Many financial calculations and estimations require a statistical analysis of the variability of past market returns. Because we have strong evidence that market returns are approximately normally distributed, we can estimate potential market movements with a given probability using two simple parameters: mean (or average return) and standard deviation (or volatility). Mean Mean describes where […]

# Statistical Foundations of VaR

## Statistical Foundations: Understanding Correlations

Whereas standard deviation shows how risky individual assets are, correlations show how asset risks are interrelated. Correlations are calculated by observing historical comovement in returns and range between –1 and 1. A correlation of 1 means that returns move together perfectly, whereas a correlation of -1 implies perfect opposite movement. A 0 (zero) correlation implies […]

## Understanding Normal Distribution

The normal distribution is the well-known bell-shaped curve depicted below. The bell-shaped curve comes from a statistical tendency for outcomes to cluster symmetrically around the mean (or average). Deviations from the mean are described in terms of standard deviations. In all normal distributions, 68% of outcomes will fall within 1 standard deviation to either side […]

## Risk of Two Cash Positions

You now have two assets: JPY 1 billion + THB 4 billion. What is the risk over a 1-day period? Solution Step Calculation Comment 1. Measure value in USD $200 million 140 JPY/USD and 40 THB/USD 2. 1-month volatility JPY/USD 1.78% THB/USD 1.96% 3. What are risks?/$ On JPY $ 1.78 million On THB $ […]

## Statistical Foundations: Predicting Volatility

Following are the major steps we take to estimate volatility, using the S&P 500 Index: Let’s look at historical market rates. We start by converting daily prices into log changes. (Daily log changes are conceptually similar to percentage returns, except they are continuously compounded.) Our goal is to use past returns to predict the volatility […]

## Parametric VaR Estimation

We will now learn how to calculate the VaR of one position and two positions by applying the concept of volatilities and correlations. The following examples of how to calculate the risk of one and two positions illustrate the basic concept of parametric (delta) VaR estimation for linear instruments. The general steps for calculating VaR […]

## Risk of a Single Cash Position

BetaSquare are a USD-based firm with one asset: JPY 14 billion in cash. What is the 95% worst-case loss over a 1-day period? You have the following information: The daily price volatility of the JPY/USD exchange rate is 1.78%, using a 95% confidence level. (Note: This implies that 1 standard deviation equals 1.78%/1.65 = 1.08%.) […]

## Time Scaling of Volatility

Discover how to scale standard deviation to different time horizons. We know that risk increases with time: The longer we hold a position, the greater the potential loss. Following is a simple approximation to help you scale volatility estimates to a longer (or shorter) -time horizon. Note, however, that this is just an approximation. Volatility […]