Market Risk Measurement and Management

• Topic 01

  Advanced VaR Models

  61 questions
• Topic 02

  Term Structure Models

  59 questions
• Topic 03

  Volatility Smiles and Surfaces

  60 questions
• Topic 04

  Correlation Risk and Trading

  32 questions
• Topic 05

  Mortgage Risk Management

  43 questions
• Topic 06

  Market Risk Regulation

  69 questions
• Topic 07

  Advanced Backtesting and Stress Testing

  81 questions
• Topic 08

  Counterparty Credit Risk

  48 questions
• Topic 09

  Interest Rate Derivatives Risk

  38 questions
• Topic 10

  Equity Risk Management

  45 questions

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

1. What is the key difference between the CIR model and the Vasicek model, and what does this difference guarantee (under a condition)?

   • A
     The CIR model replaces Vasicek's constant diffusion term σ with σ√r, making volatility proportional to the square root of the current rate. When the Feller condition 2κθ ≥ σ² is satisfied, the short rate cannot reach zero (it is a reflecting barrier), guaranteeing non-negative rates.
     Correct answer
   • B
     The CIR model adds a second stochastic factor (the long-run mean θ itself becomes stochastic), making it a two-factor model compared with Vasicek's single-factor structure.
   • C
     The CIR model replaces Vasicek's mean reversion drift term with a log-normal specification, preventing the rate from going negative while maintaining the same analytical tractability.
   • D
     The CIR model uses a time-varying drift θ(t) calibrated to fit today's yield curve exactly, while Vasicek uses a constant θ. This is the no-arbitrage extension of the Vasicek model.
   Explanation

   The CIR model SDE is: $dr = \kappa(\theta - r) dt + \sigma\sqrt{r} dW$. The only difference from Vasicek is the diffusion term: $\sigma\sqrt{r}$ instead of $\sigma$. This has important consequences: (1) when $r$ approaches zero, the diffusion term $\sigma\sqrt{r} \to 0$, reducing the volatility of the rate near zero; (2) the Feller condition $2\kappa\theta \geq \sigma^2$ensures that the drift ($\kappa(\theta - r) > 0$near zero) is strong enough to push the rate away from zero before the diffusion can reach it, guaranteeing$r > 0$ at all times. If the Feller condition fails, zero can be reached but not crossed (reflecting barrier).

2. What is the FRTB output floor (Basel IV) and what is its final level?

   • A
     The output floor requires that risk-weighted assets (RWAs) calculated using internal models must be at least 72.5% of RWAs calculated under the standardised approach. This limits the capital benefit of using advanced models.
     Correct answer
   • B
     The output floor requires that IMA capital must be at least 50% of the stressed VaR capital from Basel 2.5
   • C
     The output floor is a minimum VaR multiplier of 3.0 that applies to all IMA banks regardless of backtesting results
   • D
     The output floor sets a minimum ES capital charge of 97.5% confidence regardless of the bank's model results
   Explanation

   The Basel IV output floor (finalised January 2023, phase-in 2025–2030) requires that RWAs under any internal model approach cannot fall below 72.5% of RWAs calculated under the standardised approach. The floor applies at the firm level across all risk types (market risk, credit risk, operational risk). For FRTB specifically, even an IMA-approved bank's market risk capital cannot be less than 72.5% of the SA capital. The phase-in starts at 50% in 2025, rising to 72.5% by January 2030.

3. A GARCH(1,1) model has parameters α = 0.10 and β = 0.85. Yesterday's equity return was –5% (a large negative shock) and the previous conditional variance estimate was σ²_{t-1} = 0.0004. How does the GARCH model update today's variance estimate and what key property of equity returns does this capture?

   • A
     σ²_t = ω + 0.10 × (0.05)² + 0.85 × 0.0004. The large negative shock feeds into the ARCH term, sharply raising today's variance. This captures volatility clustering — large moves tend to be followed by more large moves — which is a well-documented property of equity returns.
     Correct answer
   • B
     σ²_t = ω + 0.85 × (0.05)² + 0.10 × 0.0004. The GARCH parameter multiplies the new shock and the ARCH parameter multiplies the persistent term, reflecting the model's emphasis on persistence over news.
   • C
     The GARCH model does not update the variance after a single observation; it requires a full rolling window of squared returns to recompute σ²_t.
   • D
     σ²_t = (ω + 0.10 × (0.05)² + 0.85 × 0.0004) × (1 – α – β) / ω. A normalisation factor must be applied to ensure the conditional variance sums to the unconditional variance over time.
   Explanation

   The GARCH(1,1) update: $\sigma^2_t = \omega + \alpha \varepsilon^2_{t-1} + \beta \sigma^2_{t-1}$. Substituting: $\sigma^2_t = \omega + 0.10 \times (0.05)^2 + 0.85 \times 0.0004 = \omega + 0.00025 + 0.00034$. The large squared shock $(0.05)^2 = 0.0025$ amplifies the ARCH term. The GARCH term $(0.85 \times 0.0004)$ contributes persistence. Together they raise today's conditional variance significantly above the previous level. This mechanism — high variance following a large shock — is precisely volatility clustering (Mandelbrot 1963, Engle 1982): large absolute returns tend to cluster together in time, making periods of high volatility self-reinforcing for several days or weeks.