Risk-free Rate

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.

The risk-free rate is most commonly proxied by:

1. A.Corporate bondsCorporate bonds carry default risk, which is precisely what a risk-free proxy must lack. Their yields include a credit spread over Treasuries that would contaminate the measure.
2. B.GoldGold is often called a safe haven, which is where the appeal comes from. Its price is volatile and it pays no interest, so it supplies neither the certainty nor the yield a risk-free rate requires.
3. C.Short-term U.S. Treasury billsCorrect - T-bills approximate risk-free.
4. D.The S&P 500The S&P 500 is the market portfolio in CAPM, sitting at the opposite end of the risk spectrum. Subtracting the risk-free rate from the market return produces the equity risk premium, so the two cannot be the same thing.

Why: Short-term U.S. Treasury bills are the standard proxy for the risk-free rate.

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.