Sharpe Ratio

With an 8% return, a 2% risk-free rate, and a 10% standard deviation, the Sharpe ratio is:

1. A.0.6Correct - 6 / 10.
2. B.0.2This divides the risk-free rate by the standard deviation (2/10) instead of the excess return. The numerator must be the return earned above the risk-free rate, which is 6%.
3. C.0.8This divides the total return by standard deviation (8/10) and skips subtracting the risk-free rate. Sharpe measures reward per unit of risk for the portion of return that required taking risk at all, so the 2% earned risk-free must come out first.
4. D.1.0This adds the risk-free rate rather than subtracting it, giving 10/10. Adding it inflates the ratio, but the risk-free return is the baseline an investor could have had without any volatility.

Why: Sharpe = (8 - 2) / 10 = 0.6.

A portfolio returns 10% with a 12% standard deviation; the risk-free rate is 2%. The Sharpe ratio is:

1. A.0.67Correct - 8 / 12 = 0.67.
2. B.0.20This comes from putting the risk-free rate over the return (2/10) rather than dividing excess return by risk. Standard deviation, the measure of risk, must be the denominator in a reward-per-unit-of-risk ratio.
3. C.0.83This is 10/12, the total return over standard deviation with the risk-free rate never subtracted. Because 2% was available with no volatility at all, only the 8% earned above it deserves credit for the risk taken.
4. D.1.20This inverts the fraction, dividing 12 by 10. A Sharpe ratio above 1.0 would mean excess return exceeded volatility, which cannot be true when the excess return here is 8% against 12% of risk.

Why: Sharpe = (return - risk-free) / standard deviation = (10 - 2) / 12 = 0.67.

Between two portfolios, a higher Sharpe ratio indicates:

1. A.Higher feesFees do reduce the return that feeds into the numerator, so heavy fees push Sharpe down rather than up. The ratio itself contains only return, the risk-free rate, and standard deviation.
2. B.Lower returnReturn sits in the numerator, so a lower return pushes the ratio down. A higher Sharpe means more excess return was earned for each unit of volatility endured.
3. C.Higher total riskStandard deviation is the denominator, so more total risk lowers the ratio unless return rises faster. The whole point of a risk-adjusted measure is that raw risk-taking earns no credit on its own.
4. D.Better risk-adjusted returnCorrect - more return per unit of risk.

Why: A higher Sharpe ratio means better return per unit of total risk (risk-adjusted return).

With a 12% return, a 3% risk-free rate, and a 15% standard deviation, the Sharpe ratio is:

1. A.0.8This is 12/15, the full return divided by risk with the risk-free rate never removed. The 3% was available without accepting any volatility, so only the 9% above it counts as compensation for risk.
2. B.1.25This inverts the ratio, dividing 15 by 12. A Sharpe above 1.0 would require excess return larger than volatility, but here 9% of excess return sits against 15% of risk.
3. C.0.6Correct - 9 / 15.
4. D.0.2This puts the risk-free rate over the standard deviation (3/15), using the wrong numerator entirely. The numerator must be excess return: total return minus the risk-free rate.

Why: Sharpe = (12 - 3) / 15 = 0.6.