Monte Carlo Simulation, Back-Testing & Regulatory Capital

Recap of Risk-Modeling Alternatives

  • Internal-model space usually offers three core techniques for market-risk metrics such as Value-at-Risk (VaR)
    • Variance–Covariance (VCV) / delta-normal method
    • Historical Simulation (HS)
    • Monte-Carlo Simulation (MC) ⇐ focus of the current lecture

Monte Carlo Simulation for Market Risk

  • Name origin
    • Borrowed from the famous "Monte Carlo" casino in Monaco
    • Conjures images of dice-throwing → randomness, repetition, unpredictability
  • Conceptual essence
    • Generate thousands (often millions) of possible future states for each risk factor or portfolio
    • Each path is drawn from a known probability distribution rather than ad-hoc guessing
  • Simple dice metaphor
    • Every path ≈ one dice roll; repeating rolls produces an empirical distribution of outcomes
  • Choice of distribution (basic version)
    • Assume i.i.d. normal returns with
      f(x)=1σ2πe−(x−μ)22σ2f(x)=\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
      • μ\mu = mean (expected) return
      • σ\sigma = standard deviation (volatility)
    • Properties: bell-shaped, symmetric, fully characterised by μ\mu and σ\sigma
  • Minimum required inputs
    • Historical time-series of returns to estimate μ\mu and σ\sigma
    • Sampling algorithm (e.g., Box–Muller, Ziggurat) to convert uniform pseudo-random numbers into N(μ,σ2)N(\mu,\sigma^2) draws
  • Simulation workflow (basic, one-factor illustration)
    1. Estimate μ^,σ^\hat\mu, \hat\sigma from data set (e.g., last 250 daily returns).
    2. Create MM random draws r<em>1,…,r</em>M{r<em>1,\dots,r</em>M} from N(μ^,σ^2)N(\hat\mu,\hat\sigma^2).
    3. Mark-to-market portfolio under each draw.
    4. Build empirical loss distribution → read VaR/ES percentiles.
  • Extensions when “markets aren’t always normal”
    • Replace normal with Student-tt, skewed, or mixture distributions
    • Stochastic volatility (e.g., GARCH, Heston) ⇒ time-varying σt\sigma_t
    • Jump-diffusion models to insert rare, large moves
    • Multi-factor correlations by simulating vector processes, often via Cholesky decomposition of a covariance matrix
  • Complexity vs simplicity
    • Although conceptually heavier than HS or VCV, the starting ingredient list is surprisingly short: just μ\mu and σ\sigma.

Regulatory Back-Testing & the Traffic-Light Framework

  • Regulators evaluate VaR models by counting exceptions (days where actual loss > model VaR)
  • Typical benchmark: 99 % one-day VaR over ≈ 260 trading days
    • Expected exceptions:
      E[N]=(1−0.99)×260≈3E[N]= (1-0.99)\times260 \approx 3
  • Colour zones (Basel traffic light)
    • Green = 0–4 exceptions → model acceptable
    • Yellow = 5–9 → model usable but penalty factor applied
    • Red = 10+ → model deemed unreliable
  • Illustrative case from lecture
    • Observed 6 exceptions ⇒ Yellow zone
    • Scaling factor K is raised above its base of 3 (exact increment depends on table; 6 exceptions implies +0.2 in this example)
Worked example (teacher’s numbers)
  1. Observed exceptions: 8 (i.e., 252 days within VaR, 8 outside).
  2. Excess over expected: 8−3=58-3=5.
  3. Regulatory table dictates +1.7 to the scaling factor.
  4. Resulting capital multiplier: K=3+1.7=4.7K=3+1.7=4.7 (slide verbally rounded as “~3.2” in one snippet—understand the principle, not the exact rounding).

Basel II Internal-Model Capital Formula

  • Banks on the internal-model approach compute daily market-risk capital as
    Capital=max⁡(VaR<em>t−1,  K VaR‾</em>60)\text{Capital}=\max\Big(\text{VaR}<em>{t-1},\; K\,\overline{\text{VaR}}</em>{60}\Big)
    where
    • VaR<em>t−1\text{VaR}<em>{t-1} = yesterday’s 10-day, 99 % (or 99.9 %) VaR • VaR‾</em>60\overline{\text{VaR}}</em>{60} = average of the same VaR measure over last 60 trading days
    • KK = traffic-light penalty (≥3)
  • Rationale / ethical dimension
    • Prevents capital relief from a lucky calm day; bank must hold whichever estimate is larger
    • Penalises chronic under-prediction, incentivising model improvement

Basel III & Post-Crisis Enhancements

  • 2008 crisis exposed limitations of VaR‐only capital
  • Two additional charges introduced:
    1. Stressed VaR (sVaR): VaR calibrated on a historical high-volatility period
    2. Incremental Default & Migration Risk (IRC): capital for jump-like credit events in trading portfolios
  • Ongoing evolution → Fundamental Review of the Trading Book (FRTB)
    • Consolidates VaR into Expected Shortfall (ES) at 97.5 % over 10 days
    • Keeps back-testing & traffic-light ideas but with different thresholds and modellability tests

Key Take-Aways & Forward Linkage

  • Monte-Carlo is the flexible workhorse; normal-assumption version needs only μ\mu & σ\sigma.
  • Regulatory back-testing translates statistical misses into real money via K-factors.
  • Capital today involves a suite of metrics (VaR, sVaR, IRC, ES) reflecting both calm and stressed conditions.
  • Next lecture will ⇒ hands-on capital calculations using VCV and Historical Simulation methods.