So I’ve got the formula down: M^2 = Rfr + SR*(Mkt Std Dev) But I’m not sure I really understand how this incorporates leverage. Could somebody who understands this risk adjusted performance measure a little better help me out with that? Is M^2 just an ex-ante RETURN measure approximation for the portfolio? whereas (SR) Sharpe Ratio is a measure of the SLOPE of the line connecting the Rfr and the portfolio? So for Sharpe Ratio you are comparing whether the SLOPE is higher than the CML, and for M^2 you are comparing whether the ex-ante portfolio RETURN is greater than the associated return on the CML for a given level of total risk?
M^2 appears to be just some tool that allows you to directly compare your account performance to the market. Not sure what it’s advantages are over your general sharpe ratio, treynor ratio etc. It incorporates leverage in that imagine your account has a standard deviation of 10% and a return of 5% while the market has a standard deviation of 15% and a return of 8%. in order to have the same amount of risk as the market you would borrow at the risk free rate to increase your account holding by 50% so that your standard deviation is now 15% while your return is hypothetically 7.5%. because your return is less than the market, your account has performed poorly and suggests you dont have any skill in generating alpha.
i would also expect to see a question asking you to compare M^2, Treynor, Jensen, and Sharpe and to be able to determine which one is better/worse diversified.
SkipE99 Wrote: ------------------------------------------------------- > i would also expect to see a question asking you > to compare M^2, Treynor, Jensen, and Sharpe and to > be able to determine which one is better/worse > diversified. Agree. You also found the trick to what M2 really is. Simple decomposition when worked out.
The CFAI text, volume 6, page 174 states that M2 “allows us to compare the return on the account to that of the market index at the same level of risk”. To try to illustrate this point, let’s say that the managed account A has 1.5 times the std dev of the market portfolio. If we sell 1/3 of A and invest in t-bills, the new portfolio A* will have the same std dev as the market portfolio. And then we’ll be able to make the comparison suggested in the text. M2 is now the return on A* minus the return on the market portfolio. If M2 is positive it follows that 1. the CAL of the managed portfolio A is steeper than the CML, and 2. the Sharpe ratio of the managed portfolio A is higher than the market portfolio’s Sharpe Ratio The opposite will be the case when M2 is negative
i believe you would need to compare M^2 to the market return. It is not a direct measure relative to 0 as you are suggesting.
Why is M2 anymore useful than Sharpe? If you are using it as a slope you already have Sharpe. If you are using it to graph I could maybe see it but what is the point?
Paraguay Wrote: ------------------------------------------------------- > Why is M2 anymore useful than Sharpe? If you are > using it as a slope you already have Sharpe. If > you are using it to graph I could maybe see it but > what is the point? Key point is M2 is a percentage return. which is a 2 dimension measure vs Sharpe. Sharpe - compare two ratios and see which one is better M2 - compare two returns - you get comparison and also return % in one.
sk22 Wrote: ------------------------------------------------------- > Paraguay Wrote: > -------------------------------------------------- > ----- > > Why is M2 anymore useful than Sharpe? If you > are > > using it as a slope you already have Sharpe. > If > > you are using it to graph I could maybe see it > but > > what is the point? > > Key point is M2 is a percentage return. which is a > 2 dimension measure vs Sharpe. > > Sharpe - compare two ratios and see which one is > better > > M2 - compare two returns - you get comparison and > also return % in one. Explain to me the second part. I still don’t get it. How does RFR + Sharpe(mkt SD) yield any better information than Sharpe?
M2 and Sharpe both work in risk=SD world. so results for ranking will be the same. But if you compare two ports a and b; Sa= 0.7 Sb=0.9 M2a = 10% M2b=12% Portfolio B is better from both we know. Shapre Ratio comment is “Port B is better than A as it has a higher excess return per unit of risk as measured by SD” But M2 has given us a direct interpretable number since its is actual return saying “Portfolio B has a Risk Adjusted Return of 12% and is better as it earns 2% more on a Risk Adjusted Basis vs A”
I think what sk22 is saying is that the sharpe ratio compares a ratio to a ratio but M^2 compares a return to a return at specific risk levels. So it may be easier to show how at a specific risk level the market return x while the portfolio returns x+a instead of saying this slope is higher than the mkts slope. I don’t think the info is any better, they both indicate the same thing, it’s just that the data may be easier to interperet.
sk22 Wrote: ------------------------------------------------------- > M2 and Sharpe both work in risk=SD world. so > results for ranking will be the same. > > But if you compare two ports a and b; > > Sa= 0.7 Sb=0.9 > M2a = 10% M2b=12% > > Portfolio B is better from both we know. > > Shapre Ratio comment is “Port B is better than A > as it has a higher excess return per unit of risk > as measured by SD” > > But M2 has given us a direct interpretable number > since its is actual return saying “Portfolio B has > a Risk Adjusted Return of 12% and is better as it > earns 2% more on a Risk Adjusted Basis vs A” Thanks for the well worded response. I understand it better but still from a practical standpoint don’t understand it. Most of the things in the curriculum I try to think of a practical application so I can remember them. Just remember it as one shows risk adjusted return and may be easier to understand.
The practical application of M2 is that it gives us an easy to interpret differential return relative to the benchmark market index. Also, with a bit of manipulation of formulas we can state that M2 = (sharpe ratio of portfolio - sharpe ratio of market index)* std dev of market index. This shows how the M2 and Sharpe measures are directly related.
Jose - something is off in your formula. does not add up to what was posted above that M^2 = RFR + SRp.StdDevMarket
I think he is trying to show slope CAL compared to slope of SML?
Yes, when the slope of CAL > slope of CML then the Sharpe ratio of the portfolio > Sharpe ratio of market index and M2 is positive.
CP is correct and Jose G. is wrong M^2 = RFR + SRp.StdDevMarket M^2 = RFR + (SRp - SPm + SPm )* StdDevMarket M^2 = RFR + SPm* StdDevMarket + (SRp - SPm)* StdDevMarket = RFR + (Market return - RFR)/StdDevMarket* StdDevMarket + (SRp - SPm)* StdDevMarket = Market return + (SRp - SPm)* StdDevMarket (SRp - SPm)* StdDevMarket is the positive return Jose G is talking about. M2 is the return of the portfolio, if it HAD the risk profile of the market, i.e., if you do a linear combination between risk free and the portfolio so that the combo has the stddev/risk of the market --> this combo will have a return = M2. The idea is to “normalize” the stddev of all portfolios (to be equal the stddev of market) for apple-to-apple comparison of portfolios with widely different stddev, so if M2 > market return --> the manager delivers beyond market return independent of his systematic risk profile.
I think M^2 would be useful if a client had a stated risk objective, like no more than 10% standard deviation. So the manager can plot the market return at 10% SD and the portfolio’s return at 10% SD and tell them, “Look, we beat the market by 5% on a risk adjusted basis. Now give us all your money.”
bpdulog Just a small side comment: M^2 gives the same insight as Sharpe ratio, so you can explain the same thing by using Sharpe ratio (your portfolio’s SF > market portfolio’s SF), and Sharpe ratio is much more used/available than M2.
Just to sum it up. (M^2 - return of the market index) = (sharpe ratio of portfolio - sharpe ratio of market index)* std dev of market index So when SR of portfolio > SR market index, M^2 > return on index And when SR of portfolio < SR market index, M^2 < return on index