Introduction

In this chapter, we delve into the calculation of Value at Risk (VaR) specifically for linear derivatives. Linear derivatives are financial instruments whose price movement is directly proportional to the price movement of an underlying asset. We will explore the methodology and calculations required to estimate VaR for portfolios involving linear derivatives. Understanding how to assess risk for these instruments is essential for effective risk management and informed decision-making.

Linear derivatives, such as forward contracts and futures contracts, exhibit a linear relationship between their prices and the underlying asset’s prices. The concept of linearity simplifies the calculation of risk exposure, making it a crucial consideration in VaR estimation.

Calculating VaR for Linear Derivatives

Calculating VaR for linear derivatives involves two primary components: the position’s market value and the risk factor’s volatility. The formula for calculating VaR is:

$$VaR = \text{Position Market Value} \times \text{Z-score} \times Volatility$$

Where:

• Position Market Value represents the value of the position in the derivative.
• Z-score is the standard score corresponding to the desired confidence level.
• Volatility indicates the volatility of the risk factor.

Example: Consider a portfolio that holds a futures contract on a stock index. The contract size is USD 100 per index point, and the current index level is 1500. The portfolio holds a long position with a market value of USD 300,000. The historical volatility of the index is 0.18.

To calculate a one-day 95% VaR for this position, we use the formula:

$$VaR = 300,000 \times \text{Z-score(95\%)} \times 0.18 $$

Assuming a Z-score of 1.645 (corresponding to the 95% confidence level), the VaR is:

$$VaR = 300,000 \times 1.645 \times 0.18 = {USD }8,190\text$$

This implies that there is a 5% chance of the portfolio losing more than USD 8,190 in one day due to the price movement of the futures contract.