio; so the presumption, I think, is that the partial portfolio is a fraction of a larger, well-diversified portfolio. i.e., it is not your total portfolio, so we (dubiously?) assume the specific risk will be diversifed away in your total portfolio and therefore need not be counted.

(But, again, I still cannot fully disagree with you...nor can i find a conclusive disagreement in my library)

Re: "how can one get a zero beta asset by short selling. As per the definition, a short seller sells an asset that is not owned by him/her but anyhow s/he is looking for its price to fall down which is an effect of market forces/Interest rate volatility i.e. systematic risk."

If your portfolio has beta, say, +1.2 then you short market index futures, you can lower the portfolio beta to zero. If the equrity risk premium is, say 3%, your portfolio will increase by (3%)(1.2) but your short futures will decline in roughly offsetting value; and vice-versa. That's shorting the index, but similarly you could short a portfolio of stocks: they hedge to the extent they have a beta, but (in a short) their beta implies that market-based (beta-based) gains for the underlying portfolio are offset by losses in the (shorted) hedged portfolio.

re: does risk free asset have a systematic risk. No, the risk free asset is the only portfolio/asset on either the CML/SML without systematic risk. We can verify with beta = cov(riskless asset, market)/variance(market). Note Covariance (constant, X) = 0. See http://en.wikipedia.org/wiki/Covariance

re otherwise why does merton model take a third portfolio into consideration? I don't follow, apologies?

David