Finance
Risk management
How much a portfolio could lose on a bad day, whether the model has been right, and how a company’s credit looks.
Market risk first: how much a portfolio could lose, measured three ways and checked against what actually happened. Then credit risk: a screen for a company in financial distress.
Value at Risk
The loss a portfolio should exceed only rarely: at 99% confidence, on one day in a hundred. Expected Shortfall answers the next question, how bad those days are on average. Three ways to estimate both, side by side.
The lines mark each method’s VaR as a return. Everything to their left is the tail they summarise.
- Historical replays past returns as they were; it knows nothing worse than its sample. Normal assumes the bell curve, which understates fat tails. Monte Carlo here simulates 20,000 outcomes from a Student-t fitted to the returns’ mean, volatility and tail weight (kurtosis), compounding across the horizon.
- Longer horizons scale the historical numbers by the square root of time, the Basel convention. That assumes returns are independent from one period to the next, which calm-then-stormy markets break.
Backtest
Was the model right? Each day, estimate VaR from the window of returns before it, then check whether that day’s loss broke it. A 99% VaR should be broken on about 1% of days, and the breaks shouldn’t bunch together.
The confidence level is the one set under Value at Risk.
Red dots are days that lost more than the forecast allowed.
- Kupiec’s test asks whether the share of breaks is consistent with the confidence level. Christoffersen’s test asks whether a break makes another more likely the next day; bunched breaks mean the model reacts too slowly.
- The Basel traffic light grades a bank’s 99% model on its last 250 days: up to 4 breaks is green, 5 to 9 yellow, 10 or more red. Here it is applied to the whole backtest, scaled by the same binomial probabilities.
Stress scenarios
VaR describes ordinary bad days. A stress test asks what a specific shock would do: shares fall, rates jump, credit spreads widen. Each position is revalued under the scenario.
- Bonds are revalued with duration and convexity: price change ≈ −duration × change in yield + ½ × convexity × change in yield². Corporate bonds feel rates and spreads together.
- A first-order revaluation: no second-round effects, no options, no liquidity costs of selling into a falling market.
Altman Z-score
A five-ratio screen for financial distress, from Edward Altman’s 1968 study of manufacturers, with his later versions for private companies and for non-manufacturers.
Each ratio multiplied by its weight. Bars to the left pull the score down.
- It is a screen, not a forecast. Altman built it on 66 US manufacturers from 1946–1965; treat the zones as prompts for a closer look.
- Not meant for banks, insurers or other financial firms, whose balance sheets work differently.
- Each version has its own cut-offs. Use Z″ for retailers and service companies, which carry fewer assets per dollar of sales.
Sources
- Altman, E. I. (1968). Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. The Journal of Finance, 23(4), 589–609.
- Altman, E. I. (2000). Predicting financial distress of companies: revisiting the Z-score and ZETA models (Z′ and Z″ coefficients and zones).
- J.P. Morgan/Reuters (1996). RiskMetrics: Technical Document, 4th ed. (EWMA volatility, λ = 0.94).
- Kupiec, P. H. (1995). Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives, 3(2), 73–84.
- Christoffersen, P. F. (1998). Evaluating interval forecasts. International Economic Review, 39(4), 841–862.
- Basel Committee on Banking Supervision (1996). Supervisory framework for the use of “backtesting” in conjunction with the internal models approach to market risk capital requirements.
- S&P 500 monthly prices and dividends: Robert Shiller’s data, via github.com/datasets/s-and-p-500. His prices are monthly averages of daily closes.