## **Learning Outcomes**

This article explains how variance–covariance (delta-normal) VaR works, how to compute it for portfolios and linear derivatives, and how to extend the same framework to expected shortfall (ES). After reading, you will be able to:

- Calculate portfolio volatility using a variance–covariance matrix and portfolio weights (or factor sensitivities)
- Compute delta-normal VaR and ES for a normally distributed portfolio P&L at a specified confidence level and holding period
- Explain the key assumptions, common implementation choices, and limitations of the delta-normal approach (including when it can misstate risk)

## **GARP FRM Part 1 Syllabus**

For the GARP FRM Part 1 exam, you are expected to understand parametric market risk measurement with a focus on the following syllabus points:

- Define and interpret VaR, including the roles of confidence level, holding period, and distributional assumptions
- Explain and calculate ES, and compare VaR vs. ES (including coherence and subadditivity)
- Apply the delta-normal method to linear portfolios and linear derivatives using variance–covariance inputs (volatilities and correlations)

## **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. The delta-normal (variance–covariance) VaR method relies most directly on which assumption?  
   a) Asset prices are lognormal  
   b) Portfolio P&L is normally distributed because the portfolio is linear in risk factors  
   c) The loss distribution is uniform in the tail  
   d) Correlations are zero

2. A two-asset portfolio has weights $w_1=0.7$, $w_2=0.3$, daily volatilities $\sigma_1=1.4\%$, $\sigma_2=0.9\%$, and correlation $\rho_{12}=0.25$. Compute the daily portfolio volatility $\sigma_p$.

3. In one sentence, explain why VaR provides no information about “how bad it gets” beyond the VaR threshold.

4. True/False (brief explanation required): VaR is always subadditive, so the VaR of a diversified portfolio cannot exceed the sum of stand-alone VaRs.

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

- Can you convert between a confidence level (e.g., 99%) and the correct one-tailed $z$-value (e.g., 2.326)?
- Can you compute portfolio variance using weights, volatilities, and correlations (or $w^\top \Sigma w$)?
- Do you know when the square-root-of-time rule is used, and what assumptions make it reasonable?

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

## **Introduction**

Variance–covariance VaR (often called delta-normal VaR) is a parametric method for estimating downside market risk. It is popular because it is fast: once you have portfolio volatility (from volatilities and correlations), VaR is obtained with a single standard normal quantile.

The method sits naturally beside the mean–variance framework. If returns (or factor changes) are elliptically distributed—most commonly assumed to be normal—then a portfolio that is linear in its risk factors has a normally distributed change in value. Under that assumption, VaR and ES become simple closed-form calculations using the portfolio mean, standard deviation, and a $z$-score.

However, the FRM exam also expects you to understand what can go wrong. Risky asset returns frequently show fatter tails and more extreme outcomes than the normal distribution predicts. In addition, correlation patterns can change in stress, and non-linear instruments (options) are only approximately handled by a delta-based method.

> **Key Term: Variance–covariance (delta-normal) VaR**  
> A parametric VaR method that assumes portfolio value changes are normally distributed, with variance computed from a variance–covariance matrix (and either portfolio weights or factor sensitivities).

## **Delta-Normal VaR and Its Assumptions**

Delta-normal VaR is best understood as two linked ideas:

1. **Linearity of portfolio P&L in risk factors**  
   Over the holding period (often one day), approximate the portfolio change as a linear function of movements in the risk factors.

2. **Normal (or elliptical) distribution for factor changes**  
   Assume the vector of risk-factor changes is multivariate normal. A linear combination of jointly normal variables is normal, so portfolio P&L is normal.

A common representation uses a vector of risk-factor changes $\Delta x$ and a vector of sensitivities (deltas) $\delta$:
$$\Delta P \approx \delta^\top \Delta x$$
If $\Delta x \sim N(\mu_x, \Sigma_x)$, then:
$$\Delta P \sim N(\mu_p, \sigma_p^2)$$
with:
$$\mu_p = \delta^\top \mu_x$$
$$\sigma_p^2 = \delta^\top \Sigma_x \delta$$
When the portfolio is simply a set of cash positions in assets (e.g., equities), you often work with **returns** instead of factor changes. If $R$ is the vector of asset returns and $w$ are portfolio weights:
$$R_p = w^\top R$$
$$\mu_p = w^\top \mu$$
$$\sigma_p^2 = w^\top \Sigma w$$

> **Key Term: Linear portfolio**  
> A portfolio whose value change over a short horizon can be approximated as a linear function of changes in its risk factors (e.g., stocks, forwards, futures; options are not linear).

### Why the normality assumption matters

Under normality, tail quantiles are determined entirely by $\mu$ and $\sigma$. But empirical returns on risky assets often have:

- **fat tails** (more large moves than normal)
- **time-varying volatility** (conditional vs. unconditional distributions)
- **correlation shifts** during market stress

These features can cause delta-normal VaR to understate losses during adverse periods, even if it performs reasonably in calmer conditions.

## **Portfolio Volatility Using the Variance–Covariance Matrix**

The “variance–covariance” label reflects that the method first computes portfolio variance from a covariance structure.

### Two-asset portfolio volatility

For two assets with weights $w_1, w_2$, volatilities $\sigma_1, \sigma_2$, and correlation $\rho_{12}$:
$$\sigma_p = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2\rho_{12}}$$
This highlights diversification: if $\rho_{12} \lt 1$, the third term is smaller than the "no diversification" case.

### Multi-asset portfolio volatility

For $n$ assets:
$$\mu_p = \sum_{i=1}^n w_i\mu_i$$

$$\sigma_p = \sqrt{\sum_{i=1}^n\sum_{j=1}^n w_i w_j \sigma_i \sigma_j \rho_{ij}}$$
In matrix notation:
$$\sigma_p = \sqrt{w^\top \Sigma w}$$
where $\Sigma$ is the covariance matrix with elements $\Sigma_{ij}=\sigma_i\sigma_j\rho_{ij}$.

> **Key Term: Variance–covariance matrix**  
> A matrix (Σ) whose diagonal elements are variances and off-diagonal elements are covariances. It summarizes how risk factors move together and determines portfolio variance via:
>
> $$
> w^\top \Sigma w
> $$
>
> or:
>
> $$
> \delta^\top \Sigma \delta
> $$

### Worked Example 1.1

A two-asset portfolio has $w_1=0.6$, $w_2=0.4$, daily volatilities $\sigma_1=1.2\%=0.012$, $\sigma_2=0.8\%=0.008$, and correlation $\rho_{12}=0.30$. 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:
>
> $$
> w_1^2\sigma_1^2 = 0.6^2 \times 0.012^2 = 0.36 \times 0.000144 = 0.00005184
> $$
>
> $$
> w_2^2\sigma_2^2 = 0.4^2 \times 0.008^2 = 0.16 \times 0.000064 = 0.00001024
> $$
>
> $$
> 2w_1w_2\sigma_1\sigma_2\rho_{12} = 2(0.6)(0.4)(0.012)(0.008)(0.30)
> $$
>
> $$
> = 0.48 \times 0.000096 \times 0.30 = 0.000013824
> $$
>
> Sum:
> $$\sigma_p^2 = 0.00005184 + 0.00001024 + 0.000013824 = 0.000075904$$
>
> Take the square root:
> $$\sigma_p = \sqrt{0.000075904} \approx 0.00872 = 0.872\% \text{ per day}$$

## **Computing Delta-Normal VaR in Money Terms**

Once $\mu_p$ and $\sigma_p$ are known, the delta-normal method uses a standard normal quantile.

### VaR as a quantile return (Kaplan convention)

A common FRM-friendly expression for VaR in **return space** at confidence level $x$ is:
$$\text{VaR}_{\text{return}} = \mu_p - z\sigma_p$$
where $z$ is the standard normal value with tail probability $(1-x)$ (e.g., $z=1.645$ for 95% one-tailed).

To convert to a money amount for a portfolio with value $V_0$:
$$\text{VaR}_{\$} = (\mu_p - z\sigma_p)\times V_0$$
This will often be a negative number because it is a left-tail return. Risk reports usually quote **loss magnitude** as a positive number:
$$\text{VaR}_{\text{loss}} = (z\sigma_p - \mu_p)\times V_0$$

> **Key Term: Confidence level and tail probability**  
> The confidence level x (e.g., 99%) corresponds to a tail probability (1-x) (e.g., 1%). VaR focuses on that tail region of adverse outcomes.

### Worked Example 2.1

Using $\sigma_p=0.872\%$ from Worked Example 1.1, assume:

- portfolio value $V_0$ = \$50,000,000
- expected daily return $\mu_p=0.02\%=0.0002$
- confidence level $x=99\%$ so $z=2.326$

Compute the 1-day delta-normal VaR.

> **Answer:**  
> Compute the 1-day VaR quantile return:
> $$\text{VaR}_{\text{return}} = \mu_p - z\sigma_p = 0.0002 - 2.326(0.00872)$$
> $$= 0.0002 - 0.02028 = -0.02008$$
>
> Convert to dollars:
> $$\text{VaR}_{\$} = (-0.02008)\times \$50{,}000{,}000 = -\$1{,}004{,}000$$
>
> Report as loss magnitude:
> $$\text{VaR}_{\text{loss}} = \$1{,}004{,}000$$
>
> Interpretation: under the delta-normal model, the manager is 99% confident that the portfolio will not lose more than about \$1.004 million over one day (under “normal” conditions implied by the model).

### **Exam Warning**

> Common exam errors include (1) using a two-tailed z instead of a one-tailed z, (2) mixing percent returns with currency P&L, (3) reporting the negative quantile return as “the VaR” without converting to a positive loss amount, and (4) using (1-x) incorrectly in ES formulas.

## **Expected Shortfall Under the Normality Assumption**

VaR tells you a quantile threshold. It does not describe the average loss beyond that threshold. Expected shortfall (ES) addresses this by taking the conditional mean of tail losses.

Conceptually, at confidence level $x$:

- VaR: “loss threshold exceeded with probability $(1-x)$”
- ES: “average loss given that we are in the worst $(1-x)$ of outcomes”

For a normal distribution, ES has a closed form. Using the notation in the syllabus readings:
$$\text{ES} = \mu_p + \sigma_p \frac{e^{-z^2/2}}{(1-x)\sqrt{2\pi}}$$
A useful identity is the standard normal density $\phi(z)$:
$$\phi(z) = \frac{1}{\sqrt{2\pi}}e^{-z^2/2}$$
so:
$$\text{ES} = \mu_p + \sigma_p\frac{\phi(z)}{1-x}$$
When ES is computed as a **loss magnitude**, it will exceed VaR for the same $x$ and horizon.

> **Key Term: Expected shortfall**  
> The expected (average) loss conditional on losses exceeding the VaR threshold; also called conditional VaR or expected tail loss.

### Worked Example 3.1

Continue Worked Example 2.1 with $\mu_p=0.0002$, $\sigma_p=0.00872$, $x=0.99$, and $z=2.326$. Compute 1-day ES (loss magnitude) and compare it to VaR.

> **Answer:**  
> First compute φ(z):
> $$\phi(2.326)=\frac{1}{\sqrt{2\pi}}e^{-(2.326^2/2)}$$
> Compute the exponent:
> $$2.326^2/2 \approx 5.411/2 = 2.7055$$
> $$e^{-2.7055}\approx 0.0667$$
> $$\frac{1}{\sqrt{2\pi}}\approx 0.3989$$
> So:
> $$\phi(2.326)\approx 0.3989 \times 0.0667 = 0.0266$$
>
> Now compute ES (as a tail loss rate in decimal form):
> $$\text{ES} \approx \mu_p + \sigma_p\frac{\phi(z)}{1-x} = 0.0002 + 0.00872\frac{0.0266}{0.01}$$
> $$= 0.0002 + 0.00872(2.66) = 0.0002 + 0.0232 = 0.0234$$
>
> Convert to dollars:
> $$\text{ES}_{\text{loss}} \approx 0.0234 \times \$50{,}000{,}000 = \$1{,}170{,}000$$
>
> Comparison: VaR loss was about \$1.004 million, while ES loss is about \$1.17 million. ES is larger because it averages losses deeper into the tail than the VaR cutoff.

## **Applying Delta-Normal VaR to Linear Derivatives**

For derivatives whose value moves linearly with a single risk factor (or can be treated as linear over the horizon), VaR can be obtained by scaling the factor’s VaR by a sensitivity.

For a single risk factor $S$:
$$\delta = \frac{\Delta P}{\Delta S}$$
For a long position in a linear derivative, a common relationship is:
$$\text{VaR}_P = \delta \,\text{VaR}_S$$
Here, $\text{VaR}_S$ is the VaR of that risk factor (in the same units as $\Delta S$), and $\delta$ converts a factor move into a P&L move.

This is the same variance–covariance logic written in a one-factor sensitivity form: if $P=\delta S$ (linear), then the standard deviation of $P$ is $|\delta|$ times the standard deviation of $S$, and quantiles scale the same way.

### Worked Example 4.1

A long equity index futures position gains \$500 for every 1-point increase in the index. The 1-day 99% VaR of the index level is 80 points (interpreted as an adverse 80-point move). Estimate the 1-day 99% VaR of the futures position.

> **Answer:**  
> The delta (dollar change per index point) is:
> $$\delta = \$500 \text{ per point}$$
> The futures VaR is:
> $$\text{VaR}_{\text{fut}} = \$500 \times 80 = \$40{,}000$$
> Interpretation: with 99% confidence (under the model producing the 80-point index VaR), the 1-day loss on the futures position should not exceed about \$40,000.

## **Scaling VaR Across Holding Periods**

VaR depends on the holding period. Under a common simplifying assumption that daily returns are independent and identically distributed with mean approximately zero, volatility scales with the square root of time:
$$\sigma(T) \approx \sigma(1)\sqrt{T}$$
This leads to the standard scaling rules:
$$\text{VaR}(T, x) = \text{VaR}(1, x)\sqrt{T}$$
$$\text{ES}(T, x) = \text{ES}(1, x)\sqrt{T}$$
In practice, this scaling can be poor when volatility is time-varying (mean-reverting) or when returns are not independent across days.

### Worked Example 5.1

A portfolio has 1-day 99% VaR of \$1.2 million and 1-day 99% ES of \$1.5 million (both computed under a delta-normal framework with $\mu\approx 0$). Estimate 10-day VaR and ES using square-root-of-time scaling.

> **Answer:**  
> Compute sqrt(10) ≈ 3.1623.
>
> $$\text{VaR}(10,0.99) \approx \$1.2\text{ million} \times 3.1623 = \$3.795\text{ million}$$
>
> $$\text{ES}(10,0.99) \approx \$1.5\text{ million} \times 3.1623 = \$4.743\text{ million}$$

## **Model Risk and Method Limitations**

Delta-normal VaR is fast and transparent, but its accuracy depends on assumptions and implementation choices.

### Distributional mismatch and tail risk

Risky asset returns are often more peaked with fatter tails than a normal distribution. If the true distribution has more tail probability than normal, a normal-based VaR will tend to understate losses at high confidence levels (e.g., 99%).

### Correlation instability

Variance–covariance VaR relies heavily on correlations. During crises, correlations often rise and diversification benefits shrink. If correlations used in the covariance matrix are estimated from calm periods, VaR can be too low exactly when it is most needed.

### Non-linearity and large moves

The delta-normal method is exact for linear portfolios. For portfolios with options and other non-linear payoffs, delta is only a local linear approximation. For large moves, ignoring curvature (gamma) can distort the loss distribution.

### Model risk and implementation risk

VaR and ES are subject to:

- **model risk** (wrong distributional assumptions, unstable correlations, inappropriate mapping from positions to risk factors)
- **implementation risk** (coding errors, data issues, incorrect $z$ values, wrong time scaling)

These issues help explain why institutions often use VaR/ES together with stress testing and scenario analysis, especially to examine extreme but plausible environments.

> **Key Term: Model risk and implementation risk**  
> Model risk is error from incorrect model assumptions; implementation risk is error from how the model is built, coded, parameterized, or run (including data handling).

## **Connecting VaR and ES to Coherent Risk Measurement**

A coherent risk measure satisfies four properties:

- Monotonicity: $Y \ge X \Rightarrow \rho(Y) \le \rho(X)$
- Subadditivity: $\rho(X+Y) \le \rho(X) + \rho(Y)$
- Positive homogeneity: $\rho(\beta X)=\beta \rho(X)$ for $\beta>0$
- Translation invariance: $\rho(c+X)=\rho(X)-c$

ES satisfies all four properties under general conditions and is therefore coherent. VaR, however, can fail subadditivity when returns are not elliptical (for example, in certain skewed or fat-tailed settings). This matters because subadditivity formalizes diversification: portfolio risk should not exceed the sum of stand-alone risks.

> **Key Term: Subadditivity**  
> The requirement that a diversified portfolio’s risk is no greater than the sum of risks of its components:
>
> $$
> \rho(X+Y)\le \rho(X)+\rho(Y)
> $$
>
> ES satisfies this property; VaR may not.

From a portfolio construction viewpoint, subadditivity is linked to convexity of the “risk surface” as confidence level and holding period change. ES tends to behave better in optimization settings because it respects diversification more reliably than VaR.

## **Summary**

Variance–covariance (delta-normal) VaR is a parametric risk measure computed from a portfolio mean and standard deviation, with the standard deviation obtained from a variance–covariance matrix. When portfolio P&L is linear in its risk factors and factor changes are multivariate normal (or approximately so), portfolio P&L is normal and VaR is a straightforward $z$-score calculation.

Expected shortfall extends VaR by estimating the average loss beyond the VaR threshold. Under the normality assumption, ES also has a closed-form formula and, unlike VaR, it satisfies subadditivity and the other coherence properties.

For FRM Part 1, you should be able to compute portfolio volatility from weights, volatilities, and correlations; calculate delta-normal VaR and ES; and explain when and why the method can misstate risk (fat tails, correlation shifts, and non-linear instruments).

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Delta-normal (variance–covariance) VaR assumes linear P&L and normally distributed risk-factor changes
- Portfolio volatility can be computed via $w^\top\Sigma w$ (or $\delta^\top\Sigma\delta$)
- VaR at confidence level $x$ uses a one-tailed $z$ and is often reported as a positive loss magnitude
- ES estimates average tail loss and exceeds VaR for the same $x$ and horizon
- Square-root-of-time scaling is a simplifying rule that depends on independence and stable volatility
- VaR may violate subadditivity for non-elliptical distributions; ES is a coherent risk measure
- Model risk and implementation risk are practical concerns for any VaR/ES system

## **Key Terms and Concepts**

- Variance–covariance (delta-normal) VaR
- Linear portfolio
- Variance–covariance matrix
- Confidence level and tail probability
- Expected shortfall (ES)
- Subadditivity
- Model risk
- Implementation risk
- Square-root-of-time rule
- Elliptical distributions


Value-at-Risk (VaR) and Expected Shortfall - Variance–covariance (Delta-Normal) VaR