Intro to Value at Risk (VaR)

Value at Risk (VaR) – Part 1 (LOs 7.1 – 8.7 , 12.1 – 13.9, 15.1-15.5) Intro to VaR (Allen Chapter 1) 1 VaR Mapping 4 VaR Methods 5 Cash flow at Risk (CFaR) 2 Putting VaR to Work (Allen Chapter 3) 3 Stress Testing 6

Value at Risk (VaR) in the Readings We are reviewing here (Sec II) Was reviewed in Quant (Sec I) To be reviewed in Investments (Sec V) Learning Outcome Location in Study Guide Reading LO 7.1 to 7.6 II. Market 1.A. Intro to VaR Allen Ch. 1 LO 7.7 to 7.15 II. Market 1.B. Putting VaR to work Allen Ch. 3 LO 8.1 to 8.7 II. Market 6.A. Firm-wide Approach to Risk Stulz Ch. 4 LO 9.1 to 9.11 V. Investment 6.A. Portfolio Risk Jorion Ch. 7 LO 10.1 10.7 I. Quant 3.A. Forecasting Risk and Correlation Jorion Ch. 9 LO 11.1 to 11.10 I. Quant 1. Quantifying Volatility Allen Ch. 2

We are reviewing here (Sec II) Was reviewed in Quant (Sec I) To be reviewed in Investments (Sec V) Value at Risk (VaR) in the Readings Learning Outcome Location in Study Guide Reading LO 12.1 to 12.6 II. Market 3.A. VaR Methods Jorion Ch. 10 LO 13.1 to 13.9 II. Market 3.B. VaR Mapping Jorion Ch. 11 LO 14.1 to 14.7 I. Quant 3.B. MCS Jorion Ch. 12 LO 15.1 to 15.5 I. Market 3.C Stress Testing Jorion Ch. 14 LO 16.1 to 16.3 I. Quant 4. EVT Kalyvas Ch. 4 LO 17.1 17.13 V. Investment 6.B. Budgeting in I/M Jorion Ch. 17

Value at Risk (VaR) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Value at Risk (VaR) ,[object Object],Single-Period VaR (n=1)

Value at Risk (VaR) ,[object Object]

Value at Risk (VaR) - % Basis -1.645  One-Period VaR (n=1) and 95% confidence (5% significance)

Value at Risk (VaR) – Dollar Basis -1.645  One-Period VaR (n=1) and 95% confidence (5% significance)

10-period VaR @ 5% significance -1.645  10 10-Period VaR (n=10) and 95% confidence (5% significance)

10-period (10-day) VaR 5% significance, annual  = +12% $100(1+  ) +0.48 ($95.30)

Absolute versus Relative VaR $100(1+  ) $100  $5.20  $4.73

Absolute VaR $100(1+  ) $100

Value at Risk (VaR) ,[object Object],What is one-day VaR with 95% confidence, dollars and percentage terms?

Value at Risk (VaR) ,[object Object],What is one-day VaR with 95% confidence?

Value at Risk (VaR) ,[object Object],Square-root rule J-day VaR = 1-day VaR  Square Root of Delta Time Assumes i.i.d. Independent  not (auto/serial) correlated Identically distr.  constant variance (homoskedastic)

Value at Risk (VaR) ,[object Object],Daily VaR is (-)$10,000. What is 5-day VaR?

Value at Risk (VaR) ,[object Object],10-day VaR is (-)$1 million. What is annual VaR (assume 250 trading days)?

Value at Risk (VaR) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Value at Risk (VaR) ,[object Object],[object Object],[object Object],[object Object],Except for interest rate variables: absolute 

Value at Risk (VaR) ,[object Object],Directional Impacts Factor Impact on Portfolio volatility Higher variance  Greater asset concentration  More equally weighted assets  Lower correlation  Higher systematic risk  Higher idiosyncratic risk Not relevant

Value at Risk (VaR) ,[object Object],[object Object],[object Object],All assets are locally linear . For example, an option: the option is convex in the value of the underlying. The delta is the slope of the tangent line. For small changes, the delta is approximately constant.

Value at Risk (VaR) ,[object Object],[object Object],[object Object],[object Object]

Value at Risk (VaR) ,[object Object],[object Object],[object Object],[object Object],[object Object]

Value at Risk (VaR) ,[object Object],Advantage Disadvantage Structured Monte Carlo ,[object Object],[object Object],[object Object]

Value at Risk (VaR) ,[object Object],Advantage Disadvantage Stress Testing ,[object Object],[object Object],[object Object],[object Object],[object Object]

Value at Risk (VaR) ,[object Object],[object Object]

Value at Risk (VaR) ,[object Object],[object Object],[object Object]

Value at Risk (VaR) ,[object Object],VAR CFAR Balance sheet (asset values) Statement of cash flows Banks, financial services firm, funds Non-financial corporations  External markets for capital  Internal growth provides capital

Value at Risk (VaR) ,[object Object],Impact of “small” project: Buy #1 and Sell #3 1% of portfolio

Value at Risk (VaR) ,[object Object],Valuation Method Risk factor Local Full Analytical Delta-normal Not used Delta-gamma-delta Simulated Delta-gamma-Monte-Carlo Monte Carlo Grid Monte Carlo Historical

Value at Risk (VaR) ,[object Object],Advantage Disadvantage Delta-normal Easy to implement Computationally fast Can be run in real-time Amenable to analysis (can run marginal and incremental VaR) Normality assumption violated by fat-tails (compensate by increasing the confidence interval) Inadequate for nonlinear assets

Value at Risk (VaR) ,[object Object],Advantage Disadvantage Historical Simulation Simple to implement Does not require covariance matrix Can account for fat-tails Robust because it does not require distributional assumption (e.g., normal) Can do full valuation Allows for horizon choice Intuitive Uses only one sample path (if history does not represent future, important tail events not captured) High sampling variation (data in tail may be small) Assumes stationary distribution (can be addressed with filtered simulation)

Value at Risk (VaR) ,[object Object],Advantage Disadvantage Monte Carlo Most powerful Handles fat tails Handles nonlinearities Incorporates passage of time; e.g., including time decay of options Computationally intensive (need lots of computer and/or time) Can be expensive Model risk Sampling variation

Value at Risk (VaR) ,[object Object],[object Object],[object Object],[object Object],[object Object],The second component is the “mapping system” which transforms (or maps) the portfolio positions into weights on each of the securities for which risk is measured.

Value at Risk (VaR) ,[object Object],VAR vs. Stress Testing VAR Stress Testing No information on magnitude of losses in excess of VaR Captures the “magnitude effect” of large market moves. Little/no information on direction of exposure; e.g., is exposure due to price increase or market decline Simulates changes in market rates and prices, in both directions Says nothing about the risk due to omitted factors; e.g., due to lack of data or to maintain simplicity Incorporates multiple factors and captures the effect of nonlinear instruments.

Value at Risk (VaR) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Value at Risk (VaR) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Value at Risk (VaR) ,[object Object],Advantage Disadvantage Prospective Scenarios in MDA Relies on input of managers to frame scenario and therefore may be most realistic vis-à-vis actual extreme exposures May not be well-suited to “large, complex” portfolios FACTOR PUSH METHOD: ignores correlations Historical scenarios in MDA Useful for measuring joint movements in financial variables Typically, limited number of events to draw upon

Value at Risk (VaR) ,[object Object],Advantage Disadvantage Conditional Scenario Method More realistically incorporates correlations across variables: allows us to predict certain variables conditional on movements in key variables Relies on correlations derived from entire sample period. Highly subjective

VaR: Question ,[object Object],[object Object],[object Object]

VaR: Answer 1 ,[object Object],[object Object]

VaR: Answer 2 ,[object Object],[object Object]

VaR: Question ,[object Object],[object Object]

VaR: Answer ,[object Object],[object Object]

VaR of nonlinear derivative ,[object Object],[object Object],[object Object],[object Object]

Taylor approximation AIM: Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives 1. Constant approximation 2. First-order (linear) approximation 3. Second-order (quadratic) approximation

Taylor approximation AIM: Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives

Taylor approximation AIM: Discuss why the Taylor approximation is ineffective for certain types of securities Does not perform well when the derivative shows extreme nonlinearities . For example: ,[object Object],[object Object],When beta/duration can change rapidly, Taylor approximation (delta-gamma approximation) is ineffective – need more complex models.

Versus Full Re-value AIM: Explain the differences between the delta-normal and full-revaluation methods for measuring the risk of non-linear derivatives Delta-normal: linear approximation that assumes normality ,[object Object],[object Object],Full-revaluation: linear approximation that assumes normality Computationally fast but… approximate Accurate but… computationally burdensome ,[object Object]

Structured Monte Carlo ,[object Object],[object Object],[object Object],Simulate with one variable (e.g., GBM) or several (Cholesky)

Structured Monte Carlo ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Scenario Analysis Evaluate Correlation Matrix Under Scenarios ERM Crisis (92) Mexican Crisis (94) Crash Of 1987 Gulf War (90) Asian Crisis (’97/8)

Scenario Analysis AIM: Discuss the implications of correlation breakdown for scenario analysis Severe stress events wreak havoc on the covariance matrix

Scenario Analysis AIM: Describe the primary approaches to stress testing and the advantages and disadvantages of each approach ,[object Object],[object Object],[object Object],(a) Plugs-in historical events , or (b) Analyzes predetermined scenarios

Scenario Analysis Historical events + Can inform on portfolio weaknesses - But could miss weaknesses unique to the portfolio Stress Scenarios + Gives exposure to standard risk factors - But may generate unwarranted red flags - May not perform well in regard to asset-class-specific risk AIM: Describe the primary approaches to stress testing and the advantages and disadvantages of each approach

Summary ,[object Object],[object Object],[object Object],[object Object]

Random Variables + + + Short-term Asset Returns Probability distributions are models of random behavior + + + - - - - - - - ? ? ?

Random Variables ,[object Object],[object Object]

Random Variables ,[object Object],[object Object],[object Object]

Random Variables ,[object Object],[object Object],[object Object],[object Object]

Random Variables ,[object Object],[object Object],[object Object],[object Object],[object Object]

Random Variables ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Random Variables ,[object Object],One Event: Roll a seven Six outcomes

Conditional ,[object Object],Unconditional

Conditional ,[object Object],Conditional

Joint probability ,[object Object],S= $10 S= $15 S=$20 Total T=$15 0 2 2 4 T=$20 3 4 3 10 T=$30 3 6 3 12 Total 6 12 8 26

Theorems ,[object Object],Independent = not correlated

Theorems ,[object Object],What is the variance of a single six-sided die?

Covariance & correlation ,[object Object]

Covariance & correlation ,[object Object],X Y 3 5 2 4 4 6

Covariance & correlation ,[object Object],X Y (X-X avg )(Y-Y avg ) 3 5 0.0 2 4 1.0 4 6 1.0 Avg = 3 Avg = 5 Avg = .67

Covariance & correlation X Y (X-X avg )(Y-Y avg ) 3 5 0.0 2 4 1.0 4 6 1.0 Avg = 3 Avg = 5 Avg = .67 s.d. = SQRT(.67) s.d. = SQRT(.67) Correl. = 1.0

Bayes’ formula ,[object Object]

Permutations & combinations ,[object Object]

Permutations & combinations ,[object Object],Given a set of seven letters: {a, b, c, d, e, f, g} How many permutations of three letters? How many combinations of three letters?

Distributions ,[object Object],LO 2.2 Discuss a probability function, a probability density function, and a cumulative distribution function

Comparison Probability Density Function (pdf) Cumulative Distribution Discrete variable Continuous variable

Distributions ,[object Object],[object Object],[object Object],[object Object],[object Object]

Distributions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Distributions ,[object Object]

Normal distribution ,[object Object]

Normal distribution ,[object Object],% of all (two-tailed) % “to the left” (one-tailed) Critical values Interval –math (two-tailed) VaR ~ 68% ~ 34% 1 ~ 90% ~ 5.0 % 1.645 (~1.65) ~ 95% ~ 2.5% 1.96 ~ 98% ~ 1.0 % 2.327 (~2.33) ~ 99% ~ 0.5% 2.58

Normal distribution ,[object Object],[object Object],[object Object],[object Object]

For Parametric VAR ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Poisson: Question ,[object Object],[object Object]

Poisson: Answer ,[object Object]

Distributions - Poisson ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Compared ,[object Object],In Poisson, the expected value (the mean) = variance Variance is standard deviation 2

Lognormal Transform x-axis To logarithmic scale

Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Sampling ,[object Object],[object Object],[object Object],[object Object],[object Object],We take a sample (from the population) in order to draw an inference about the population.

Frequencies ,[object Object],[object Object]

Sampling ,[object Object],[object Object],[object Object]

Sampling ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Geo & Arithmetic mean 2003 5.0% 2004 8.0% 2005 (3.0%) 2006 9.0% Geo. Arith.

Geo & Arithmetic mean 2003 5.0% 1.05 2004 8.0% 1.08 2005 (3.0%) 0.97 2006 9.0% 1.09  1.199 Geo. 4.641% Arith. 4.75%

Sampling ,[object Object],The sampling distribution is the probability of the sample statistic

Sampling ,[object Object],Variance of sampling distribution of means: Infinite population or with replacement

Sampling ,[object Object],Variance of sampling distribution of means: Finite population (size N) and without replacement Variance of sampling distribution of means: Infinite population or with replacement

Sampling ,[object Object],Standardized Variable: “ Asymptotically normal” even when population is not normally distributed !!

Sampling ,[object Object],Standardized Variable: “ Asymptotically normal” even when population is not normally distributed !! Central limit theorem: Random variables are not normally distributed, But as sample size increases -> Average (and summation) tend toward normal

Sampling ,[object Object],Sampling distribution of proportions Where p = probability of success

Sampling ,[object Object],Sampling distribution of differences Two populations, Two samples, difference of the means

Sampling ,[object Object],Sampling distribution of differences Two populations, Two samples, sum of the means

Variance ,[object Object],Population Variance

Variance ,[object Object],Sample Variance

Variance ,[object Object],Sample Standard Deviation

Chebyshev’s ,[object Object],What is the probability that random variable X (with finite mean and variance) will differ by more than three (3) standard deviations from its mean?

Chebyshev’s ,[object Object],If k = 3, then P() = 1/(3 2 ) = 1/9 = 0.1111

Skewness & Kurtosis ,[object Object]

Intro to Value at Risk (VaR)

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