Market Risk Measurement - VaR and Beyond › Parametric VaR: The Variance-Covariance Method

The parametric approach assumes portfolio returns follow a known distribution, typically the normal distribution. This allows closed-form calculation of VaR using only the mean and standard deviation of returns.

The parametric method is simple and computationally efficient. If we accept the assumption that returns are normally distributed, we can exploit the well-known properties of the normal distribution to derive an explicit formula for VaR. This avoids the need for simulation or extensive historical databases, making the method particularly attractive for large portfolios where computational speed matters.

For a single asset with normally distributed returns having mean $\mu$ and standard deviation $\sigma$, we can determine any quantile of the distribution using the standard normal transformation. The key insight is that any normal distribution can be expressed as a linear transformation of the standard normal distribution, which has mean zero and standard deviation one.

The $(1-\alpha)$ quantile of returns is:

$$R_{1-\alpha} = \mu + \sigma \cdot z_{1-\alpha}$$

where:

• $R_{1-\alpha}$: $(1-\alpha)$ quantile of returns
• $\mu$: mean return
• $\sigma$: return standard deviation
• $z_{1-\alpha}$: $(1-\alpha)$ quantile of the standard normal distribution

This formula follows directly from the properties of normal distributions. If $R \sim N(\mu, \sigma^2)$, then $Z = (R - \mu)/\sigma$ follows the standard normal $N(0,1)$. Solving for $R$ at the point where $Z = z_{1-\alpha}$ gives us the quantile formula above.

For common confidence levels, $z_{0.05} \approx -1.645$ and $z_{0.01} \approx -2.326$. These values indicate how many standard deviations below the mean we must go to reach the specified tail probability. The negative sign reflects that we are looking at the left tail of the distribution, where the worst returns reside.

The parametric VaR for a portfolio of value $V$ is derived by scaling the return quantile by the portfolio value:

$$\begin{aligned} \text{VaR}_{\alpha} &= -V \cdot R_{1-\alpha} && \text{(scale return quantile by value)} \\ &= -V(\mu + \sigma \cdot z_{1-\alpha}) && \text{(substitute parametric return model)} \end{aligned}$$

where:

• $\text{VaR}_{\alpha}$: Value-at-Risk
• $V$: portfolio value
• $\mu$: expected return
• $\sigma$: volatility
• $z_{1-\alpha}$: standard normal quantile

Let us trace through this derivation more carefully. We start with the relationship $\text{VaR}_{\alpha} = -V \cdot R_{1-\alpha}$, which converts the worst expected return into a dollar loss. Substituting our expression for the return quantile, we obtain $\text{VaR}_{\alpha} = -V \cdot (\mu + \sigma \cdot z_{1-\alpha})$. Since $z_{1-\alpha}$ is negative for tail quantiles (we are looking at returns below the mean), the product $\sigma \cdot z_{1-\alpha}$ is also negative, and subtracting this negative quantity effectively adds a positive risk term.

For short time horizons like one day, the expected return $\mu$ is typically negligible compared to volatility ($\mu \approx 0$). To see why, consider that a typical stock might have an expected daily return of 0.04% (about 10% annualized), while daily volatility might be 2%. The volatility term dominates the calculation by roughly a factor of 50. Since $z_{1-\alpha}$ is negative for tail quantiles, $-z_{1-\alpha} = |z_{1-\alpha}|$, allowing us to simplify:

$$\text{VaR}_{\alpha} \approx V \cdot \sigma \cdot |z_{1-\alpha}|$$

where:

• $\text{VaR}_{\alpha}$: simplified Value-at-Risk
• $V$: portfolio value
• $\sigma$: volatility
• $|z_{1-\alpha}|$: absolute value of the quantile (e.g., 1.645 for 95% confidence)

This simplified formula reveals the intuitive structure of parametric VaR. The risk measure is proportional to portfolio value, volatility, and a multiplier that depends on the confidence level. For 95% confidence, the multiplier is 1.645; for 99% confidence, it increases to 2.326. This scaling reflects the deeper penetration into the tail required for higher confidence levels.

Let's implement this for a stock portfolio using historical data to estimate volatility:

For a portfolio of multiple assets, we need to account for correlations between asset returns. As we discussed in Part IV on Modern Portfolio Theory, the variance of a portfolio depends on the covariance matrix of its constituents. When assets move together, portfolio risk is higher; when they move independently or inversely, diversification reduces total risk below the weighted sum of individual risks.

The mathematical framework for multi-asset VaR builds directly on portfolio theory. Consider a portfolio with $n$ assets, where we invest fraction $w_i$ in asset $i$. The portfolio return is the weighted sum of individual returns: $R_p = \sum_{i=1}^{n} w_i R_i$. Since the sum of normally distributed random variables is also normal, the portfolio return inherits normality from its constituents.

For a portfolio with weight vector $\mathbf{w}$ and asset covariance matrix $\boldsymbol{\Sigma}$, the portfolio variance is:

$$\sigma_p^2 = \mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w}$$

where:

• $\sigma_p^2$: portfolio variance
• $\mathbf{w}$: vector of portfolio weights
• $\boldsymbol{\Sigma}$: covariance matrix of asset returns
• $^T$: transpose operator

This quadratic form captures all pairwise interactions between assets. The diagonal elements of $\boldsymbol{\Sigma}$ contribute weighted variances, while the off-diagonal elements contribute weighted covariances. When assets are less than perfectly correlated, the cross-product terms partially cancel, reducing portfolio variance below what a simple weighted average would suggest.

The parametric VaR extends naturally by substituting portfolio volatility $\sigma_p = \sqrt{\mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w}}$ into our single-asset formula:

$$\text{VaR}_{\alpha} = V \cdot \sqrt{\mathbf{w}^T \boldsymbol{\Sigma} \mathbf{w}} \cdot |z_{1-\alpha}|$$

where:

• $\text{VaR}_{\alpha}$: Value-at-Risk
• $V$: total portfolio value
• $\mathbf{w}$: weight vector
• $\boldsymbol{\Sigma}$: covariance matrix
• $z_{1-\alpha}$: standard normal quantile

This formula demonstrates that portfolio VaR depends on the entire covariance structure, not just individual asset volatilities. Two portfolios with identical asset volatilities can have dramatically different VaR depending on their correlation structure. A portfolio of highly correlated assets concentrates risk, while a portfolio of uncorrelated or negatively correlated assets disperses it.

The parametric method is computationally fast, requires only estimates of means, volatilities, and correlations, and provides intuitive decomposition of risk contributions. Financial institutions with thousands of positions can compute VaR almost instantaneously.

However, the normality assumption is problematic. As we explored in the chapter on Stylized Facts of Financial Returns, actual return distributions exhibit fat tails and excess kurtosis. The normal distribution underestimates the probability of extreme events. A return of 4 standard deviations should occur about once every 126 years under normality, yet markets regularly experience such moves several times per decade.