Market-Risk Measurement CHAPTER 5 Market-Risk Measurement

Introduction Market risk for a bank Trading group in a bank can trade the financial instruments (for example, bonds and stocks) in the market When the values of traced instruments changes, the bank might get a loss The five common approaches to measuring market risk Sensitivity analysis Stress testing Scenario testing Capital Asset Pricing Model (CAPM) Value at Risk (VaR) In this class, we focus on VaR

VALUE AT RISK Value at Risk (VaR) is a measure of market risk that tries objectively to combine the sensitivity of the portfolio to market changes and the probability of a given market change. As we will discuss over the next few chapters, VaR has some significant limitations that require the continued use of stress and scenario tests as a backup In overall, VaR is the best single risk-measurement technique available As such, VaR has been adopted by the Basel Committee to set the standard for the minimum amount of capital to be held against market risks

VALUE AT RISK Value at Risk is defined as the value that can be expected to be lost during severe adverse market fluctuations. Typically, a severe loss is defined as a loss that has a 1% chance of occurring on any given day If we are measuring daily losses, this is equivalent to saying, "On average, we will lose VaR or more on two to three days per year

VALUE AT RISK A common assumption is that movements in the market have a Normal probability distribution, meaning there is a 1% chance that losses will be greater than 2.32 standard deviations. Assuming a Normal distribution, 99% VaR can be defined as follows: standard deviation of the portfolio's value The subscript T in the VaR expression refers to the time period over which the standard deviation of returns is calculated. VaR can be calculated for any time horizon. For trading operations, a one-day horizon is typically used.

VALUE AT RISK For an example of a VaR statement, consider an equity portfolio with a daily standard deviation of $10 million. Using the assumption of a Normal distribution, the 99% confidence interval VaR is $23 million. We would expect that the losses would be greater than $23 million on 1% of trading days, or 2 to 3 days per year.

VALUE AT RISK Senior management should clearly understand that VaR is not the worst possible loss. Losses equal to the size of VaR are expected to happen several times per year VaR is therefore not equal to capital We will discuss the relationship between VaR and capital in great depth in later chapters but a very rough rule of thumb is that the capital should be 10 times VaR

Sensitivity or Duration Analysis

Sensitivity Analysis for Bonds

Sensitivity Analysis for Bonds

Sensitivity Analysis for Equities

Sensitivity Analysis for forwards and Futures

Sensitivity Analysis for Options

STRESS TESTING The sensitivity analyses discussed above give decent approximations for the change in the value of the portfolio when the change in the market-risk factors is small However, if the change in a risk factor is large (e.g., in a crisis), the linear sensitivity will not give a good estimate to the change in the value of a portfolio

STRESS TESTING In stress testing, large changes are made in the risk factors, and full, nonlinear pricing is used to revalue the portfolio and estimate the loss The purpose of stress testing is to provide a clear, objective measure of risk that is easily understood and everyone buys into. For stress testing, a standard set of changes in the risk factors is set, and the subsequent change in portfolio value is calculated. For example, a typical stress statement would be "If interest rates move up by 2%, we would lose $15 million; if they move by 4%, we would lose $28 million."

STRESS TESTING Typically, the movements are standardized in order to communicate them easily throughout the organization For example, the changes in all equity values may be set at -20%, -10%, and +10% and +20% It also makes sense to decide which factors should be moved together to analyze the results more easily. This is called "blocking."

SCENARIO TESTING Stress testing and scenario testing are similar in that both use specified changes in the market-risk factors and reprice the portfolio with full, nonlinear pricing models However, in stress testing, the changes in risk factors are very uniform and objective. In scenario testing, the changes are tailored and subjectively chosen.

SCENARIO TESTING In scenario testing, informed opinion is used to create a limited set of worst-case scenarios. Each scenario corresponds to a specific type of market crisis, such as U.K. equities market crashes, a default by China, or the raising of oil prices by OPEC. Typically, 5 to 10 "worst-case" scenarios are chosen

SCENARIO TESTING The scenarios are typically derived from one of three sources: previous crises, the bank's current portfolio, and the opinion of the bank's experts such as the head trader, bank economists, and the risk management group In using previous crises, the risk management group looks at historical data from many markets and asks: what if those events were to happen here and now? For example, if a 20 one-day drop in the U.S. market happened in 1987, one scenario could be that the same happens for all the euro markets

VaR for Bonds For a bond, VaR can be approximated by multiplying the dollar duration by the "worst-case" daily interest move. This gives the value change in the "worst case."

VaR for Bonds the "worst-case" daily interest move for 1% chance in one day If we assume that interest-rate movements have a Normal probability distribution, then the 1% worst case will correspond to2.32 standard deviations of the daily rate movements

VaR for Bonds As an example. If the duration is 7 years (time duration in term of year), the current price is $100 (the dollar duration is 7X100), and the daily standard deviation in the absolute level of interest rates is 0.2%, then the VaR is approximately $3.24: VaR = $100 x 7 x 0.2 x 2.32 = $3.24 The approximations that we made here were as follows: the changes in the rate is Normally distributed The change in the price can be well-approximated by the linear measure of duration.

VaR for Equities The VaR for an equity is easy to calculate If we assume that equity prices have a Normal distribution. The VaR is then the number of shares held (N), multiplied by 2.32 and the standard deviation of the equity price (σE): VaR = 2.32X σE X N So, for example, if we held 100 shares of IBM, and the daily standard deviation of the price was 10 cents, the VaR would be $23.2: VaR = 2.32 x $0.1 x 100 = $23.2

VaR for Options A simple approximation of the VaR to an option can be obtained using the linear sensitivities The standard deviation of the option price caused by changes in the stock price is simply the standard deviation of the stock price multiplied by delta

VaR for Options The derivative of value of option with respect to the price of underlying stock The standard error of the underlying stock’s price The second derivative The pricing function for a option, for example, the Black-Scholes equations

General Considerations in Using VaR In the discussion above, we gave approximations for calculating the one-day 99% VaR. However, there are several conventions in use for the VaR probability, which implies a different multiplication factor for the standard deviation. The most common alternative is to set the tail probability at 2.5%. If a Normal distribution is assumed, this implies a multiplier of 1.96 rather than 2.32

General Considerations in Using VaR In some cases, we may wish to know the VaR for the potential losses over multiple days. A reasonable approximation to the multi day VaR is that it is equal to the one-day VaR multiplied by the square root of the number of days:

General Considerations in Using VaR This relationship requires the following assumptions: 1. Changes in market factors are Normally distributed. 2. The one-day VaR is constant over the time period. 3. There is no serial correlation. Serial correlation is present if the results on one day are not independent of the results on a previous day.

General Considerations in Using VaR In general, for trading operations it is safe to assume that if the term VaR is used without a specified time, it means one-day VaR. the term VaR is also used to refer to the potential loss from asset liability management, in which case a monthly or yearly horizon is used. Also, the term "credit VaR" is sometimes used to describe the loss distribution from a credit portfolio. This is quite different from the VaR used for trading portfolios.

General Considerations in Using VaR The major limitation of VaR is that it describes what happens on bad days (e.g.,twice a year) rather than terrible days (e.g., once every 10 years). VaR is therefore good for avoiding bad days, but to avoid terrible days you still need stress and scenario tests.