###### Bayesian Analysis

After completing this reading you should be able to: Describe Bayes’ theorem and... **Read More**

After completing this reading, you should be able to:

- Calculate, compare, and evaluate the Treynor measure, the Sharpe measure, and Jensen’s alpha.
- Compute and interpret tracking error, the information ratio, and the Sortino ratio.

*Exam tip: Be sure to understand the calculations from this chapter because there is a strong likelihood you will be getting a question on this in your FRM part 1 exam.*

The Sharpe ratio is equal to the risk premium divided by the standard deviation:

$$ { S }_{ P }=\frac { E\left( { R }_{ p } \right) -{ R }_{ f } }{ \sigma \left( { R }_{ P } \right) } $$

Where:

\(E\left( { R }_{ p } \right)\) Indicates the portfolio’s expected return

\({ R }_{ f }\) Indicates the risk-free rate

\(\sigma \left( { R }_{ P } \right)\) Indicates the standard deviation of returns of the portfolio

The Sharpe ratio, or reward-to-variability ratio, is the slope of the capital allocation line (CAL). The greater the slope (higher number), the better the asset. Note that the risk being used is the total risk of the portfolio, not its systematic risk, which is a limitation of the measure. The portfolio with the highest Sharpe ratio has the best performance, but the Sharpe ratio by itself is not informative. To rank portfolios, the Sharpe ratio for each portfolio must be computed.

A further limitation occurs when the numerators are negative. In this instance, the Sharpe ratio will be less negative for a riskier portfolio resulting in incorrect rankings.

The Treynor measure (1965) of a portfolio is explained by the expression below.

$$ { T }_{ P }=\frac { E\left( { R }_{ p } \right) -{ R }_{ f } }{ { \beta }_{ p } } $$

Where:

\(E\left( { R }_{ p } \right)\) Indicates the portfolio’s expected return

\({ R }_{ f }\) Indicates the risk-free rate

\({ \beta }_{ p }\) Indicates the beta of the portfolio

As with the Sharpe ratio, the Treynor ratio requires positive numerators to give meaningful comparative results and, the Treynor ratio does not work for negative beta assets. However, while both the Sharpe and Treynor ratios can rank portfolios, they do not provide information on whether the portfolios are better than the market portfolio or information about the degree of superiority of a higher ratio portfolio over a lower ratio portfolio.

Jensen’s alpha (Jensen, 1968) is described as an asset’s excess return over and above the return predicted by CAPM.

$$ Jensen’s\quad measure\quad of\quad a\quad portfolio={ \alpha }_{ p }=E\left( { R }_{ p } \right) -\left[ { R }_{ f }+{ \beta }_{ p }\left( E\left( { R }_{ m } \right) -{ R }_{ f } \right) \right] $$

Jensen’s alpha is based on systematic risk. The daily returns of the portfolio are regressed against the daily market returns to compute a measure of this systematic risk in the same manner as the CAPM. The difference between the actual return of the portfolio and the calculated or modeled risk-adjusted return is a measure of performance relative to the market.

If α_{p} is positive, the portfolio has outperformed the market, whereas a negative value indicates underperformance. The values of alpha can be used to rank portfolios or the managers of those portfolios, with the alpha being a representation of the maximum an investor should pay for the active management of that portfolio.

$$ Treynor\quad measure={ T }_{ p }=\frac { { \alpha }_{ p } }{ { \beta }_{ p } } +\left( E\left( { R }_{ m } \right) -{ R }_{ f } \right) $$

For well-diversified portfolios,

$$ { \beta }_{ p }\approx \frac { { \sigma }_{ p } }{ { \sigma }_{ m } } $$

$$ Sharpe\quad measure={ S }_{ P }\approx \frac { { \alpha }_{ p } }{ { \sigma }_{ p } } +\frac { \left( E\left( { R }_{ m } \right) -{ R }_{ f } \right) }{ { \sigma }_{ m } } $$

$$ { S }_{ P }\approx \frac { { T }_{ P } }{ { \sigma }_{ m } } $$

$$ E\left( { R }_{ reference } \right) ={ R }_{ f }+\left[ E\left( { R }_{ M } \right) -{ R }_{ F } \right] \left[ \frac { { \sigma }_{ P } }{ { \sigma }_{ M } } \right] $$

$$ Alpha=E\left( { R }_{ P } \right) -E\left( { R }_{ reference } \right) $$

Tracking error describes the standard deviation of the difference between the portfolio return and the benchmark return.

$$ TE=\sigma \left( { R }_{ P }-{ R }_{ B } \right) $$

The information ratio is the alpha of the managed portfolio relative to its benchmark divided by the tracking error. If we let:

$$ { e }_{ P }= { R }_{ P }-{ R }_{ B } $$

Where:

\({ R }_{ P }\)=portfolio return and

\({ R }_{ B }\)=benchmark return

Then,

$$ IR=\frac { E\left( { R }_{ P } \right) -E\left( { R }_{ B } \right) }{ \sigma \left( { R }_{ P }-{ R }_{ B } \right) } =\frac { { \alpha }_{ P } }{ \sigma \left( { e }_{ P } \right) } $$

The Sortino ratio is much like the Sharpe ratio, but there are two glaring differences:

- The risk-free rate is replaced with a minimum acceptable return, denoted as \({ R }_{ min }\).
- The standard deviation is replaced by a semi-standard deviation, which measures the variability of only those returns that fall below the minimum acceptable performance.

The measure of risk, \({ MSD }_{ min }\) is the square root of the mean squared deviation from \({ R }_{ min }\) of those observations in time period \(t\) where \({ R }_{ Pt }<{ R }_{ min }\), else zero.

$$ Sortino\quad ratio=\frac { E\left( { R }_{ P } \right) -{ R }_{ min } }{ \sqrt { { MSD }_{ min } } } $$

Where:

$$ { MSD }_{ min }=\frac { 1 }{ T } \sum _{ { R }_{ Pt }<{ R }_{ min } }^{ }{ { \left( { R }_{ Pt }-{ R }_{ min } \right) }^{ 2 } } $$

A portfolio has an expected return of 18% and a volatility of 10%. If the risk-free rate of interest is 4%, then what is the Sharpe ratio of the portfolio?

- 0.14
- 0.18
- 1.8
- 1.4

The correct answer is **D**.

$$ { S }_{ P }=\frac { E\left( { R }_{ p } \right) -{ R }_{ f } }{ \sigma \left( { R }_{ P } \right) } $$

Sharpe Ratio of the portfolio = (0.18-0.04)/0.10 = 1.4

Your portfolio had a value of EUR 1,000,000 at the start and EUR 1,150,000 at the end of the year. Over the same period, the benchmark index has had a return of 4%. If the tracking error is 11%, then what is the information ratio?

- 1
- 0.11
- 0.733
- 1.36

The correct answer is **A**.

The return of the portfolio is (1,150,000 – 1,000,000) / 1,000,000 = 0.15 or 15%

$$ IR=\frac { E\left( { R }_{ P } \right) -E\left( { R }_{ B } \right) }{ \text{Tracking error} } $$

= (15% – 4%) / 11% = 1