Since total risk = systematic+unsystematic risk (in a CAPM world), can we say that if a portfolio or security that has a lower Treynor ratio than the market, but a higher Sharpe ratio than the market has generated alpha due solely to idioayncratic risk?
Lower treynor ratio of a security ( rp - rf / Beta) than the market ( Rb-rf ---- Beta for market = 1) could be due to 2 reason 1) High systematic risk - beta ( but same Rp) or 2) Lower Rp than Rb assuming beta market = beta security But if the same security has higher sharpe ratio ( Rp- rf / std dev) than market ( Rp - Rf --std dev will be 0 for market) could be because of only one reason 1) Higher Rp. I am not sure if this means alpha due to idioayncratic risk
mwvt9 Wrote: ------------------------------------------------------- > Since total risk = systematic+unsystematic risk > (in a CAPM world), can we say that if a portfolio > or security that has a lower Treynor ratio than > the market, but a higher Sharpe ratio than the > market has generated alpha due solely to > idioayncratic risk? I wonder if that’s possible …
If you beat market sharpe ratio you plot above the CML. If you lose to marker treynor ratio you plot below the SML. If I beat the CML but lost to SML it would have to be due to unsystematic risk right? Because I would have to have a greater risk adjusted return from the unsystematic risk than the return I lost to the treynor by (in order to beat sharpe on the CML overall).
So Treynor only incorporates Systematic risk, while Sharpe incorporates both Systematic and Unsystematic risk (Total risk). The numerator for both ratio’s is the same: (Rp - Rf). If the security’s Treynor ratio is less than the market, it implies that the security is riskier than the market. In other words lesser return per unit of risk. IMO this security cannot have a higher Sharpe ratio than the market. That is because the market will not have any unsystematic risk (gets diversified away). And the security has more systematic risk than the market (as implied by Treynor ratio). So the total risk of the security should be higher than that of the market hence a lower Sharpe ratio than the market. PS: I’m just thinking out loud here. Would love to hear more thoughts. Though not sure how this helps in terms of the curriculum.
This has to be right. Doesn’t it? Ha ha, I am starting to confuse myself. Return per unit total risk = return per unit systematic risk + return per unit unsystematic risk Or Sharpe ratio = Treynor ratio + unnamed ratio (return per unit unsystematic risk)
Aren’t the two statements above and below ‘OR’ the same? Anyway, not entirely sure where you’re going with this. As I said, in my previous post. The hypothesis you’re presenting in the question cannot be true. The security cannot have a higher sharpe ratio while having a lesser treynor’s ratio IMO. The only part of that post i agree with is the getting confused part 
I will drop it now. Had to reply to you guys using my phone.