## **Learning Outcomes**

This article explains how time-varying volatility is modeled using EWMA and GARCH frameworks and how these models support risk measurement (e.g., VaR). After reading, you will be able to:

- Explain volatility clustering and conditional heteroskedasticity in financial return series
- Calculate EWMA conditional variance and interpret the decay factor and half-life
- Specify and forecast volatility using a GARCH(1,1) model, including long-run variance and multi-step forecasts

## **GARP FRM Part 1 Syllabus**

For the GARP FRM Part 1 exam, you are expected to understand time-series behavior of returns and volatility with a focus on the following syllabus points:

- Measuring and scaling volatility (variance rate vs volatility; daily-to-annual conversion)
- White noise vs dependent white noise, and why returns can have unpredictable means but predictable variances
- Conditional heteroskedasticity models used in risk measurement, including EWMA and GARCH-type volatility forecasting

## **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. In an EWMA variance model with decay factor $\lambda$, the weight on the squared return from $k$ days ago (i.e., $r_{t-k}^2$) is:  
   a) $\lambda^k$  
   b) $(1-\lambda)\lambda^{k-1}$  
   c) $(1-\lambda)^k$  
   d) $\frac{1}{k}$

2. Calculation: An analyst uses EWMA with $\lambda=0.94$. Yesterday’s conditional variance estimate was $\sigma_{t-1}^2=0.00010$ and yesterday’s return was $r_{t-1}=-1.5\%=-0.015$. Compute today’s variance estimate $\sigma_t^2$.

3. Short answer: A GARCH(1,1) model is $\sigma_t^2=\omega+\alpha \epsilon_{t-1}^2+\beta\sigma_{t-1}^2$ with $\omega=0.000002$, $\alpha=0.05$, $\beta=0.90$. What is the long-run (unconditional) variance?

4. True/False (explain): If $\alpha+\beta=1$ in GARCH(1,1), the model has a finite unconditional variance equal to $\omega/(1-\alpha-\beta)$.

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

- Can you explain why returns often look close to white noise, but squared returns show serial dependence?
- Can you compute volatility scaling (daily to annual) without mixing variance and volatility?
- Can you write a one-step-ahead conditional variance forecast for EWMA and for GARCH(1,1)?

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

## **Introduction**

Risk managers rarely treat volatility as constant. Market data commonly show **volatility clustering**: calm periods are followed by calm periods, and turbulent periods are followed by turbulent periods. This matters directly for market risk measurement because volatility is a main input to tools such as Value-at-Risk (VaR), Expected Shortfall (ES), option valuation, and margin setting.

At the same time, many asset return series have weak linear predictability in the mean (daily or weekly returns are often close to unpredictable). This combination—unpredictable mean with time-varying variance—motivates a class of models where the **conditional variance** changes over time even when the conditional mean is stable.

> **Key Term: Volatility clustering**  
> A common empirical pattern where large absolute returns tend to be followed by large absolute returns (of either sign), and small absolute returns tend to be followed by small absolute returns.

## **Volatility as a time-series object**

### Returns, variance, and scaling

Returns can be measured as simple returns or log returns. For prices $P_t$:

- Simple return: $R_t=\frac{P_t-P_{t-1}}{P_{t-1}}$
- Log return: $r_t=\ln(P_t)-\ln(P_{t-1})$

Volatility is the standard deviation of returns, $\sigma$, and variance is $\sigma^2$. Scaling across time is done using variance:

- If daily variance is $\sigma_d^2$, annual variance is approximately $252\sigma_d^2$
- Annual volatility is $\sigma_{\text{ann}}=\sqrt{252\sigma_d^2}=\sqrt{252}\,\sigma_d$

A frequent exam error is scaling volatility linearly (incorrect) instead of scaling variance linearly (correct).

> **Key Term: Variance rate**  
> The variance of returns per unit of time (e.g., daily variance). Scaling to longer horizons is done by multiplying the variance rate by the number of periods, then taking a square root to get volatility.

### White noise vs dependent white noise

A white noise process has mean zero, finite variance, and zero autocorrelation across time. In finance, returns may have little autocorrelation, which makes them resemble white noise in the mean. However, financial returns often violate independence: volatility varies over time and depends on past shocks.

This is a leading example of **dependent white noise**: shocks are uncorrelated over time, but their variance changes predictably with past information. This is the statistical basis for ARCH/GARCH-type modeling of volatility clustering.

> **Key Term: Conditional heteroskedasticity**  
> A time-series feature where the variance of the shock (or return) at time t depends on past information, so variance is not constant over time.

### Conditional variance setup used by EWMA and GARCH

A standard way to express time-varying volatility is:
$$r_t=\mu+\epsilon_t,\quad \epsilon_t=\sigma_t z_t$$
where $z_t$ is often modeled as iid with $E[z_t]=0$ and $Var(z_t)=1$ (normality is optional in the time-series structure, but often assumed for estimation).

The goal is to model the **conditional variance**:
$$\sigma_t^2 = Var(\epsilon_t \mid \mathcal{F}_{t-1})$$
where $\mathcal{F}_{t-1}$ is the information set at time $t-1$ (including past returns).

Two widely tested approaches in practice and in the FRM curriculum are:

- EWMA: a parsimonious recursion that updates variance using exponentially decaying weights
- GARCH(1,1): a parametric model that updates variance using both last period’s squared shock and last period’s variance and includes mean reversion

## **EWMA volatility model**

EWMA (exponentially weighted moving average) models conditional variance as a weighted average of the previous variance estimate and the most recent squared return (or squared residual).

A common EWMA recursion (assuming $\mu=0$ or using demeaned returns) is:
$$\sigma_t^2=\lambda \sigma_{t-1}^2 + (1-\lambda) r_{t-1}^2,\quad 0<\lambda<1$$

> **Key Term: EWMA**  
> A volatility model where the current variance estimate is an exponentially weighted average of past squared returns, implemented through a simple recursion with decay factor λ.

### EWMA weights and effective memory

Expanding the recursion shows the exponential weighting:
$$\sigma_t^2=(1-\lambda)\sum_{i=1}^{\infty}\lambda^{i-1} r_{t-i}^2$$

So the weight on $r_{t-k}^2$ is:
$$w_k=(1-\lambda)\lambda^{k-1}$$

A larger $\lambda$ means slower decay (longer memory); a smaller $\lambda$ means faster decay (more weight on recent moves).

### Half-life interpretation

A useful way to interpret $\lambda$ is the **half-life**: the number of periods until a weight falls to half its initial value. The ratio of weights $k$ periods apart is $\lambda^k$, so:
$$\lambda^{h}=0.5 \quad\Rightarrow\quad h=\frac{\ln(0.5)}{\ln(\lambda)}$$

This is commonly tested qualitatively and occasionally quantitatively.

### Multi-step variance forecasts under EWMA

EWMA is often used as a one-step-ahead volatility forecast:
$$\sigma_{t+1|t}^2=\lambda \sigma_{t}^2 + (1-\lambda) r_{t}^2$$

For horizons beyond one step, a common exam-relevant point is that EWMA has no mean reversion term (no intercept). As a result, once you move beyond the next period, the forecast typically remains flat at the last conditional variance estimate (under the basic RiskMetrics-style EWMA).

### Worked Example 1.1

An analyst uses EWMA with $\lambda=0.94$. Yesterday’s conditional variance estimate was $\sigma_{t-1}^2=0.00010$ and yesterday’s return was $r_{t-1}=-0.015$. Compute today’s variance estimate $\sigma_t^2$ and today’s annualized volatility (assume 252 trading days).

> **Answer:**  
> Step 1: Square the return.
>
> $$
> r_{t-1}^2=(-0.015)^2=0.000225
> $$
>
> Step 2: Apply EWMA recursion.  
> $$\sigma_t^2=0.94(0.00010)+0.06(0.000225)$$  
> $$\sigma_t^2=0.000094+0.0000135=0.0001075$$
>
> Step 3: Convert variance to daily volatility.  
> $$\sigma_t=\sqrt{0.0001075}\approx 0.01037=1.037\%$$
>
> Step 4: Annualize volatility.  
> $$\sigma_{\text{ann}}=\sqrt{252}\,\sigma_t\approx 15.874 \times 0.01037 \approx 0.1646$$  
> Annualized volatility ≈ 16.46%.

### Worked Example 1.2

Compute the half-life (in days) of an EWMA model with $\lambda=0.94$.

> **Answer:**  
> $$h=\frac{\ln(0.5)}{\ln(0.94)}$$
> Using:
>
> $$
> \ln(0.5)\approx -0.6931,\quad \ln(0.94)\approx -0.06188
> $$
>
> $$h\approx \frac{-0.6931}{-0.06188}\approx 11.2\text{ days}$$
> Interpretation: after about 11 trading days, the weight on a squared return is about half the weight on the most recent squared return.

## **GARCH volatility models**

EWMA is simple and widely used, but it has a structural limitation: it does not include a long-run variance level. GARCH models extend ARCH ideas by allowing variance to depend on both recent squared shocks and past variance, typically leading to volatility that gradually reverts toward a long-run level.

### From ARCH to GARCH

In an ARCH(1) model, conditional variance depends on the last squared shock:
$$\sigma_t^2=\omega+\alpha \epsilon_{t-1}^2$$

In a GARCH(1,1) model, conditional variance depends on the last squared shock and the last variance:
$$\sigma_t^2=\omega+\alpha \epsilon_{t-1}^2+\beta\sigma_{t-1}^2$$

> **Key Term: GARCH(1,1)**  
> A conditional variance model where today’s variance depends on a constant term, yesterday’s squared shock, and yesterday’s variance.

Here, $\epsilon_t=r_t-\mu$ if a mean is modeled, or $\epsilon_t=r_t$ if returns are treated as mean-zero.

### Parameter constraints and stationarity

For the variance to remain non-negative (a basic requirement), typical constraints are:

- $\omega>0$
- $\alpha\ge 0$
- $\beta\ge 0$

A standard covariance-stationary condition for GARCH(1,1) (finite unconditional variance) is:
$$\alpha+\beta<1$$

When $\alpha+\beta$ is close to 1, volatility is highly persistent: shocks to variance decay slowly.

> **Key Term: Volatility persistence**  
> The degree to which a volatility shock carries forward into the future. In GARCH(1,1), persistence is largely summarized by α + β.

### Long-run (unconditional) variance and mean reversion

If $\alpha+\beta<1$, the long-run variance is:
$$\sigma^2_{\text{LR}}=\frac{\omega}{1-\alpha-\beta}$$

This provides an anchor level. When current variance is above $\sigma^2_{\text{LR}}$, the model tends to forecast declining variance; when below, it tends to forecast rising variance.

### One-step and multi-step forecasting

Given information at time $t$, the one-step-ahead variance forecast is:
$$\sigma_{t+1|t}^2=\omega+\alpha \epsilon_t^2+\beta\sigma_t^2$$

A common multi-step result (for $h\ge 2$) is:
$$\sigma_{t+h|t}^2=\sigma^2_{\text{LR}}+(\alpha+\beta)^{h-1}\left(\sigma_{t+1|t}^2-\sigma^2_{\text{LR}}\right)$$

This shows explicit mean reversion toward $\sigma^2_{\text{LR}}$.

### Worked Example 2.1

A desk models daily returns with GARCH(1,1):
$$\sigma_t^2=\omega+\alpha \epsilon_{t-1}^2+\beta\sigma_{t-1}^2$$
Given $\omega=0.000002$, $\alpha=0.05$, $\beta=0.90$, yesterday’s variance $\sigma_t^2=0.00010$, and today’s shock $\epsilon_t=-0.012$, compute:

1. The long-run variance
2. Tomorrow’s variance forecast $\sigma_{t+1|t}^2$
3. The 10-day-ahead variance forecast $\sigma_{t+10|t}^2$

> **Answer:**  
> Step 1: Long-run variance.  
> First compute:
>
> $$
> 1-\alpha-\beta=1-0.05-0.90=0.05
> $$
>
> $$\sigma^2_{\text{LR}}=\frac{0.000002}{0.05}=0.00004$$
>
> Step 2: One-step-ahead forecast.  
> Compute today’s squared shock:
>
> $$
> \epsilon_t^2=(-0.012)^2=0.000144
> $$
>
> $$\sigma_{t+1|t}^2=0.000002+0.05(0.000144)+0.90(0.00010)$$  
> $$\sigma_{t+1|t}^2=0.000002+0.0000072+0.00009=0.0000992$$
>
> Step 3: 10-step-ahead forecast.  
> Persistence is α + β = 0.95. Use:  
> $$\sigma_{t+10|t}^2=\sigma^2_{\text{LR}}+(\alpha+\beta)^{9}\left(\sigma_{t+1|t}^2-\sigma^2_{\text{LR}}\right)$$  
> $$\sigma_{t+10|t}^2=0.00004+0.95^{9}(0.0000992-0.00004)$$
> Compute the gap:
>
> $$
> 0.0000992-0.00004=0.0000592
> $$
>
> Compute (rounded):
>
> $$
> 0.95^9\approx 0.630
> $$
>
> $$\sigma_{t+10|t}^2\approx 0.00004+0.630(0.0000592)=0.00004+0.0000373=0.0000773$$
> Interpretation: variance is forecast to move toward the long-run level 0.00004, but slowly because persistence is high.

### **Exam Warning**

> Many candidates confuse εₜ² with rₜ² and also confuse variance with volatility. In GARCH, εₜ is the mean-corrected shock. When asked for volatility, take the square root at the end, and only then annualize.

## **Model comparison and risk management applications**

### EWMA vs GARCH in practice

EWMA and GARCH can both fit volatility clustering, but they differ structurally:

- **EWMA**
  - Very simple recursion, typically no parameter estimation in exam-style questions
  - No explicit long-run variance level (no mean reversion term)
  - Common in “RiskMetrics-style” VaR systems and as a fast-moving risk signal

- **GARCH(1,1)**
  - Adds a constant term $\omega$ so volatility mean-reverts to $\sigma^2_{\text{LR}}$
  - Requires parameter estimation (often maximum likelihood in practice)
  - Useful when a risk manager wants a model-based long-run volatility anchor

A useful conceptual link:

- EWMA is closely related to an “integrated” GARCH form where the persistence is effectively 1 (no mean reversion).

### Using conditional volatility forecasts in VaR

If a position has value \$$V$ and one-day return volatility forecast is $\sigma_{t+1|t}$, a normal VaR estimate at confidence level $c$ is:
$$\text{VaR}_{c} = z_c \,\sigma_{t+1|t}\, V$$
where $z_c$ is the standard normal quantile (e.g., about 2.33 for 99%).

In exam questions, you are usually given $z_c$ (or it is implied) and asked to compute VaR using a volatility forecast coming from EWMA or GARCH.

### Diagnostics and model risk checks

Even for Part 1, you should know the logic of checking whether a volatility model has captured dependence:

- Raw returns may show little autocorrelation
- Squared returns or squared standardized residuals often show autocorrelation if volatility clustering remains
- Ljung-Box or Box-Pierce style Q-statistics can be applied to residual autocorrelations (commonly extended in practice to squared residuals for volatility diagnostics)

Model risk issues to remember:

- Parameter instability during regime changes
- Underestimation of tail risk if conditional normality is assumed when returns are heavy-tailed
- Forecast error increases materially out-of-sample, so backtesting (especially VaR exceptions) matters

## **Summary**

Financial return series often have limited predictability in the mean but strong time-variation in volatility. This leads to models where shocks are uncorrelated but not independent, and where conditional variance changes with recent history.

EWMA provides a simple way to update variance using exponentially decaying weights on past squared returns. GARCH(1,1) generalizes this idea by adding a constant (long-run variance anchor) and by combining the effects of recent shocks and past variance. For the FRM exam, you should be able to compute one-step forecasts under both models, interpret persistence, and scale volatility correctly for risk measures such as VaR.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Volatility differs from variance, and scaling is done on variance (then square-root for volatility)
- Returns can resemble white noise in mean while still displaying conditional heteroskedasticity
- EWMA variance recursion: $\sigma_t^2=\lambda\sigma_{t-1}^2+(1-\lambda)r_{t-1}^2$
- EWMA weights decay exponentially; half-life is $h=\ln(0.5)/\ln(\lambda)$
- GARCH(1,1) variance recursion: $\sigma_t^2=\omega+\alpha\epsilon_{t-1}^2+\beta\sigma_{t-1}^2$
- Stationarity and finite unconditional variance typically require $\alpha+\beta<1$
- Long-run variance in GARCH(1,1): $\sigma^2_{\text{LR}}=\omega/(1-\alpha-\beta)$
- Conditional volatility forecasts feed directly into VaR calculations

## **Key Terms and Concepts**

- Volatility clustering
- Variance rate
- Dependent white noise
- Conditional heteroskedasticity
- Conditional variance
- EWMA
- Decay factor $\lambda$
- Half-life
- ARCH
- GARCH(1,1)
- Positivity constraints ($\omega>0$, $\alpha\ge 0$, $\beta\ge 0$)
- Volatility persistence ($\alpha+\beta$)
- Long-run (unconditional) variance
- One-step-ahead variance forecast
- Multi-step-ahead variance forecast
- VaR based on conditional volatility


Time-series and volatility modeling - EWMA and GARCH volatility models