FRM Part I

• Unit 01

  Foundations of Risk Management

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  717 questions · 10 topics
• Unit 02

  Quantitative Analysis

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  432 questions · 10 topics
• Unit 03

  Financial Markets and Products

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  489 questions · 12 topics
• Unit 04

  Valuation and Risk Models

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  500 questions · 12 topics

A few questions from this module, with the answer and a full explanation. The complete bank is available when you start practising.

1. What does gamma measure in the context of option hedging?

   • A
     Gamma measures the second-order sensitivity of an option portfolio to interest rate changes; duration-neutral hedges must also be gamma-neutral for complete protection
   • B
     Gamma measures the rate of change of an option's price with respect to the passage of time; high gamma means the option loses value quickly as expiry approaches
   • C
     Gamma measures the sensitivity of an option's delta to changes in implied volatility; options with high gamma are more sensitive to volatility changes than options with low gamma
   • D
     Gamma measures the rate of change of an option's delta with respect to the underlying price; a high gamma means delta changes rapidly as the underlying price moves
     Correct answer
   Explanation

   Gamma = ∂Δ/∂S, the second derivative of the option's value with respect to the underlying price. A high gamma means that for a given price move, delta changes substantially, which means the delta hedge requires frequent rebalancing. Long option positions (long calls or long puts) have positive gamma; short option positions have negative gamma. A negative gamma position is particularly dangerous because large price moves in either direction cause losses — the delta hedge falls behind the actual risk as the price moves.

2. What is tracking error (active risk), and how is it calculated?

   • A
     Tracking error is the beta of the portfolio with respect to the benchmark, measuring systematic deviation.
   • B
     Tracking error measures how closely the portfolio replicates the benchmark at the position level (not the return level).
   • C
     Tracking error is the difference between the portfolio's average return and the benchmark's average return, without squaring or taking standard deviation.
   • D
     Tracking error (TE) is the standard deviation of the active return (portfolio return minus benchmark return). TE = $\sigma(r_P - r_B)$, where $r_B$ is the benchmark return.
     Correct answer
   Explanation

   Active return = $r_P - r_B$ (the return difference between the portfolio and its benchmark). Tracking error = standard deviation of active returns: $TE = \text{StdDev}(r_P - r_B)$. A low TE indicates the portfolio closely tracks the benchmark (index fund). A high TE means large deviations from the benchmark. TE is also called 'active risk.' It is used in the information ratio: IR = Active return / Tracking error = $\alpha / TE$, which measures the risk-adjusted value added by active management.

3. Which of the four coherence axioms does VaR violate, and under what distribution conditions?

   • A
     VaR violates monotonicity: a worse outcome in all scenarios does not necessarily produce a higher VaR
   • B
     VaR violates translation invariance: adding riskless cash does not in all cases reduce VaR by the cash amount
   • C
     VaR violates subadditivity for all distributions, including the normal distribution
   • D
     VaR violates subadditivity for non-elliptical loss distributions (e.g., heavy-tailed or skewed credit distributions); it satisfies subadditivity for elliptical distributions such as the normal or t-distribution
     Correct answer
   Explanation

   VaR fails subadditivity specifically for non-elliptical distributions (heavy-tailed, skewed, or distributions with discrete jump components like credit default models). For elliptical distributions (e.g., multivariate normal, student-t), VaR is subadditive because the tail quantiles combine in a predictable way. The subadditivity failure is most pronounced for credit risk (concentrated, jump-to-default) and complex structured products. This is why regulators and academics favour Expected Shortfall (ES), which satisfies all four coherence axioms regardless of the distribution.