Value at Risk Ch.12. PAK Study Manual

Transcription

1 Value at Risk Ch.12

2 Related Learning Objectives 3a) Apply and construct risk metrics to quantify major types of risk exposure such as market risk, credit risk, liquidity risk, regulatory risk etc., and tolerances in the context of an integrated risk management process 3c) Analyze quantitative financial data and construct measures from insurance data using modern statistical methods (including asset prices, credit spreads and defaults, interest rates, incidence, causes and losses). Contrast the available range of methods with respect to scope, coverage and application Key Points of This Reading 1) Understand how to conduct Monte Carlo simulation 2) Understand the tradeoff between speed and accuracy

3 Monte Carlo Simulation Advantages 1. It is the most powerful approach to VAR due to its flexibility 2. It accounts for a wide range of risks and complex interactions Disadvantages 1. This approach involves costly investments in intellectual and systems development 2. It requires substantially more computing power than simpler methods 3. It accounts for nonlinear exposures and complex pricing patterns 4. Simulations can be extended to longer horizons 5. It can be used for operational risk measurement, and integrated risk management

4 Steps of Using the Monte Carlo Simulation to Calculate VAR 1. Simulate repeatedly a random process for the financial variable(s) of interest covering a wide range of possible situations 2. Draw these variables from pre-specified probability distributions that are assumed to be known, including the analytical function and its parameters 3. Simulations recreate the entire distribution of portfolio values, from which VAR can be derived

5 Geometric Brownian Motion (GBM) Model The model assumes that innovations in the asset price are uncorrelated over time and that small movements in prices can be described by dst =µ tstdt +σtstdz, where dz is a random variable distributed normally with mean zero and variance dt S t dst =µ Stdt +σstdz ds t = µ S dt t dz Time Stock Price Formula: t+ 1 t t t t ( ) S = S + S = S + S µ t+ σε t

6 Step i Previous Price S t + i 1 Example Random Variable ε i Increment t ( ) S = S µ t+ σε t Current Price S = S + S t+ i t+ i 1 t+ i Interval =100 steps, t = 1, µ= 0 σ Total = 10% local volatility = over the total interval σ= 10% 1/100 = ( ) ( ) S = S µ t+ σε t t t S = (0.199) 1 = ( ) S = (1.665) 1 = S = t 1 S + + t St S = = S 1 2 = =

7 Simulate Price Paths Price 95% Upper Limit Path #1 $100 Path #2 0 95% Lower Limit Steps into the future 100

8 Code: Create Random Numbers Uniform 1 Cumulative Standard Normal X=0.31 Two Steps To Create Random Numbers 1. Assume the random-number generator follows a uniform distribution over the interval [0,1] that produces a random variable x 0-3 Standard Normal Transform the uniform random number x into the desired distribution ( y=n -1 (x) ) through the inverse cumulative probability distribution function (pdf) -0.50

9 Bootstrap This method generates random numbers by sampling from historical data (empirical distribution) Example: Suppose we want to generate 100 stock prices into the future Bootstrap Procedure 1. Project returns by randomly picking one return at a time from the sample over the past M = 500 days, with replacement 2. Define the index choice as m(1), a number between 1 and The selected return then is R m(1), and the simulated next-day return is S t+1 = S t (1+R m(1) ) 4. Repeating #1 - #3 for a total of 100 draws yields a total of 100 pseudo-values: S t+1,, S t+100

10 Bootstrap Advantages 1. It can include fat tails jumps, or any departure from the normal distribution 2. It accounts for correlations across series because one draw consists of the simultaneous returns for N series Disadvantages 1. For small sample sizes M, the bootstrapped distribution may be a poor approximation of the actual one so sufficient data points are needed 2. It relies heavily on the assumption that returns are independent so by resampling at random, any pattern of time variation is broken

11 Speed vs. Accuracy Price Distribution VAR Number of replications

12 Acceleration Methods 1. Antithetic variable technique 2. Control variates technique 3. Important sampling technique 4. Stratified sampling technique

13 From Independent to Correlated Variables If the variables (ε) are uncorrelated, the randomization can be performed independently for each variable St = St( µ t+ σε t), where the ε values are independent across time period and series Correlations between Variables To account for correlations between variables, we start with a set of independent variables η, which then are transformed into the ε ε 1 =η1 ε = ρη + (1 ρ ) η 2 1/

14 Cholesky Decomposition It is used to generate correlated variables (ε) R = TT ' 2 1 ρ a11 0 a11 a12 a11 a11a ρ 1 = a12 a 22 0 a = 22 a11a12 a12 + a22 a 2 a11 = 1 a11a12 = ρ + a = a a a = 1 = ρ = (1 ρ ) 2 1/2 1 ρ ρ 2 1/2 2 1/2 ρ 1 = ρ (1 ρ ) 0 (1 ρ ) R = TT ' ε1 1 0 η1 ε = Tη: ε = 2 1/2 2 ρ (1 ρ ) η 2

15 Model the Dynamics of Interest Rates Short-term interest rate process: γ dr =κ( θ r ) dt +σr dz t t t t When Formula Model γ = 0 dr =κ( θ r ) dt +σdz t t t Vasicek model γ = 0.5 dr =κ( θ r ) dt +σ r dz t t t t CIR model γ = 1 dr =κ( θ r ) dt +σrdz t t t t Lognormal model