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Teaching/Finance/Lecture Notes/Lecture 13
Finance · Lecture 13 of 15

Financial Risk Management: VaR, Expected Shortfall, and Stress Testing

Part IV · Banking and risk

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Level: Graduate / Advanced Undergraduate | Field: Finance

Learning objectives

Quantify market and credit risk; estimate VaR and expected shortfall; understand tail dependence, model risk, and stress testing.

1. Value at Risk

For loss \(L\), confidence level \(\alpha\), VaR is the loss quantile

\[ VaR_\alpha(L)=\inf\{\ell:P(L\leq\ell)\geq\alpha\}. \]

It does not describe the severity of losses beyond the quantile.

2. Expected shortfall

For continuous distributions,

\[ ES_\alpha(L)=E[L\mid L\geq VaR_\alpha(L)]. \]

For distributions with atoms, use a quantile-integral definition to handle probability mass at VaR. ES summarizes tail severity beyond the threshold.

3. Estimation methods

Historical simulation revalues positions under observed changes. Variance-covariance methods impose distributional assumptions, often multivariate normality. Monte Carlo simulation generates risk factors from a specified stochastic model and reprices the portfolio.

4. Credit and counterparty risk

Credit portfolio risk depends on default probabilities, loss severity, exposure, and dependence. Counterparty exposure can vary with market values and netting agreements. Wrong-way risk occurs when exposure and counterparty default risk increase together.

5. Stress testing and model risk

Stress tests apply coherent adverse scenarios; reverse stress testing identifies conditions that cause failure. Backtesting compares realized losses with risk forecasts. Model risk includes misspecification, parameter uncertainty, implementation error, and unstable correlations.

6. Worked example

Given ordered portfolio losses, calculate empirical 95% VaR and average losses in the worst 5%, stating the finite-sample quantile convention.

7. Computational lab

Estimate historical and parametric VaR/ES. Backtest exceedances, compare normal and heavy-tailed innovations, and run a correlated market shock scenario.

Problem set

  1. Define VaR precisely.
  2. Explain why ES captures tail severity.
  3. Compare historical and parametric simulation.
  4. Design a stress test for a leveraged portfolio.

Coding exercises

These exercises use real market data in Python: monthly Fama–French factor returns (1926–2018), daily S&P 500 and NASDAQ prices (1999–2018), the VIX, Moody's AAA and BAA corporate bond yields (1919–2018), all bundled with the arch package, plus U.S. Treasury yields and household data from the wooldridge package. Install once with pip install arch wooldridge statsmodels scipy numpy pandas. Try the starter code first, then open the solution. Every printed result below comes from running the solution code exactly as shown.

Exercise 13.1 — Value-at-Risk and Expected Shortfall for the S&P 500, with backtests

Data: Daily S&P 500 log returns, 1999–2018.

  1. Using a rolling 500-day window, compute one-day 99% VaR each day by (a) historical simulation, (b) a normal distribution with the window's mean and SD, and (c) RiskMetrics EWMA volatility (\(\lambda=0.94\)) with normal quantiles.
  2. Count VaR violations over 2001–2018 and run Kupiec's unconditional-coverage likelihood-ratio test for each method.
  3. Report the violations in 2008 alone. Why does historical simulation over a long window react slowly to a crisis?
  4. Compute the full-sample 99% Expected Shortfall by historical simulation and compare it with VaR.
import numpy as np, pandas as pd
from scipy.stats import norm, chi2
from arch.data import sp500
r = 100 * np.log(sp500.load()["Adj Close"]).diff().dropna()
# TODO: rolling quantile, rolling std, EWMA variance; violations; Kupiec LR
Solution, output and discussion 
import numpy as np, pandas as pd
from scipy.stats import norm, chi2
from arch.data import sp500
r = 100 * np.log(sp500.load()["Adj Close"]).diff().dropna()
hs = -r.rolling(500).quantile(0.01).shift(1)
nv = -(r.rolling(500).mean() - norm.ppf(0.99) * r.rolling(500).std()).shift(1)
ew = np.zeros(len(r)); ew[0] = r.iloc[:50].var()
for t in range(1, len(r)): ew[t] = 0.94 * ew[t - 1] + 0.06 * r.iloc[t - 1]**2
em = pd.Series(norm.ppf(0.99) * np.sqrt(ew), index=r.index)
def kupiec(v, p=0.01):
    n, x = len(v), v.sum(); ph = x / n
    return -2 * ((n - x) * np.log(1 - p) + x * np.log(p) - (n - x) * np.log(1 - ph) - x * np.log(ph))
for lab, var in (("historical", hs), ("normal", nv), ("EWMA", em)):
    s = pd.concat([r, var], axis=1, keys=["r", "var"]).loc["2001":].dropna(); v = (s.r < -s["var"])
    lr = kupiec(v)
    print(f"{lab:10s}: {v.sum():3d} violations in {len(v)} days ({v.mean():.2%}, expected 1%), Kupiec LR = {lr:.2f} (p = {chi2.sf(lr, 1):.3f}); in 2008: {v.loc['2008'].sum()}")
q = r.quantile(0.01)
print(f"full sample: 99% VaR {-q:.2f}%, 99% Expected Shortfall {-r[r <= q].mean():.2f}%")

Output:

historical:  73 violations in 4527 days (1.61%, expected 1%), Kupiec LR = 14.47 (p = 0.000); in 2008: 21
normal    : 113 violations in 4527 days (2.50%, expected 1%), Kupiec LR = 72.30 (p = 0.000); in 2008: 28
EWMA      :  96 violations in 4527 days (2.12%, expected 1%), Kupiec LR = 43.44 (p = 0.000); in 2008: 9
full sample: 99% VaR 3.36%, 99% Expected Shortfall 4.81%

All three methods produce too many violations over 2001–2018, and Kupiec's test rejects correct coverage for each. The normal model is worst (2.5% violations), because daily returns have fat tails. Historical simulation does better on average but clusters its violations: 21 in 2008 alone, because a 500-day window adapts slowly to a sudden jump in volatility. EWMA reacts quickly (only 9 violations in 2008) but uses normal quantiles, so it fails in the tails overall. Expected Shortfall (4.8%) is much larger than VaR (3.4%): the average loss beyond the VaR threshold is large, which is why Basel III moved to ES.

Exercise 13.2 — GARCH volatility and filtered historical simulation

Data: Daily S&P 500 log returns, 1999–2018.

  1. Estimate a GARCH(1,1) with Student-\(t\) errors using the arch package. Report the persistence \(\alpha+\beta\), the half-life of volatility shocks and the degrees of freedom.
  2. Compare the conditional volatility at its 2008 peak and at the end of 2018 with the unconditional volatility.
  3. Compute a one-day 99% VaR for 2 January 2019 by filtered historical simulation: scale the empirical quantile of standardized residuals by the forecast volatility. Compare with the GARCH-\(t\) quantile.
import numpy as np
from arch import arch_model
from arch.data import sp500
r = 100 * np.log(sp500.load()["Adj Close"]).diff().dropna()
res = arch_model(r, vol="GARCH", p=1, q=1, dist="t").fit(disp="off")
# TODO: persistence, half-life, conditional_volatility, forecast(horizon=1), std_resid quantile
Solution, output and discussion 
import numpy as np
from scipy import stats
from arch import arch_model
from arch.data import sp500
r = 100 * np.log(sp500.load()["Adj Close"]).diff().dropna()
res = arch_model(r, vol="GARCH", p=1, q=1, dist="t").fit(disp="off")
a, b, nu = res.params["alpha[1]"], res.params["beta[1]"], res.params["nu"]
print(res.params.round(4).to_dict())
print(f"persistence alpha + beta = {a + b:.4f}, half-life {np.log(0.5)/np.log(a + b):.0f} days, t degrees of freedom {nu:.1f}")
cv = res.conditional_volatility; unc = np.sqrt(res.params["omega"] / (1 - a - b))
print(f"conditional vol: 2008 peak {cv.loc['2008'].max():.2f}% ({cv.loc['2008'].idxmax().date()}), end-2018 {cv.iloc[-1]:.2f}%, unconditional {unc:.2f}% (sample sd {r.std():.2f}%)")
f = np.sqrt(res.forecast(horizon=1).variance.iloc[-1, 0])
z = res.std_resid.dropna()
var_fhs = -(res.params["mu"] + f * np.quantile(z, 0.01))
var_t = -(res.params["mu"] + f * stats.t.ppf(0.01, nu) * np.sqrt((nu - 2) / nu))
print(f"forecast vol for the next day {f:.2f}%: 99% VaR by filtered historical simulation {var_fhs:.2f}%, GARCH-t {var_t:.2f}%")

Output:

{'mu': 0.0646, 'omega': 0.0086, 'alpha[1]': 0.0995, 'beta[1]': 0.9002, 'nu': 6.5094}
persistence alpha + beta = 0.9996, half-life 1980 days, t degrees of freedom 6.5
conditional vol: 2008 peak 5.61% (2008-10-16), end-2018 2.03%, unconditional 4.97% (sample sd 1.20%)
forecast vol for the next day 1.94%: 99% VaR by filtered historical simulation 5.22%, GARCH-t 4.88%

Volatility is extremely persistent: \(\alpha+\beta\approx0.9996\), close to integrated GARCH, so the implied "unconditional" volatility (5%) is unreliable — far above the sample SD of 1.2%. In practice, shocks to volatility decay over months. The Student-\(t\) degrees of freedom (about 6.5) confirm fat tails even after conditioning on volatility. Conditional volatility peaked at about 5.6% a day in October 2008. Filtered historical simulation gives a slightly higher VaR than the GARCH-\(t\) quantile, because the empirical standardized residuals have a heavier left tail than a symmetric \(t\) distribution.

Further reading

Jorion, Value at Risk; McNeil, Frey and Embrechts, Quantitative Risk Management.