FRM (Financial Risk Manager) Valuation and Risk Models (Part I) Syllabus

Valuation and Risk Models (Part I) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Valuation and Risk Models (Part I) in FRM (Financial Risk Manager), not a summary of it.

1. 1

   Value at Risk Foundations

   4 topics

   • VaR Concept and Definition
     • Confidence level and holding period
     • Absolute vs. relative VaR
   • VaR Estimation Methods
     • Parametric (delta-normal) approach
     • Historical simulation
     • Monte Carlo VaR
   • Expected Shortfall and Coherence
     • Limitations of VaR
     • Coherent risk measures and subadditivity
   • Backtesting VaR Models
     • Exception counting and the traffic light approach
     • Basel backtesting rules
2. 2

   Option Valuation and Greeks

   3 topics

   • Binomial Trees
     • One-step and multi-step trees
     • Risk-neutral valuation
   • Black-Scholes-Merton Model
     • Assumptions and inputs
     • Implied volatility and the volatility smile
   • Option Sensitivities (The Greeks)
     • Delta, gamma, and theta
     • Vega and rho
     • Delta-gamma hedging
3. 3

   Fixed Income Risk and Pricing

   3 topics

   • Measuring Interest Rate Risk
     • Effective duration and convexity
     • Key rate durations
   • Term Structure Models
     • Yield curve construction and bootstrapping
     • Spot rate and forward rate dynamics
   • Bond Portfolio Hedging
     • Duration-based hedging
     • Barbell and bullet strategies
4. 4

   Credit Risk Models and Operational Risk

   4 topics

   • External and Internal Ratings
     • Rating transition matrices
     • Through-the-cycle vs. point-in-time
   • Credit Risk Quantification
     • Probability of default, LGD, and exposure at default
     • Expected and unexpected loss
   • Operational Risk Foundations
     • Loss event categories
     • Risk and control self-assessment
   • Stress Testing Basics
     • Scenario design and sensitivity analysis

Valuation and Risk Models (Part I) flashcards for FRM (Financial Risk Manager)

21 of 50 cards from the Valuation and Risk Models (Part I) deck — real questions with worked answers.

1. What does Value at Risk (VaR) measure?

   VaR is the maximum loss on a position or portfolio that will not be exceeded over a given holding period at a specified confidence level. Equivalently, it is the loss threshold such that the probability of a larger loss equals $1-c$ (the significance level).

2. State the formal probability statement that defines VaR at confidence level $c$.

   $$P(L > \text{VaR}) = 1 - c$$ The loss $L$ exceeds VaR only with probability $1-c$. For example, a one-day 99% VaR is exceeded on only 1% of days.

3. What are the two key parameters that must be specified for any VaR estimate?

   The confidence level $c$ (e.g., 95% or 99%) and the holding period (time horizon, e.g., 1 day or 10 days). VaR is meaningless without both.

4. How does increasing the confidence level affect the size of VaR, all else equal?

   A higher confidence level produces a larger VaR, because you are capturing a more extreme point further in the left tail of the loss distribution (e.g., 99% VaR > 95% VaR).

5. How does VaR scale with the holding period under the i.i.d. normal-returns assumption (square-root-of-time rule)?

   VaR scales with the square root of time: $$\text{VaR}_{T} = \text{VaR}_{1}\,\sqrt{T}$$ A 10-day VaR equals the 1-day VaR multiplied by $\sqrt{10}$, assuming zero mean and i.i.d. returns.

6. Distinguish absolute VaR from relative VaR.

   Absolute VaR measures loss relative to zero (an actual dollar loss): $\text{VaR}_{\text{abs}} = -\mu + \sigma\,z_{c}$. Relative VaR measures loss relative to the expected value (mean) of the distribution: $\text{VaR}_{\text{rel}} = \sigma\,z_{c}$. Relative VaR ignores the drift term $\mu$.

7. Write the formula for absolute parametric VaR over horizon and its components.

   $$\text{VaR}_{\text{abs}} = \left(-\mu_{P} + z_{c}\,\sigma_{P}\right)$$ where $\mu_{P}$ is the expected portfolio return (drift), $z_{c}$ is the standard normal deviate for confidence $c$, and $\sigma_{P}$ is the portfolio return standard deviation. In dollar terms multiply by portfolio value.

8. What are the standard normal $z$-values used for 95% and 99% confidence VaR?

   For 95% confidence, $z = 1.645$ (one-tailed). For 99% confidence, $z = 2.326$ (one-tailed).

9. Name the three principal methods for estimating VaR.

   1) Parametric (delta-normal / variance-covariance) approach; 2) Historical simulation; 3) Monte Carlo simulation.

10. Describe the parametric (delta-normal) approach to VaR.

    It assumes returns (or risk-factor changes) are normally distributed and uses estimated means, variances, and covariances to compute portfolio standard deviation, then VaR $= z_{c}\sigma_{P}$ (relative). Positions are linearized via first-order (delta) sensitivities to risk factors.

11. For a single asset, give the delta-normal dollar VaR formula.

    $$\text{VaR} = z_{c}\,\sigma\,V$$ where $\sigma$ is the return volatility over the horizon and $V$ is the position market value. With a nonzero mean, absolute VaR $=(z_{c}\sigma-\mu)V$.

12. For a two-asset portfolio, how is portfolio standard deviation computed for delta-normal VaR?

    $$\sigma_{P} = \sqrt{w_{1}^{2}\sigma_{1}^{2} + w_{2}^{2}\sigma_{2}^{2} + 2\,w_{1}w_{2}\,\rho_{12}\,\sigma_{1}\sigma_{2}}$$ where $w_i$ are weights, $\sigma_i$ volatilities, and $\rho_{12}$ the correlation.

13. What is the main weakness of the delta-normal (parametric) approach for options?

    It relies on a linear (delta) approximation and normality, so it poorly captures the nonlinear (convex/gamma) payoffs of options and underestimates risk when payoffs are nonlinear or returns have fat tails.

14. What are the key advantages of the delta-normal VaR method?

    It is computationally fast and simple, requires only estimates of means/variances/covariances, and scales easily across large portfolios. It does not require full revaluation or large historical datasets.

15. Describe the historical simulation approach to VaR.

    Apply the actual historical changes in risk factors over a look-back window to the current portfolio, generate a distribution of hypothetical P&L, and read off the VaR as the appropriate empirical percentile of the simulated losses.

16. In historical simulation with 1,000 ordered loss observations, which observation gives the 95% VaR?

    The 95% VaR is at the 5th percentile of the loss distribution, i.e., the 50th worst loss out of 1,000 ($1000 \times 0.05 = 50$).

17. State two advantages of historical simulation VaR.

    It makes no assumption about the distributional form (captures fat tails and skewness present in the data) and naturally captures actual historical correlations among risk factors without estimating them.

18. State two disadvantages of historical simulation VaR.

    It assumes the past is representative of the future (only events in the window can occur), and results are highly sensitive to the length and choice of the look-back window. It also weights all observations equally unless modified.

19. Describe the Monte Carlo approach to VaR.

    Specify a stochastic process and distribution for the risk factors, generate a large number of random scenarios, fully revalue the portfolio under each scenario, and take the appropriate percentile of the simulated P&L distribution as VaR.

20. What are the main advantages of Monte Carlo VaR?

    It can handle nonlinear instruments (full revaluation), any specified distribution (including fat tails), path dependency, and complex portfolios. It is the most flexible of the three methods.

21. What are the main drawbacks of Monte Carlo VaR?

    It is computationally intensive (many full revaluations), depends heavily on the chosen model and distributional assumptions (model risk), and may suffer sampling/simulation error if too few paths are used.

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