M-Squared (M2) and M-Squared Alpha
To repeat these steps in general form: first, find the weighting which "scales up" risk from the portfolio of interest to the benchmark market:
$$\displaystyle w_P = \frac{\sigma_M}{\sigma_P} $$
Then, scale up the return premium of the portfolio to a "mimicking portfolio" return:
$$\displaystyle R_{MP} = R_f + \frac{\sigma_M}{\sigma_P}(R_P - R_f) $$
And finally, the M2 measure is the difference of this mimicking portfolio return and that of the benchmark market portfolio:
$$\displaystyle M^2 = R_{MP} - R_M $$.
Then the weighting is used to adjust the portfolio's realized return to a risk-adjusted return, which is assumed to be a fair expected return given the total risk included in the portfolio. What does this suggest for the $$M^2$$ measure when a portfolio has relatively high variance?
To summarize this discussion:
[[summary]]
No, this statement falls short of what the M2 really is. It is a risk-adjusted measure, for one thing.
No, this statement ignores the market return, and the M2 is a function of the market return as well.
How could you best restate the meaning behind the M2 alpha measure?
First, she looks at the total risk of each portfolio. Portfolio P has a standard deviation of returns of 9.1%, compared with the market's 14.5%. Does this make P look better or worse than before?
Absolutely! So this portfolio had lower returns, but is also less risky. But what if it was adjusted to have the same return as the market?
This is what Leah wants to do now. She'll "scale up" the portfolio's risk (measured by standard deviations of returns) of 9.1% to the market's risk of 14.5%. So to get there, a factor of $$0.145 / 0.091$$ is needed in the $$M^2$$ calculation.
Fortunately, Leah is doing that right now, which should make a fine example. Leah has a portfolio P which she is trying to evaluate. It returned 6.2% last year, compared to the market's return of 8.8%. That doesn't look too good, but Leah would like to adjust this return for risk to see if she can come up with a better, more direct comparison. She's decided to follow the recipe to make the M2.
No, the initial information was that the portfolio returned less than the market, which was bad. But now think about how the portfolio having less risk as well will affect investor sentiment.
No, while returns are important to investors, risk-adjusted returns are more important. This total risk measure certainly has an effect on how the portfolio's performance is seen.
Since this is the scaling factor needed for risk, she decides to do the same for the return premium (not just raw returns). She takes the portfolio P return premium (the 6.2% return minus the risk-free rate), scales it (by that factor of 1.5934 to adjust for risk), and then adds the risk-free rate back to arrive at a risk-adjusted return for some "mimicking" portfolio of the same risk level as the market.
$$\displaystyle M^2 = \left[ E(R_P) - R_f \right] \frac{\sigma_m}{\sigma_P} + R_f $$
If the appropriate risk-free rate is 0.8%, what risk-adjusted return for the mimicking portfolio should she obtain?
No, that's the adjusted return premium, but then the risk-free rate should be added back to this.
That's right!
The unadjusted return of 6.2% indicates a return premium of $$6.2 \% - 0.8 \% = 5.4 \%$$. This is scaled to get an adjusted return premium, and the risk-free rate is added:
$$\displaystyle M^2 = \left[ 6.2 \% - 0.8 \% \right] \frac{14.5 \%}{9.1 \%} + 0.8 \% \approx 9.4 \% $$
No, this is the value of the market return adjusted by the factor calculated, but it's the portfolio P return which needs to be adjusted to match the market's risk level.
Now Leah's final step in comparing this to the market: just find the difference. Leah's risk-adjusted return of 9.4% looks pretty good now, since the market returned 8.8% with the same level of total risk. The __M2 alpha__ is just the difference of these:
$$M^2 \text{ alpha} = M^2 - R_M = 9.4 \% - 8.8 \% = 0.6 \% $$
__M2__ has no squared term. It's just a name.
It was designed by Franco Modigliani and his granddaughter, hence the name combining the Ms. This is a total risk measure, which is created by first calculating weights between the portfolio under analysis and the risk-free rate, which will produce a linear combination along its __capital allocation line (CAL)__ to arrive at a mimicking portfolio. This mimicking portfolio has the same total risk as the benchmark portfolio (perhaps just the market portfolio).
Yes!
All of the important pieces are there. The $$M^2$$ alpha is the return premium, adjusted for total risk, and then reduced by the market return premium.
What's really nice about using the $$M^2$$ alpha measure is that it not only allows Leah to compare portfolio P to other portfolios in terms of risk-adjusted returns, but the measure itself is an intuitive comparison of that portfolio to the broad market. A positive $$M^2$$ alpha means that the portfolio outperformed the market and a negative $$M^2$$ alpha means that it underperformed the market.
Oh, and Franco Modigliani's granddaughter's name: Leah.
The market return doesn't enter into the $$M^2$$ measure; that comparison will be made later on.
Exactly.
If the portfolio only gets a 30% return, that might look great. But if it achieves that with a very high risk level, that enters into the equation. Specifically, this equation:
$$\displaystyle M^2 = \left[ E(R_P) - R_f \right] \frac{\sigma_m}{\sigma_P} + R_f $$
A higher level of risk for the portfolio will decrease that fraction, causing the $$M^2$$ to "scale down" that high return into something more comparable with other portfolios. Also notice that this looks a lot like the Sharpe ratio, and this could also be stated as:
$$\displaystyle M^2 = SR \times \sigma_m + R_f $$
No, that wouldn't be fair; this very risky portfolio already shows impressive returns. The adjustment here is to make it look more comparable with other returns based on risk.
It's the risk-adjusted excess return over that of the market
It's the difference in market premiums between a portfolio and the market
It's the portfolio return adjusted for total risk minus the risk-free rate
Better
Worse
Neither; returns are most important
8.6%
9.4%
9.9%
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It should be lower than its unadjusted return
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It should be higher than its unadjusted return
It should be higher than the market return
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