# Exponential Risk

Exponentially Weighted Moving Average (EWMA)

Exponential risk is calculated using the exponentially weighted moving average (EWMA). Exponential risk has two main benefits over the equally weighted risk model (Historical Risk)

1. Recent time series observations carry a higher weight (importance) relative to observations in the past. This allows exponentially weighted risk estimates (covariance) to react faster to recent shocks in markets.

2. Risk estimates in this model have a shorter memory following a shock. The risk estimates decline smoothly and rapidly as the significance of shock observations decreases through time. In contrast, shocks observed by the equally weighted historical risk model will increase risk estimates for the full observation period and will cause an abrupt shift when they fall out of the observation window.

## Decay Factor

The decay-factor, $\lambda$ , is a parameter that controls how quickly the significance of older observations is reduced. The $n^{th}$ observation is weighted by $\lambda^n$. The decay factor ranges from 0 to 1, where 1 will equivalent to the same weighting as the equally weighted model (historical risk).

## Half-life

The half-life describes the time it takes for the weight on time series observations to fall to one-half of the weight assigned to the most recent observation. Half-life is a function of the decay-factor and can be solved by

$\text{half-life}=\frac{log(\frac{1}{2})}{log(\lambda)}$

For example, to calculate the half-life of the standard deviation of monthly returns with a decay-factor of 0.97, we calculate the log of 1/2 divided by the log of 0.97. The half-life for this example is 22.76 months.