Market-Risk Measurement CHAPTER 2 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 common approaches to measuring market risk Standard deviation, variance: total risk Capital Asset Pricing Model (CAPM): systematic risk Value at Risk (VaR) In this class, we focus on VaR

Risk vs. Uncertainty

Average (1/2) Equation: sum of probability * average dollars Compute: 20%*-2+80%*+2=1.2 (hundred million) 20% 80% -2 hundred million +2 hundred million

Average(2/2) FX: Company A Company B Determine : Low average dollars has higher risk Advantage: easy to understand Disadvantage: FX: Company A Company B The average dollars is the same (1.2 hundred million), but the risk is different 20% 80% -2 hundred million +2 hundred million 100% firm gets 1.2 hundred million

Variance (Standard Deviation)(1/2) Equation: sum of probability * (dollars-average dollars)^2 Compute: Variance: 20%*(-2-1.2)^2+80%*(+2-1.2)^2=2.56 (hundred million) Standard deviation: 2.56^0.5=1.6 (hundred million) 20% 80% -2 hundred million +2 hundred million

Variance(Standard Deviation)(2/2) Determine : high variance or standard deviation has higher risk Disadvantage: FX: Company C Company D The variance or standard deviation is the same (1.6 hundred million), but the risk is different 80% 20% -2 hundred million +2 hundred million 20% 80% -2 hundred million +2 hundred million

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. 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 VaR summarizes the predicted maximum loss (or worst loss) over a target horizon within a given confidence interval. (Jorion（2000)) How to compute VaR Know the distribution of the asset Know the confidence interval or significant level Know the time horizon (time period) Typically, a severe loss is defined as a loss that has a 1% (significant level) chance of occurring on any given day

VALUE AT RISK Feature Unit: dollars. Not standard deviation or ratio (Sharpe ratio) Estimate value. is computed by statistics, given the confidence interval to compute the estimate value, not certain value. VaR is estimated in normal market. VaR fails to tell you the maximum loss in stress market, such as 2008 financial crisis.

VALUE AT RISK A common assumption is that movements in the market have a Normal probability distribution, meaning there is a 1% (significant level) chance that losses will be greater than 2.32 standard deviations. Assuming a Normal distribution, 99% (confidence interval) 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.

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

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 7*100), and the daily standard deviation in the absolute level of interest rates is 0.2%, then the VaR is approximately $3.24: VaR = $100 *7 *2.32 * 0.2 = $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 (no consider the convexity).

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 = 2.32* σE * 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 *$0.1 * 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 Delta: the derivative of value of option with respect to the price of underlying stock The standard deviation of the underlying stock’s price

How to calculate volatility of each asset? JP Morgan's RiskMetrics system Equally-Weighted Moving Average (that is Simple Moving Average (SMA); variance) Exponentially-Weighted Moving Average (EWMA) λ: decay rate, 0<λ<1. The more the λ value, the less last observation affects the current dispersion estimation. The formula of the EWMA model can be rearranged to the following form:

How to calculate volatility of each asset? The EWMA model has an advantage in comparison with SMA, because the EWMA has a memory. Using the EWMA allows one to capture the dynamic features of volatility. This model uses the latest observations with the highest weights in the volatility estimate. However, SMA has the same weights for any observation. JP Morgan suggests: the optimal value for current daily dispersion (volatility) is =0.94; the optimal value for current monthly dispersion (volatility) is =0.97 Using moving window method to calculate the daily volatility and then calculate the daily VaR

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: Changes in market factors are Normally distributed. The one-day VaR is constant over the time period. 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.