Explanation:
The Sharpe ratio is a measure of risk-adjusted return, calculated as the excess return of the portfolio over the risk-free rate, divided by the standard deviation of the portfolio's return. The standard deviation of the portfolio's return is a measure of the total risk of the portfolio, including both systematic and unsystematic risk. When a portfolio is diversified by adding more stocks, the unsystematic risk decreases, which reduces the total standard deviation. As the denominator decreases while the numerator (excess return) remains the same, the Sharpe ratio will increase. On the other hand, the Treynor ratio uses beta (systematic risk) in the denominator. Diversification does not necessarily change the systematic risk (beta) of the portfolio, so the Treynor ratio remains unaffected.
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Q.2757 A portfolio manager increases the number of stocks in his portfolio from ten to twenty, spread out across a range of industries. Assuming, there is no change in the excess return of the portfolio, what will be the most likely effect of this on the Sharpe and Treynor ratios of the portfolio?
A
The Sharpe ratio will increase and the Treynor ratio will remain unaffected.
B
The Sharpe ratio will decrease and the Treynor ratio will remain unaffected.
C
The Sharpe ratio will remain unaffected and the Treynor ratio will increase.
D
The Sharpe ratio will remain unaffected and the Treynor ratio will decrease.
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