## **Learning Outcomes**

This article explains how correlation and covariance matrices are built, interpreted, and used in portfolio risk models (mean-variance, VaR, ES, and simulation). After reading, you will be able to:

- Convert between correlation matrices and covariance matrices and interpret each entry
- Calculate portfolio variance and volatility using both summation and matrix notation
- Explain estimation, updating (EWMA), and stress behavior of correlations and the implications for VaR and ES

## **GARP FRM Part 1 Syllabus**

For the GARP FRM Part 1 exam, you are expected to understand portfolio dependence modeling with a focus on the following syllabus points:

- Estimating standard deviations and correlations for delta-normal VaR and Monte Carlo simulation, including the use of stressed periods
- Updating correlation estimates (e.g., covariance and correlation via EWMA-style updating)
- Implications of correlation breakdown in stressed markets for VaR or ES analysis

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. Which statement is most accurate for a valid correlation matrix?  
   a) Off-diagonal entries must all be positive  
   b) It must be diagonal  
   c) It must be symmetric with ones on the diagonal and be positive semidefinite  
   d) Its entries can exceed 1 in absolute value if volatility is high

2. A two-asset portfolio has weights $w_1 = 0.4$, $w_2 = 0.6$, volatilities $\sigma_1 = 10\%$, $\sigma_2 = 20\%$, and correlation $\rho_{12} = 0.30$. Compute the portfolio volatility.

3. In one or two sentences, explain why using correlations estimated from “normal” markets can understate VaR or ES in crisis conditions.

4. True/False (explain briefly): A covariance matrix must be symmetric and must produce a non-negative variance for any portfolio of the assets.

## **What Do You Know Already?**

- Can you explain the difference between covariance and correlation, including their units and interpretation?
- Can you compute portfolio variance using $w^\top \Sigma w$ and explain what $\Sigma$ represents?
- Can you describe how correlations tend to change in stressed markets and why this matters for VaR, ES, and stress testing?

Use these prompts to gauge your starting point; then focus your revision where your answers were least confident.

## **Introduction**

Diversification is only as reliable as the dependence assumptions behind it. In portfolio models, that dependence is summarized by the correlation matrix (dimensionless pairwise dependence) and the covariance matrix (dependence scaled by volatility). These matrices sit at the center of the mean-variance framework (efficient frontier), delta-normal VaR and ES, and Monte Carlo simulation.

For FRM Part 1, you should be able to move comfortably between (i) pairwise concepts such as covariance and correlation and (ii) matrix concepts such as $w^\top \Sigma w$. You should also understand a key empirical regularity: in stressed markets, volatilities rise and correlations generally rise, weakening diversification and creating “correlation breakdown” relative to normal-market estimates. This is why stressed VaR/ES and correlation stress tests are standard risk management tools.

> **Key Term: Covariance**  
> Covariance measures how two return series move together. For returns R_i and R_j:
>
> $$
> cov(R_i, R_j) = E[(R_i-\mu_i)(R_j-\mu_j)]
> $$
>
> It has units of “return squared.”

## **From pairwise dependence to portfolio risk**

### Covariance and correlation

For asset returns $R_i$ and $R_j$ with means $\mu_i$ and $\mu_j$:
$$cov_{ij} = cov(R_i, R_j) = E[(R_i-\mu_i)(R_j-\mu_j)]$$
The correlation coefficient standardizes covariance:
$$\rho_{ij} = corr(R_i, R_j) = \frac{cov_{ij}}{\sigma_i \sigma_j}$$
where $\sigma_i = \sqrt{var(R_i)}$.

- Covariance depends on scale (it changes if you change units or volatility).
- Correlation is dimensionless and lies in $[-1, 1]$ for a valid model.

> **Key Term: Correlation coefficient**  
> The correlation coefficient is the covariance between two returns divided by the product of their standard deviations:
>
> $$
> \rho_{ij} = \frac{cov_{ij}}{\sigma_i \sigma_j}
> $$
>
> It measures linear comovement on a standardized scale.

### Two-asset portfolio volatility and the role of correlation

In the mean-variance framework, portfolio volatility depends not only on individual volatilities but also on their correlation. For a two-asset portfolio with weights $w_1, w_2$:
$$\sigma_p = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2\rho_{12}}$$
This is the core "diversification" identity: when $\rho_{12} \lt 1$, portfolio volatility is lower than the weighted average of individual volatilities (in general).

### Worked Example 1.1

A portfolio holds two assets with $w_1=0.4$, $w_2=0.6$, $\sigma_1=10\%$, $\sigma_2=20\%$, and $\rho_{12}=0.30$. Compute $\sigma_p$.

> **Answer:**  
> Use:
> $$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2\rho_{12}$$
> Compute each term:
>
> - $$
>   w_1^2\sigma_1^2 = (0.4)^2(0.10)^2 = 0.16\times 0.01 = 0.0016
>   $$
> - $$
>   w_2^2\sigma_2^2 = (0.6)^2(0.20)^2 = 0.36\times 0.04 = 0.0144
>   $$
> - Cross term:
>   $$2(0.4)(0.6)(0.10)(0.20)(0.30) = 2(0.24)(0.02)(0.30)=2(0.00144)=0.00288$$
>   Sum:
>   $$\sigma_p^2 = 0.0016 + 0.0144 + 0.00288 = 0.01888$$
>   Therefore:
>   $$\sigma_p = \sqrt{0.01888} \approx 0.1374 = 13.74\%$$

## **Correlation matrices and covariance matrices**

Pairwise dependence scales poorly: for $n$ assets there are $n(n-1)/2$ distinct pairwise correlations. Matrices organize these inputs so portfolio risk can be computed and updated systematically.

> **Key Term: Correlation matrix**  
> The correlation matrix R is an n×n matrix with entries:
>
> $$
> R_{ij} = \rho_{ij},
> $$
>
> ones on the diagonal, and symmetry:
>
> $$
> R_{ij} = R_{ji}.
> $$
>
> **Key Term: Covariance matrix**  
> The covariance matrix Σ is an n×n matrix with entries:
>
> $$
> \Sigma_{ij} = cov(R_i, R_j),
> $$
>
> diagonal entries:
>
> $$
> \Sigma_{ii} = \sigma_i^2,
> $$
>
> and symmetry:
>
> $$
> \Sigma_{ij} = \Sigma_{ji}.
> $$

### Structure and interpretation

For $n$ assets:

- **Diagonal entries**
  - Correlation matrix: $R_{ii}=1$
  - Covariance matrix: $\Sigma_{ii}=\sigma_i^2$

- **Off-diagonal entries**
  - $R_{ij}$: strength and sign of linear comovement
  - $\Sigma_{ij}$: comovement scaled by volatilities

### Converting between $R$ and $\Sigma$

Let $D$ be the diagonal matrix of volatilities:
$$D = \text{diag}(\sigma_1,\sigma_2,\ldots,\sigma_n)$$
Then the relationship is:
$$\Sigma = D R D$$
Element-by-element this is:
$$\Sigma_{ij} = \rho_{ij}\sigma_i\sigma_j$$
and conversely:
$$\rho_{ij} = \frac{\Sigma_{ij}}{\sigma_i\sigma_j}$$
This conversion is tested frequently because different VaR implementations store inputs in different forms:

- a **correlation matrix plus volatilities**, or
- a **full covariance matrix**.

### Portfolio variance in summation and matrix form

For weights $(w_1,\ldots,w_n)$, the FRM syllabus commonly presents:
$$\sigma_p = \sqrt{\sum_{i=1}^n \sum_{j=1}^n w_i w_j \sigma_i \sigma_j \rho_{ij}}$$
In matrix form:
$$\sigma_p^2 = w^\top \Sigma w$$
where $w$ is the $n\times 1$ weight vector.

### Worked Example 2.1

You manage a 3-asset portfolio with weights
$$w = \begin{bmatrix}0.50\\0.30\\0.20\end{bmatrix}$$
Annual volatilities are $(20\%,15\%,10\%)$, and the correlation matrix is:

$$
R=
\begin{bmatrix}
1 & 0.40 & 0.10\\
0.40 & 1 & 0.25\\
0.10 & 0.25 & 1
\end{bmatrix}
$$

Compute the covariance matrix $\Sigma$ and the portfolio volatility.

> **Answer:**  
> Step 1: Build:
>
> $$
> D=\text{diag}(0.20,0.15,0.10)
> $$
>
> Step 2: Use:
>
> $$
> \Sigma_{ij}=\rho_{ij}\sigma_i\sigma_j
> $$
>
> - $$
>   \Sigma_{11} = 0.20^2 = 0.0400
>   $$
> - $$
>   \Sigma_{22} = 0.15^2 = 0.0225
>   $$
> - $$
>   \Sigma_{33} = 0.10^2 = 0.0100
>   $$
> - $$
>   \Sigma_{12} = 0.40(0.20)(0.15)=0.0120
>   $$
> - $$
>   \Sigma_{13} = 0.10(0.20)(0.10)=0.0020
>   $$
> - $$
>   \Sigma_{23} = 0.25(0.15)(0.10)=0.00375
>   $$
>
> So:
>
> $$
> \Sigma=
> \begin{bmatrix}
> 0.0400 & 0.0120 & 0.0020\\
> 0.0120 & 0.0225 & 0.00375\\
> 0.0020 & 0.00375 & 0.0100
> \end{bmatrix}
> $$
>
> Step 3: Compute:
>
> $$
> \sigma_p^2 = w^\top \Sigma w
> $$
>
> First compute Σw:
>
> $$
> \Sigma w =
> \begin{bmatrix}
> 0.0400(0.50)+0.0120(0.30)+0.0020(0.20)\\
> 0.0120(0.50)+0.0225(0.30)+0.00375(0.20)\\
> 0.0020(0.50)+0.00375(0.30)+0.0100(0.20)
> \end{bmatrix}
> =
> \begin{bmatrix}
> 0.0240\\
> 0.0135\\
> 0.004125
> \end{bmatrix}
> $$
>
> Then:
>
> $$
> \sigma_p^2 = w^\top(\Sigma w)
> = 0.50(0.0240)+0.30(0.0135)+0.20(0.004125)
> $$
>
> $$
> = 0.0120+0.00405+0.000825 = 0.016875
> $$
>
> Step 4: Take the square root:
> $$\sigma_p = \sqrt{0.016875} \approx 0.1299 = 12.99\% \text{ per year}$$

## **Using covariance matrices in VaR and ES models**

Correlation and covariance matrices connect directly to VaR and ES when portfolio value changes can be written as a linear function of risk-factor changes.

### Delta-normal setup for a linear portfolio

In the delta-normal approach, portfolio change over the holding period is modeled as:
$$\Delta P = \sum_i a_i x_i$$
where:

- $x_i$ is the change in the $i$th risk factor (often a proportional change for equities/commodities, and an absolute change for rates/spreads)
- $a_i$ is the portfolio sensitivity to that risk factor (for linear positions this is a constant over the horizon)

Let $x$ be the vector of risk-factor changes and let $\Sigma_x$ be their covariance matrix. Then:
$$var(\Delta P) = a^\top \Sigma_x a$$
and:
$$\sigma_{\Delta P} = \sqrt{a^\top \Sigma_x a}$$
Assuming normality for $\Delta P$, VaR and ES are computed from $\mu_{\Delta P}$ and $\sigma_{\Delta P}$:
$$VaR = \mu_{\Delta P} - z\sigma_{\Delta P}$$

$$ES = \mu_{\Delta P} + \sigma_{\Delta P}\left(\frac{e^{-z^2/2}}{(1-\alpha)\sqrt{2\pi}}\right)$$
where $\alpha$ is the confidence level (e.g., 0.99) and $z$ is the corresponding normal quantile.

### Worked Example 3.1

A portfolio’s one-day P&L is approximated as $\Delta P = a_1 x_1 + a_2 x_2$ where:

- $a_1$ = \$5,000,000, $a_2$ = \$3,000,000
- $x_1$ and $x_2$ have daily volatilities $\sigma_1=1.2\%$, $\sigma_2=0.8\%$
- $corr(x_1,x_2)=0.25$

Assume $\mu_{\Delta P}=0$. Compute one-day 99% VaR.

> **Answer:**  
> Step 1: Build the covariance terms:
>
> - $$
>   var(x_1)=0.012^2=0.000144
>   $$
> - $$
>   var(x_2)=0.008^2=0.000064
>   $$
> - $$
>   cov(x_1,x_2)=0.25(0.012)(0.008)=0.000024
>   $$
>
> Step 2: Compute var(ΔP):
>
> $$
> var(\Delta P)=a_1^2var(x_1)+a_2^2var(x_2)+2a_1a_2cov(x_1,x_2)
> $$
>
> Compute each term:
>
> - $$
>   a_1^2var(x_1)=(5{,}000{,}000)^2(0.000144)=25\times10^{12}\times0.000144=3.60\times10^9
>   $$
> - $$
>   a_2^2var(x_2)=(3{,}000{,}000)^2(0.000064)=9\times10^{12}\times0.000064=5.76\times10^8
>   $$
> - Cross term:
>   $$2(5{,}000{,}000)(3{,}000{,}000)(0.000024)=2(15\times10^{12})(0.000024)=7.20\times10^8$$
>
> Sum:
> $$var(\Delta P)=3.60\times10^9+5.76\times10^8+7.20\times10^8=4.896\times10^9$$
>
> Step 3: Standard deviation of ΔP:
> $$\sigma_{\Delta P}=\sqrt{4.896\times10^9}\approx 69{,}971$$
>
> Step 4: 99% VaR with:
>
> $$
> z\approx 2.33,\quad \mu_{\Delta P}=0
> $$
>
> $$VaR = 2.33(69{,}971)\approx 163{,}035$$
> One-day 99% VaR ≈ \$163,000.

### **Exam Warning**

> Common exam mistakes with matrices in VaR and ES:
>
> - Mixing **time units** (daily vol with annual correlations, or annual vol with one-day VaR). Keep the covariance matrix and holding period consistent.
> - Using **correlations directly** in:
>   $$
>   a^\top \Sigma a
>   $$
>   without converting to covariances when a is in currency units.
> - Forgetting that correlations can change materially in stress; using a normal-market matrix for tail risk often understates VaR or ES.

## **Estimating and updating correlation and covariance matrices**

### Historical estimation and parameter choice

To implement delta-normal VaR or Monte Carlo simulation, risk managers must estimate:

- standard deviations of risk-factor changes, and
- correlations (or covariances) between risk factors.

A common approach is to estimate them from historical data over a chosen window. The window choice is a model input:

- shorter windows respond more to current conditions but are noisy,
- longer windows are smoother but may lag regime changes.

For regulatory capital, firms may also estimate parameters from a deliberately stressed period (stressed VaR and stressed ES). The logic is: “If a specific stress repeats, is the firm resilient?”

### EWMA-style updating of covariance and correlation

Because volatility and correlation change over time, firms often update covariances with exponentially decaying weights. For mean-zero returns $X_t$ and $Y_t$, a standard EWMA covariance update is:
$$cov_t = \lambda cov_{t-1} + (1-\lambda)X_tY_t$$
Given updated covariance and updated volatilities, correlation is:
$$corr_t = \frac{cov_t}{\sigma_{X,t}\sigma_{Y,t}}$$
This aligns with the FRM idea that correlations can be estimated from covariances, and covariances are updated in a way similar to EWMA volatility updating.

> **Key Term: EWMA covariance**  
> An EWMA covariance estimate updates yesterday’s covariance with today’s cross-product, using decay factor λ so recent data receives higher weight than older data.

### Worked Example 4.1

You track two daily return series $X$ and $Y$ (assume mean zero). Yesterday’s covariance estimate was $cov_{t-1}=0.000020$. Use $\lambda=0.94$. Today’s returns are $X_t=+1.0\%=0.010$ and $Y_t=-0.5\%=-0.005$. Compute the updated covariance. If updated volatilities are $\sigma_{X,t}=1.2\%=0.012$ and $\sigma_{Y,t}=0.9\%=0.009$, compute the updated correlation.

> **Answer:**  
> Step 1: Update covariance:
> $$cov_t = 0.94(0.000020) + 0.06(0.010)(-0.005)$$
> $$= 0.0000188 + 0.06(-0.000050)$$
> $$= 0.0000188 - 0.0000030 = 0.0000158$$
>
> Step 2: Convert to correlation:
> $$corr_t = \frac{0.0000158}{(0.012)(0.009)} = \frac{0.0000158}{0.000108} \approx 0.1463$$
> The updated correlation is about 0.15.

## **Correlation breakdown and stress behavior of matrices**

### What “correlation breakdown” means in risk management

Empirically, correlations are not constant. In periods of heightened volatility, correlations often rise across many asset classes. Practitioners sometimes summarize this as “correlations go to one” in stressed markets; this is exaggerated, but the direction (higher correlations) is widely observed.

This creates a specific model risk problem:

- VaR and ES target tail outcomes.
- If correlations are estimated from calm markets, the portfolio’s diversification benefit may be overstated.
- VaR or ES can be materially understated exactly when capital and liquidity are most needed.

> **Key Term: Correlation breakdown**  
> Correlation breakdown refers to the change (often an increase) in correlations during stressed markets relative to normal periods, reducing diversification benefits and affecting VaR and ES estimates.

### Stress testing correlation and using stressed periods

Stress testing commonly uses:

1. **Historical stress events** (e.g., 1987 equity crash, 1998 Russia default, 2007–2009 crisis, 2008 Lehman default, 2020 COVID-19)
2. **Hypothetical scenarios** (e.g., a 3% rise in short rates, a 4% currency depreciation)

For dependence, stress design often includes correlation shocks because crisis events can trigger a contagion effect: volatility and correlation rise together, weakening diversification.

A related lesson appears in structured credit: pre-crisis models often assumed optimistic default correlations (for example, among mortgage defaults). When those correlations rose, loss distributions shifted sharply to the left tail.

### Governance angle for FRM candidates

Enterprise-wide stress testing is not only a calculation exercise. Governance practices typically require:

- clear policies and documentation of scenario design and correlation assumptions,
- validation and independent review, including recognition that normal-market correlation estimates may not apply in stress.

## **Making matrices usable in Monte Carlo simulation**

Monte Carlo simulation can model many risk factors jointly, but it requires a coherent way to generate correlated random shocks. In practice, this means your estimated correlation or covariance matrix must be mathematically valid.

### Positive semidefinite requirement

For any portfolio weights (or exposure vector) $q$, the variance must be non-negative:
$$q^\top \Sigma q \ge 0$$
This property is guaranteed when $\Sigma$ is **positive semidefinite** (PSD). Estimation error (especially with many assets and limited data) can produce matrices that are not PSD, which can break simulation methods.

> **Key Term: Positive semidefinite matrix**  
> A symmetric matrix Σ is positive semidefinite if, for every vector q:
>
> $$
> q^\top \Sigma q \ge 0
> $$
>
> Covariance matrices must be PSD because variances cannot be negative.

### Cholesky factorization idea (high level)

A common simulation step is to map independent standard normals into correlated normals. If $R$ is a valid correlation matrix, one may compute a matrix $L$ such that:
$$R = LL^\top$$
Then if $z$ is a vector of independent $N(0,1)$ variables, $y = Lz$ has correlation matrix $R$. (Implementation details vary; the exam focus is typically on understanding the role of the matrix and why validity matters.)

> **Key Term: Cholesky decomposition**  
> For a PSD matrix (often a correlation matrix), Cholesky decomposition expresses it as:
>
> $$
> R=LL^\top
> $$
>
> enabling efficient generation of correlated shocks in Monte Carlo simulation.

## **Summary**

Correlation and covariance matrices are the main tools used to represent dependence across many assets or risk factors. They convert large sets of pairwise relationships into a structure that can be used directly in portfolio variance ($w^\top \Sigma w$), delta-normal VaR and ES ($a^\top \Sigma_x a$), and Monte Carlo simulation (generation of correlated scenarios).

For FRM Part 1, you should be ready to (i) compute portfolio risk using matrix or summation formulas, (ii) convert between correlation and covariance matrices via $\Sigma = DRD$, and (iii) explain why correlation estimates must be monitored and stressed. The empirical behavior of markets in stress—higher volatility and generally higher correlations—means that normal-market matrices can understate tail risk unless stressed estimation or scenario analysis is used.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Covariance and correlation definitions and how they differ (scale vs dimensionless)
- Two-asset portfolio volatility and the impact of correlation on diversification
- Correlation matrix vs covariance matrix structure and interpretation
- Conversion formulas: $\Sigma_{ij}=\rho_{ij}\sigma_i\sigma_j$ and $\Sigma = DRD$
- Portfolio variance in matrix form: $\sigma_p^2 = w^\top \Sigma w$
- Delta-normal VaR and ES dependence on the covariance matrix of risk-factor changes
- EWMA-style updating of covariance and converting updated covariance to correlation
- Correlation breakdown in stress and why stressed matrices may be needed for VaR/ES
- Validity conditions (symmetry, PSD) and why they matter for Monte Carlo

## **Key Terms and Concepts**

- Covariance
- Correlation coefficient
- Correlation matrix
- Covariance matrix
- EWMA covariance
- Correlation breakdown
- Positive semidefinite matrix
- Cholesky decomposition


Volatility, correlation, and portfolio models - Correlation and covariance matrices