Value at Risk

“What loss level is such that we are X% confident it will not be exceeded in N business days?” Value at Risk is an attempt to provide a single number summarizing the total risk in a portfolio of financial assets.

THE VaR MEASURE I am X percent certain there will not be a loss of more than V dollars in the next N days The variable V is the VaR of the portfolio. It is a function of two parameters: the time horizon (N days) and the confidence level (X%). For example, when N=5 and X=97, VaR is the third percentile of the distribution of changes in the value of the portfolio over the next 5 days.

THE VaR MEASURE All senior managers are very comfortable with the idea of compressing all the Greek letter for all the market variables underlying a portfolio into a single number.

THE VaR MEASURE VaR is the loss level that will not be exceeded with a specified probability C-VaR (or expected shortfall) is the expected loss given that the loss is greater than the VaR level Although C-VaR is theoretically more appealing, it is not widely used

THE VaR MEASURE Regulators base the capital they require banks to keep on VaR The market-risk capital is k times the 10-day 99% VaR where k is at least 3.0

THE VaR MEASURE Time Horizon Instead of calculating the 10-day, 99% VaR directly analysts usually calculate a 1-day 99% VaR and assume This is exactly true when portfolio changes on successive days come from independent identically distributed normal distributions

Historical Simulation Create a database of the daily movements in all market variables. Suppose that VaR is to be calculated for portfolio using a 1-day time horizon, a 99% confidence level, and 501 days of data. Scenario 1 is where the percentage changes in the values of all variable are the same as they were between Day0 and Day1.

Historical Simulation This defines a probability distribution for daily changes in the value of the portfolio.

Historical Simulation Today is Day 500, tomorrow is Day501 The ith trial assumes that the value of the market variable tomorrow (i.e., on day m+1) is In our example, m=500. For the first variable, the value today, V500 , is 25.85. Also V0=20.33 and V1=20.78. It follows that the value of the first market variable in the first scenario is 25.85×20.78÷20.33=26.42

Historical Simulation

Historical Simulation We are interested in the 1-percentile point of the distribution of changes in the portfolio value. Because there are total of 500 scenarios in Table 20.2 we can estimate this as the fifth-worst number in the final column of the table.

Model-Building Approach Daily Volatilities In option pricing we measure volatility “per year” In VaR calculations we measure volatility “per day

Model-Building Approach Strictly speaking we should define sday as the standard deviation of the continuously compounded return in one day In practice we assume that it is the standard deviation of the percentage change in one day

Model-Building Approach Single-Asset Case Microsoft Example (page 448) We have a position worth $10 million in Microsoft shares The volatility of Microsoft is 2% per day (about 32% per year) We use N=10 and X=99

Model-Building Approach Microsoft Example (page 448) The standard deviation of the change in the portfolio in 1 day is $200,000 The standard deviation of the change in 10 days is

Model-Building Approach Microsoft Example continued We assume that the expected change in the value of the portfolio is zero (This is OK for short time periods) We assume that the change in the value of the portfolio is normally distributed Since N(–2.33)=0.01, the VaR is

Model-Building Approach AT&T Example (page 449) Consider a position of $5 million in AT&T The daily volatility of AT&T is 1% (approx 16% per year) The S.D per 10 days is

Model-Building Approach AT&T Example (page 449) The VaR is

Model-Building Approach Two-Asset Case Now consider a portfolio consisting of both Microsoft and AT&T Suppose that the correlation between the returns is 0.3 A standard result in statistics states that

Model-Building Approach In this case sX = 200,000 and sY = 50,000 and r = 0.3. The standard deviation of the change in the portfolio value in one day is therefore 220,227 The 10-day 99% VaR for the portfolio is The benefits of diversification are (1,473,621+368,405)–1,622,657=$219,369

Model-Building Approach Less than perfect correlation leads to some of the risk being “diversified away”

The Linear Model We assume The daily change in the value of a portfolio is linearly related to the daily returns from market variables The returns from the market variables are normally distributed

The Linear Model

The Linear Model In the example considered in the previous section, S1=0.02, S2=0.01, and the correlation between the returns is 0.3. The 10-day 99% VaR is 0.22×2.33×√10=1.623 million

The Linear Model Handling Interest Rates: One possibility is to assume that only parallel shifts in the yield curve occur △P = -DPΔy This approach does not usually give enough accuracy.

The Linear Model Handling Interest Rates: Cash Flow Mapping We choose as market variables bond prices with standard maturities (1mth, 3mth, 6mth, 1yr, 2yr, 5yr, 7yr, 10yr, 30yr) a simple example of a portfolio consisting of a long position in a single Treasury bond with a principal of 1million maturing in 0.8 year We suppose that the bond provides a coupon of 10% per annum payable semiannually

The Linear Model

The Linear Model When Linear Model Can be Used Portfolio of stocks Portfolio of bonds Forward contract on foreign currency Interest-rate swap

The Linear Model and The Linear Model and Options Consider a portfolio of options dependent on a single stock price, S. Define and

The Linear Model As an approximation Similarly when there are many underlying market variables where di is the delta of the portfolio with respect to the ith asset

The Linear Model Example Consider an investment in options on Microsoft and AT&T. Suppose the stock prices are 120 and 30 respectively and the deltas of the portfolio with respect to the two stock prices are 1,000 and 20,000 respectively As an approximation where Dx1 and Dx2 are the percentage changes in the two stock prices

Quadratic Model Gamma is defined as the rate of change of the delta with respect to the market variable. Gamma measures the curvature of the relationship between the portfolio value and an underlying market variable.

Quadratic Model

Quadratic Model

Quadratic Model For a portfolio dependent on a single stock price it is approximately true that this becomes

Quadratic Model With many market variables we get an expression of the form where This is not as easy to work with as the linear model

Monte Carlo Simulation To calculate VaR using M.C. simulation we Value portfolio today Sample once from the multivariate distributions of the Dxi Use the Dxi to determine market variables at end of one day Revalue the portfolio at the end of day

Monte Carlo Simulation Calculate DP Repeat many times to build up a probability distribution for DP VaR is the appropriate percentile of the distribution times square root of N For example, with 1,000 trial the 1 percentile is the 10th worst case.

Monte Carlo Simulation Speeding Up Monte Carlo Use the quadratic approximation to calculate DP

Comparison of Approaches Model building approach assumes normal distributions for market variables. It tends to give poor results for low delta portfolios Historical simulation lets historical data determine distributions, but is computationally slower

Stress Testing And Back Testing Stress Testing This involves testing how well a portfolio performs under some of the most extreme market moves seen in the last 10 to 20 years Back-Testing Tests how well VaR estimates would have performed in the past We could ask the question: How often was the actual 10-day loss greater than the 99%/10 day VaR?

Principal Components Analysis One approach to handing the risk arising from groups of highly correlated market variable is principal components analysis. This takes historical data on movements in the market variables and attempts to define a set of components or factors that explain the movement The approach is best illustrated with an example. The market variable we will consider are 10 US Treasury rates with maturities between 3 months and 30 years.

Principal Components Analysis

Principal Components Analysis The first factor is a roughly parallel shift (83.1% of variation explained) The second factor is a twist (10% of variation explained) The third factor is a bowing (2.8% of variation explained)

Principal Components Analysis Example: Sensitivity of portfolio to rates ($m) Sensitivity to first factor is from Table 18.3: 10×0.32 + 4×0.35 − 8×0.36 − 7 ×0.36 +2 ×0.36 = −0.08 Similarly sensitivity to second factor = − 4.40

Principal Components Analysis As an approximation The f1 and f2 are independent The standard deviation of DP (from Table 20.4) is The 1 day 99% VaR is 26.66 × 2.33 = 62.12