How do AIC and BIC work for comparing risk models, and when would they give different recommendations?

AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are both tools for comparing models that balance goodness-of-fit against complexity, but they differ in how aggressively they penalize additional parameters.

The Formulas

> AIC = -2 * ln(L) + 2k

> BIC = -2 ln(L) + k ln(n)

Where L is the maximized likelihood, k is the number of estimated parameters, and n is the sample size. Lower values indicate a better model.

The key difference is the penalty term:

• AIC penalty: 2k (constant per parameter)
• BIC penalty: k * ln(n) (grows with sample size)

For n > 8 (i.e., ln(n) > 2), BIC penalizes complexity more heavily than AIC. With typical risk management datasets of thousands of observations, the BIC penalty can be dramatically larger.

Example: Hawthorne Risk Group's VaR Model Selection

Hawthorne Risk Group is choosing among three GARCH variants for its equity portfolio VaR:

• AIC selects EGARCH(2,1): lowest at 2571.6 — the improved fit justifies the extra parameters.
• BIC selects GARCH(1,1): lowest at 2591.5 — the simpler model wins because BIC's heavier penalty outweighs the modest fit improvement.

When They Disagree — What to Do

This disagreement reflects a fundamental tradeoff:

• AIC is asymptotically efficient: it minimizes prediction error for the true model. It tends to select slightly more complex models that capture real but subtle features.
• BIC is consistent: as n grows, it converges to the true model if it is among the candidates. It prefers parsimony and avoids overfitting.

For risk management:

• Use BIC when robustness matters most (regulatory capital models, long-horizon forecasts). Overfitting a VaR model to recent data can produce dangerously optimistic risk estimates.
• Use AIC when predictive accuracy on new data is the priority (short-term trading risk, dynamic hedging). The extra complexity may capture real dynamics.
• Best practice: Report both. If they agree, you have strong evidence. If they disagree, investigate whether the extra parameters capture economically meaningful features (like asymmetric volatility in GJR-GARCH) or just noise.

FRM exam tip: Be able to calculate both AIC and BIC given log-likelihood, k, and n. Know that BIC penalizes more heavily for large samples and understand the consistency vs. efficiency distinction. A common exam question presents two or three models and asks which criterion selects which model.

Practice model selection problems in our FRM Part I question bank.