CFA Level II Exam · Economics and Investment Markets

Volatility, Correlation and Statistical Methods in Forecasting

Q: What is a shrinkage estimator for a covariance matrix?

It is a weighted average of the sample covariance matrix and a structured target matrix. The sample matrix is unbiased but noisy. The target is stable but biased. Blending them reduces overall estimation error.

Q: What is the difference between EWMA and GARCH?

EWMA gives weights that decline exponentially on past data and has no long-run variance it reverts to. GARCH includes a constant term, so forecasts revert toward a long-run variance when γ₁ + β < 1.

Q: Why use ARCH models in portfolio construction?

Risk is not constant over time, because volatility clusters. ARCH-type models give updated variance forecasts that feed into risk estimates and portfolio decisions.

Q: How do multifactor models help estimate correlations?

They explain returns using a few common factors, so you estimate factor sensitivities and factor covariances rather than every pairwise covariance. This cuts the number of inputs and reduces estimation error.

Updated 7 October 2026 · Fact-checked

This topic covers how to estimate risk and correlation for capital market expectations. You use shrinkage estimators to stabilise large covariance matrices, EWMA and ARCH/GARCH models to forecast time-varying volatility, and multifactor models to cut the number of inputs. In the exam, read the vignette data and apply the formula.

Understand Volatility, Correlation and Statistical Methods in Forecasting

Portfolio construction needs forecasts of risk and correlation. The simple approach is the sample covariance matrix from historical returns. It has problems. With N assets, the covariance matrix needs N variances plus N(N − 1) ÷ 2 distinct covariances, which is N(N + 1) ÷ 2 parameters in total. If the number of observations is small compared with N, the sample matrix is noisy and can even be unusable for optimisation.

One fix is a shrinkage estimator. You blend the sample covariance matrix with a structured target matrix, such as one built from a single-factor model or constant correlations. The blend is: shrinkage estimate = (weight on target × target matrix) + (weight on sample × sample matrix), with the two weights summing to 1. The target is stable but biased. The sample matrix is unbiased but noisy. Blending reduces estimation error. The more weight you put on the target, the more you shrink toward it.

Another problem is that volatility is not constant. Large moves tend to cluster, and this is called volatility clustering. Time-series models capture it. A EWMA (exponentially weighted moving average) puts more weight on recent data. An ARCH model makes the current variance depend on the past squared error. A GARCH model also lets it depend on the past variance. These forecasts can be used in portfolio risk models and option pricing, and they mean-revert toward a long-run variance.

A third tool is a multifactor model. Instead of estimating every pairwise covariance, you estimate each asset's sensitivities to a few factors and the factor covariances. This sharply reduces the number of inputs. Also remember that correlations often rise in market stress, so using calm-period estimates can understate risk. Smoothed or appraisal-based data can understate volatility and correlation too.

Key formulas to remember

Number of pairwise covariances: N(N − 1) ÷ 2; Distinct covariances (or correlations) among N assets. It grows fast as N rises. A full covariance matrix also needs N variances, so N(N + 1) ÷ 2 parameters in total.
Shrinkage estimator: Σ_shrunk = δ × Σ_target + (1 − δ) × Σ_sample; δ is the weight on the target matrix, between 0 and 1. Higher δ means more shrinkage.
EWMA variance: σ²_t = λ × σ²_(t−1) + (1 − λ) × r²_(t−1); λ is between 0 and 1. A higher λ gives more weight to old variance and a smoother estimate. The form shown assumes mean return is about zero.
ARCH(1) variance: σ²_t = γ₀ + γ₁ × ε²_(t−1); The variance depends on the previous squared error. Requires γ₀ > 0 and γ₁ ≥ 0.
GARCH(1,1) variance: σ²_t = γ₀ + γ₁ × ε²_(t−1) + β × σ²_(t−1); Stationary if γ₁ + β < 1.
GARCH long-run variance: σ² = γ₀ ÷ (1 − γ₁ − β); The level to which forecasts revert. Valid only when γ₁ + β < 1.
Covariance from correlation: Cov(A,B) = ρ × σ_A × σ_B; Use to convert a correlation forecast into a covariance.

How to solve Volatility, Correlation and Statistical Methods in Forecasting questions

Use this approach for any item-set question on estimating risk and correlation.

1Identify what the question asks: a number (variance, covariance, shrunk estimate) or a judgement (which method fits, what the bias is).
2Find the inputs in the vignette and exhibits. Check whether they are variances or standard deviations, and whether returns are monthly or annual.
3Match the tool to the problem: shrinkage for noisy covariance matrices with many assets, EWMA or GARCH for volatility clustering, a multifactor model to reduce inputs.
4Write the formula and substitute carefully. Square returns or errors where required.
5Compute step by step, keeping variance and standard deviation separate. Take the square root only at the end if volatility is asked.
6Check reasonableness: shrunk values lie between sample and target, and GARCH forecasts move toward the long-run variance.
7For judgement options, test each against the conditions, such as γ₁ + β < 1 for stationarity.

Quickest way: Plug in, then sanity-check

When to use it: Use when the question gives a model with all parameters and asks for one forecast.

Write the formula in the form shown in the vignette and label each input.
Substitute and compute variance first.
For shrinkage, the answer must lie between the sample and target values. Eliminate options outside that range.
For GARCH, check that the answer sits between the last variance and the long-run variance.
Convert to standard deviation only if asked, and annualise only if the question says so.

Common mistakes in Volatility, Correlation and Statistical Methods in Forecasting

Reporting variance when the question asks for volatility
The formulas produce variance and the last step is easy to forget.
Fix: Underline what is asked and take the square root as the final step.
Applying shrinkage weight to the wrong matrix
Candidates mix up which matrix δ multiplies.
Fix: Read the vignette wording. Weight on target is δ and weight on sample is 1 − δ.
Assuming a higher EWMA λ reacts faster to new data
Candidates think a bigger number means more responsiveness.
Fix: A higher λ puts more weight on past variance, so the estimate is smoother and slower.
Ignoring the stationarity condition in GARCH
Candidates compute the long-run variance without checking parameters.
Fix: Check γ₁ + β < 1 first. If not, there is no finite long-run variance.
Treating historical correlations as stable
Calm-period data looks reliable.
Fix: Remember that correlations tend to rise in crises, so diversification may be overstated.
Confusing ARCH with GARCH
The names are similar.
Fix: ARCH uses only the past squared error. GARCH adds the past variance term.

Worked examples

Example 1

An analyst builds a covariance estimate for two assets. The sample covariance is 0.0060 and the target (constant-correlation) covariance is 0.0036. The analyst applies a shrinkage weight of 0.40 on the target. (1) What is the shrunk covariance? (2) The sample correlation is 0.20 with volatilities of 20% and 15%. Is the sample covariance consistent with these inputs?

Show the solution

Shrunk covariance = 0.40 × 0.0036 + 0.60 × 0.0060.
0.40 × 0.0036 = 0.00144 and 0.60 × 0.0060 = 0.00360.
Sum = 0.00504, which lies between 0.0036 and 0.0060, as it should.
Consistency check: Cov = ρ × σ_A × σ_B = 0.20 × 0.20 × 0.15.
0.20 × 0.20 = 0.04, and 0.04 × 0.15 = 0.0060, which matches the sample covariance.

Answer: (1) 0.00504. (2) Yes, 0.20 × 0.20 × 0.15 = 0.0060, so the sample covariance is consistent.

Example 2

A GARCH(1,1) model has γ₀ = 0.000002, γ₁ = 0.10 and β = 0.85. Yesterday's daily variance forecast was 0.0001 and yesterday's squared error was 0.0004. (1) What is today's variance forecast? (2) What is the long-run variance? (3) Is the model stationary?

Show the solution

Today's variance = 0.000002 + 0.10 × 0.0004 + 0.85 × 0.0001.
0.10 × 0.0004 = 0.00004 and 0.85 × 0.0001 = 0.000085.
Sum = 0.000002 + 0.00004 + 0.000085 = 0.000127.
γ₁ + β = 0.95, which is below 1, so the model is stationary.
Long-run variance = 0.000002 ÷ (1 − 0.95) = 0.000002 ÷ 0.05 = 0.00004.

Answer: (1) 0.000127 daily variance (about 1.13% daily volatility). (2) 0.00004. (3) Yes, because γ₁ + β = 0.95 < 1.

Exam tips

Volatility clustering in a vignette is the cue for ARCH, GARCH or EWMA.
Check whether the vignette gives variance or standard deviation before substituting.
For shrinkage, always confirm the answer lies between the sample and target values.
Expect conceptual options about bias and noise: sample matrices are noisy with many assets, and targets add bias.
Questions are answered from the vignette, so take the parameters exactly as given.