I.A.1.7.1 The Sharp Ratio

Question 1

Static Portfolio Selection Problem (2 additonal conditions for the Sharpe Ratio)

Answer 1

• There is a risk-free asset returning the risk-free rate r. It is possible to buy or sell the risk-free asset in any amount - in other words, it is possible to borrow as well as invest any amount at the risk-free rate
• Investors’ risk preference can be described in terms of expected value and standard deviation only; they are risk-averse in the minimal sense that for any given level of risk, investors prefer the portfolio yielding maximum expected return (or, equivalently, for any level of expected return they prefer the portfolio yielding minimum risk)

Question 2

Sharpe Ratio (SR)

Answer 2

• Sharpe (1964) demonstrated under these condition optimal portfolios maximise the ratio
• μ (mu) - expected return of a portfolio
• (μ-r) - excess return over the risk-free rate
• σ - the standard deviation of excess return

Question 3

Choice of either of 2 risky assets (using SR)

Answer 3

• Position 2 risky and mutually exclusive portfolios A and B on an expected return versus standard deviation diagram. Also position the risk-free rate asset R and draw straight lines RAa and RBb.
• Any point along these lines represent a portfolio that is attainable by combining long positions in either A or B respecively and position - either long or short - in R.
• For any portfolio on the RAa lin, there are portfolios on the RBb line that have both less risk and larger expected return and which therefore ought to be preferred to the portfolio on the RAa line
• This is true as long as the slope of the RBb line is greater than the slope of the RAa lin or as long as the Sharpe ratio of portfolio B is greater than the Sharpe ratio of portfolio A

Question 4

Mean-Variance Criterion (P.60)