IntermediateRisk & PerformancePython

Run this module

cd "Value at Risk (VaR)"
python "var_calculator.py"

Value at Risk (VaR) & Conditional VaR¶

Value at Risk is the single most widely quoted number in financial risk management. It compresses "how bad could it get?" into one figure:

There is a 5% chance of losing more than this amount over the next day.

Banks size capital around it, desks set limits with it, and regulators demand it. This module implements the three standard ways to estimate VaR, the tail measure that fixes VaR's biggest flaw, and a backtest to check your number is actually honest.

Functions¶

Function	Description
`parametric_var(returns, confidence_level)`	Variance–covariance VaR (assumes normal returns)
`historical_var(returns, confidence_level)`	Empirical-quantile VaR — no distribution assumed
`monte_carlo_var(returns, confidence_level, n_sims)`	Simulation-based VaR from a fitted model
`conditional_var(returns, confidence_level)`	Conditional VaR / Expected Shortfall (mean tail loss)
`kupiec_pof_test(returns, var_level, confidence_level)`	Proportion-of-failures backtest
`value_at_risk(...)`	Backwards-compatible alias for `parametric_var`

The three methods, and when to trust them¶

Method	Assumption	Strength	Weakness
Parametric	Returns ~ Normal	Fast, closed-form	Underestimates fat tails
Historical	History repeats	Captures real skew/kurtosis	Bounded by worst observed day
Monte Carlo	A chosen model	Flexible, forward-looking	Only as good as the model

Parametric:  VaR = -(mu + sigma * Phi^-1(1 - c))
Historical:  VaR = -quantile(returns, 1 - c)

Why Conditional VaR (Expected Shortfall)?¶

VaR tells you the threshold of the tail but says nothing about how bad things get beyond it — and it is not sub-additive, so it can punish diversification. Conditional VaR averages the losses past the VaR point, is a coherent risk measure, and is what Basel III now favours.

import numpy as np
from var_calculator import parametric_var, historical_var, conditional_var, kupiec_pof_test

returns = np.random.normal(0.0005, 0.02, 1000)

print(parametric_var(returns, 0.95))
print(historical_var(returns, 0.99))
print(conditional_var(returns, 0.99))   # Expected Shortfall

# Is the model adequate? Backtest it.
var95 = parametric_var(returns, 0.95)
print(kupiec_pof_test(returns, var95, 0.95))

Backtesting with the Kupiec test¶

A VaR model is only credible if reality agrees with it. The Kupiec proportion-of-failures test counts how often losses breached your VaR and runs a likelihood-ratio test against the expected breach rate. A trustworthy model is one you fail to reject — too many exceptions means the model understates risk; too few means it wastes capital.

Practical notes¶

VaR scales with horizon roughly as sqrt(t) under the normal assumption — a 1-day VaR becomes a 10-day VaR by multiplying by sqrt(10).
The parametric and historical numbers diverge most in crises — that gap is the fat-tail risk. For the deep tail (99%+) prefer Quantitative Methods - Extreme Value Theory.
Pair this with Finance - Expected Shortfall, Risk Metrics and Risk Metrics - Stress Testing for a complete risk picture.
VaR is a probabilistic statement, not a worst case. The maximum loss is always larger than VaR — that is the whole point of also tracking CVaR.

Continue in Risk & Performance¶

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Risk Metrics

This module gives you quick, professional stats about risk in any list or array of investment returns. It's used by investors, analysts, and students everywhere!
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Comprehensive drawdown metrics for quantifying portfolio loss risk over time. Drawdown measures capture both the depth and duration of losses — dimensions VaR ignores.
Risk Metrics - Stress Testing

Stress tests answer: "What happens if 2008 repeats?" or "How big a shock kills the portfolio?" Required by Basel III, CCAR, and most institutional risk frameworks.

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