Key Concepts for FRM Part 1 (Quantitative Analysis / Risk Management)

The Sharpe ratio measures excess return (portfolio return minus risk-free rate) per unit of total risk (standard deviation of portfolio returns).
Formula:
Sharpe Ratio = (R_p – R_f) / σ_p

The Sortino ratio is a downside-risk-adjusted version of the Sharpe ratio. It uses the same numerator but replaces total volatility with downside deviation (semi-standard deviation), which only penalizes returns below the minimum acceptable return (MAR).
Formula (when MAR = R_f):
Sortino Ratio = (R_p – MAR) / Semi-standard deviation

Both ratios can be negative when the portfolio underperforms the risk-free rate (as in this case).
Beta and tracking error are distractors here—they are not used in either calculation (beta is for Treynor ratio; tracking error is for information ratio).

Given data (all in percent, but calculations are identical whether kept in percent or converted to decimals):

• R_p = 2.5%
• R_f = MAR = 3.5%
• σ_p (standard deviation) = 21%
• Semi-standard deviation (downside deviation) = 16%

Step-by-step calculation
Sharpe Ratio = (2.5% – 3.5%) / 21% = (–1%) / 21% = –0.047619 (≈ –0.0476)

Sortino Ratio = (2.5% – 3.5%) / 16% = (–1%) / 16% = –0.0625

Difference = Sortino Ratio – Sharpe Ratio
= –0.0625 – (–0.047619)
= –0.0625 + 0.047619
= –0.014881 ≈ –0.015 (rounded to three decimal places, standard in FRM-style questions)

Explanations for Each Option

A: 0.563 – Incorrect
This appears to be a common distractor from completely misapplying formulas (e.g., confusing Sortino with Treynor + beta adjustment, or incorrectly using positive excess returns, or dividing by tracking error/beta in some combination). No legitimate calculation using the given data produces 0.563. The correct difference is negative and very small.

B: 0.347 – Incorrect
This might result from incorrectly adding or subtracting the volatilities (21% – 16% = 5%, then some erroneous division) or confusing the numerators/denominators. It has no basis in the Sharpe or Sortino formulas and ignores the actual excess return of –1%.

C: –0.053 – Incorrect
This is close to a simple average of the two ratios (≈ (–0.0476 – 0.0625)/2 ≈ –0.055) or a mistaken calculation such as –1% / (21% – 16% × some factor). While it has the right sign, it does not match the precise Sortino-minus-Sharpe arithmetic required by the question.

D: –0.015 – Correct
This matches the exact calculation above (–0.014881 rounded to three decimals). The question explicitly asks for the difference Sortino ratio minus Sharpe ratio, and the negative value is expected because the portfolio return is below the risk-free rate, making both ratios negative and the Sortino more negative (due to the smaller denominator).

Reference Answer: D
(The provided reference explanation in the query that suggested A is incorrect; standard FRM prep materials such as Kaplan Schweser confirm D as the answer using the precise formulas shown here.)

This question tests your ability to (1) recall the exact definitions of Sharpe vs. Sortino, (2) ignore irrelevant data (beta, tracking error), and (3) compute the signed difference correctly even when ratios are negative—common pitfalls in the FRM Part 1 exam. Practice similar questions by always writing out “Sortino – Sharpe” explicitly.