3. Interpretation and Performance Measurement

We have determined that returns should always be compared in a risk-adjusted manner. Regarding risk, we have developed two fundamental measures: total risk, which arises from the standard deviation of returns, and systematic risk, which is reflected in the "beta" of an investment. These considerations have been supported by two graphics:

 

Capital Market Line (CML)
  • The CML represents investments in the return-standard deviation space.
  • This representation helps investors identify efficient portfolios.
  • Investors should select portfolios that lie on the CML.
  • However, the CML and the return-standard deviation space do not allow us to draw conclusions about whether individual investments are fairly valued for investors adding these investments to a well-diversified portfolio. For the analysis of investment j in the previous discussions, for example, we simply assumed it to be fairly valued.
  • For evaluating the valuation of individual investments (again from the perspective of well-diversified investors), only their systematic risk, not their total risk, is relevant.

 

Security Market Line (SML)
  • The SML provides the necessary perspective. It represents investments in the return-beta space, focusing on the systematic risk of investments.
  • In efficient markets, all investments should be fairly valued and consequently lie on (or very close to) the SML.
  • Portfolios on the CML are, by definition, also on the SML.
  • If a single investment notably deviates from the SML, this opens up (for well-diversified) investors attractive trading opportunities:
    • An investment above the SML is undervalued and should be bought.
    • An investment below the SML is overvalued and should be sold.
    • In efficient markets, however, such misvaluations should be rare and short-lived, as many active investors compete, leading to a rapid correction of prices.

  

Sharpe and Treynor Ratios

In both examined return-risk spaces, ultimately, the slope of the line between the risk-free return and the corresponding investment is relevant. The steeper this slope, the higher the return an investor receives per unit of risk (standard deviation in the CML, systematic risk in the SML).

This slope coefficient is summarized in two popular measures for evaluating investment performance: the Sharpe Ratio and the Treynor Ratio.

 

Sharpe Ratio

The so-called Sharpe Ratio (S) expresses the slope coefficient in the return-standard deviation space. It results from an investment's risk premium relative to the risk-free rate of return \((\mu_j-R_F)\), divided by the investment's standard deviation \((\sigma_j)\):

 

Sharpe Ratio: \(S_j = \frac{\mu_j - R_F}{\sigma_j} \)

 

In our previous example, the market portfolio had an expected return of \(\mu_M = 7\%\) with a standard deviation of \(\sigma_M = 15\%\). The risk-free return \(R_F\) was 2%. Accordingly, this market portfolio has a Sharpe Ratio (\(S_M\)) of 0.333:

 

Sharpe Ratio Market Portfolio: \(S_M = \frac{\mu_M - R_F}{\sigma_M} = \frac{0.07 - 0.02}{0.15}=0.333\)

 

In contrast, investment j had an expected return of \(\mu_j = 9.5\%\), with a standard deviation of \(\sigma_j = 45\%\). Consequently, investment j has a Sharpe Ratio (\(S_j\)) of 0.1667:

 

Sharpe Ratio Investment j: \(S_M = \frac{\mu_j - R_F}{\sigma_j} = \frac{0.095 - 0.02}{0.45}=0.1667\)

 

Therefore, investment j generates only half the return per unit of risk compared to the market portfolio.

Recalling how we derived the Capital Market Line: We searched for the point on the Efficient Frontier that guarantees maximum slope with a line through the risk-free return. Now, we can call a spade a spade: The market portfolio is the portfolio of risky assets with the highest Sharpe Ratio. To find it, we can run a portfolio optimization that aims to maximize the Sharpe Ratio. The attached Excel file containing the example from the section "Portfolios with multiple assets" section shows the result of this optimization on the "Market Portfolio" worksheet.

 

Treynor Ratio

Unlike the Sharpe Ratio, the Treynor Ratio (T) measures an investment's risk premium \((\mu_j-R_F)\) not relative to the total risk \(\sigma_j\) but relative to the systematic risk \(\beta_j\):

 

Treynor Ratio = \(T_j =  \frac{\mu_j - R_F}{\beta_j} \)

 

By definition, the market portfolio has a beta of 1. Consequently, the Treynor Ratio of the market portfolio, by definition, equals the market risk premium. In the previous example, the market risk premium was 5%, so the Treynor Ratio of the market portfolio is also 5%:

 

Treynor Ratio Market Portfolio = \(T_M =  \frac{\mu_M - R_F}{\beta_M}=\frac{0.07 -0.02}{1}=0.05 \)

 

In contrast, investment j offered a risk premium of 7.5% with a beta of 1.5. Therefore, the Treynor Ratio of investment j is also 5%:

 

Treynor Ratio Investment j = \(T_j =  \frac{\mu_j - R_F}{\beta_j}=\frac{0.095 -0.02}{1.5}=\frac{0.075}{1.5}=0.05 \)

 

As previously shown in the graphical derivation, we therefore conclude that investment j is fairly valued and promises a higher return for the higher systematic risk.

In efficient markets, all investments should have the same Treynor Ratio.