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Outline what Modigliani-squared and Treynor's measure are used for when comparing two portfolios, and how each is shown on a graph.

FRM® Part II · Risk Management and Investment Management · Portfolio Performance Evaluation · FRM2.IM.7.c

The idea

Modigliani-squared (M²) puts the Sharpe ranking into return units. Mix the portfolio with T-bills, or borrow at the risk-free rate, until its standard deviation equals the market's. The weight in the portfolio is then σm ÷ σp. M² is the return of this adjusted portfolio minus the market return. On a graph with standard deviation on the horizontal axis, M² is the vertical gap at σm between the line from Rf through the portfolio and the capital market line. Treynor's measure is the excess return per unit of beta, calculated as follows. Treynor = (Rp − Rf) ÷ β. On a graph of expected return against beta, it is the slope of the line from Rf to the portfolio. The security market line has slope Rm − Rf, so a steeper line means the portfolio plots above the SML. M² ranks portfolios exactly as the Sharpe ratio does, but gives its answer in percentage points.

Formulas

Modigliani-squared (M2)

M² = Rf + (σm ÷ σp) × (Rp − Rf) − Rm

Rf
the risk-free rate
σm
the market's standard deviation
σp
the portfolio's standard deviation
Rp
the portfolio's return
Rm
the market's return

The weight σm ÷ σp deleverages a risky portfolio and levers up a safe one. Using σp ÷ σm, or leaving out the T-bill or borrowing leg, is the trap.

A practice question

From the question bank, with the reason each wrong option is wrong.

Modigliani-squared (M2) is best described as:

  1. excess return per unit of beta, shown as the slope of a line in return-beta space.Why not: This is Treynor's measure, a slope in return-beta space.
  2. alpha per unit of residual risk.Why not: This is the information ratio, not a leveraged return gap.
  3. the portfolio's Sharpe ratio expressed as a unitless number.Why not: M2 is stated in return units, the Sharpe difference multiplied by the market's volatility.
  4. the risk-matched portfolio's return, after mixing with T-bills or borrowing, minus the market return. - correct

M2 leverages or deleverages the portfolio with the risk-free asset to match the market's standard deviation and compares the result with the market return, in percentage points.

A harder one

A portfolio returned 15% with a standard deviation of 24%. The market returned 11% with a standard deviation of 18%. The risk-free rate is 3%. M2 is closest to:

  1. 0.25%Why not: Forgets the T-bill leg: 0.75 x 15% = 11.25% less 11%.
  2. 4.0%Why not: Compares raw returns (15% - 11%) without matching the volatility.
  3. 1.0% - correct
  4. 8.0%Why not: Levers the portfolio up (24/18 weight) instead of down, so the adjusted portfolio is riskier than the market.

Weight in the portfolio = 18/24 = 0.75, T-bills 0.25. Adjusted return = 0.25 x 3% + 0.75 x 15% = 12.0%. M2 = 12.0% - 11% = 1.0%.

The hardest tier

A low-risk portfolio returned 9% with a standard deviation of 10%. The market returned 8% with a standard deviation of 15%. The risk-free rate is 2%. M2 is closest to:

  1. 5.5%Why not: Leverages 1.5 times but forgets the cost of borrowing at the risk-free rate.
  2. 4.5% - correct
  3. -1.3%Why not: Uses a weight of sigma_p/sigma_m (0.67) and so deleverages the portfolio, moving it away from the market's risk.
  4. 1.0%Why not: Compares raw returns (9% - 8%) without adjusting the portfolio to the market's risk.

Weight = 15/10 = 1.5, so borrow 0.5 at 2%. Adjusted return = 1.5 x 9% - 0.5 x 2% = 12.5%. M2 = 12.5% - 8% = 4.5%.

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