0   # Value-at-Risk (VaR) Introduction

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#### Executive summary

A bank will wish to estimate the extent of its risk exposures so that it can better manage its P&L account, and gain an accurate estimate of its future capital requirements. VaR is a methodology that can be used to value this risk exposure. VaR measures the potential loss in market value of a portfolio using assumed or estimated volatility.

#### Key learning objectives:

• What is VaR?
• What are the main VaR methodologies?
• How do we calculate VaR using the variance-covariance method?

#### What is risk, and what are the types of risk a bank faces?

• Risk - The uncertainty of the future total cash of value of an investment on the investor’s horizon date
• Market Risk - The risk arising from movements in prices in financial markets. Examples include FX risk, interest rate risk and basis risk
• Credit Risk - Also referred to as default risk or counterparty risk, refers to risk that a consumer to whom a bank has lent money, will default on the loan

#### What is Value-at-Risk (VaR)?

• VaR measures the volatility of the change in value of a bank’s specified balance sheet assets, and so the greater the volatility the higher the probability of loss
• It is a measurement of market risk or credit risk. It is the maximum loss that can occur, stated to a degree of confidence, say X%, over a period of specified holding period of t days
• So, for example if a daily VaR was calculated to be £10,000 to a 95% level of confidence, this means that during the day there is a 5% chance that the loss will be greater than £100,000

#### What are some common misconceptions about VaR?

• It is a unified method for measuring risk
• Measures other risks that a bank will be exposed to
• It is the maximum amount of money that a balance sheet position can lose

#### What are the main VaR methodologies?

• The correlation method
• The historical simulation method
• The Monte Carlo simulation

#### What are the assumptions behind the correlation method?

• Assumes that the returns on risk factors are normally distributed
• Assumes the correlations between risk factors are constant
• Assumes the delta of each portfolio constituent is constant
• Volatility of each risk factor is extracted from the historical observation period

#### How do we calculate the relevant risk factors?

1. Simple historic volatility (correlation) - However, the effects of a large one-off market move can significantly distort volatilities over the required forecasting period
2. Weighting past observations unequally - This is done to give more weight to recent observations so that large jumps in volatility are not caused by events long ago

#### What does the historical simulation method consist of?

• The simplest method and avoids some of the pitfalls of the correlation method
• The model calculates potential losses using actual historical returns in the risk factors
• Captures the non-normal distribution of risk factor returns
• As the risk factor returns used for revaluing the portfolio are actual past movements, the correlations in the calculation are also past correlations
• Runs a portfolio through actual historical price data to create a hypothetical time series of returns in estimating VaR

#### What are the pros and cons of the Monte Carlo method?

• More flexible than the correlation method and the historical simulation method
• Allows the risk manager to use actual historical distributions for risk factor returns
• A large number of randomly generated simulations are then run forward in time using volatility and correlation estimates
• However, its implementation requires greater processing power, and there may be a trade-off in that the time taken to perform calculations is longer

#### How do we calculate VaR using the variance-covariance method?

1. Determine the time horizon over which the firm wishes to estimate a potential loss
• Time horizons of one day to one year have been used
• For a trading book, it may be a one-day period
• Regulators and participants in illiquid markets may want to estimate exposures to market risk over a longer period
2. Select the degree of certainty required
• The largest likely loss a bank will suffer 95 times out of 100, or 1 working day out of 20 (95% confidence interval) may be sufficient
• For regulatory requirements and senior management/shareholders, a 99% confidence interval may be more appropriate
3. Create a probability distribution of likely returns for the instrument or portfolio under consideration
• The easiest to understand is a distribution of recent historical returns for the asset or portfolio
4. Calculating the VaR estimate
• Value the current portfolio using today’s prices
• Revalue the portfolio using alternative prices based on changed market factors and calculate the change in the portfolio value that would result
• Revaluing the portfolio using a number of alternative prices gives a distribution of changes in value. Given this, a portfolio VaR can be specified in terms of confidence levels
• The risk manager can then calculate the maximum the firm can lose over a specified time horizon at a specified probability level

#### What are the pros and cons of the variance-covariance method?

1. Advantages - It is simple to apply, and fairly straightforward to explain; datasets for its use are immediately available
2. Disadvantages - It assumes stable correlations and measures only linear risk; it also places excessive reliance on the normal distribution
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