#### 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?

**Simple historic volatility (correlation)**- However, the effects of a large one-off market move can significantly distort volatilities over the required forecasting period**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?

**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

**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

**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

**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?

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