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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Persistence
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Probability that the random variables take on certain values simultaneously
Choose parameters that maximize the likelihood of what observations occurring
In EWMA - the lambda parameter - In GARCH(1 -1) - sum of alpha and beta - Higher persistence implies slow decay toward the long - run average variance

2. Mean reversion in asset dynamics
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Price/return tends to run towards a long - run level
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

3. Weibul distribution
Independently and Identically Distributed
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

4. Antithetic variable technique
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Average return across assets on a given day
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

5. Priori (classical) probability
Peaks over threshold - Collects dataset in excess of some threshold
Based on an equation - P(A) = # of A/total outcomes
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)

6. Historical std dev
We reject a hypothesis that is actually true
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance reverts to a long run level

7. Extreme Value Theory
Sample mean +/ - t*(stddev(s)/sqrt(n))
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Adjusted R^2 does not increase from addition of new independent variables -Adjusted R^2 = 1 - (n - 1)/(n - k - 1) * (SSR/TSS) = 1 - su^2/sy^2

8. Cholesky factorization (decomposition)
Sampling distribution of sample means tend to be normal
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Observe sample variance and compare it to hypothetical population variance (sample variance/population variance)(n - 1) = chi - squared - Non - negative and skewed right - approaches zero as n increases - mean = k where k = degrees of freedom - Varia

9. Unconditional vs conditional distributions
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

10. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
When one regressor is a perfect linear function of the other regressors
P - value
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

11. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Nonlinearity
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

12. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Transformed to a unit variable - Mean = 0 Variance = 1
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Average return across assets on a given day

13. Implied standard deviation for options
More than one random variable
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Special type of pooled data in which the cross sectional unit is surveyed over time
i = ln(Si/Si - 1)

14. Simulation models
Variance(y)/n = variance of sample Y
Flexible and postulate stochastic process or resample historical data - Full valuation on target date - More prone to model risk - Slow and loses precision due to sampling variation
P(Z>t)
We reject a hypothesis that is actually true

15. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Z = (Y - meany)/(stddev(y)/sqrt(n))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

16. GEV
Summation((xi - mean)^k)/n
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Use historical simulation approach but use the EWMA weighting system
Var(X) + Var(Y)

17. Implications of homoscedasticity
Only requires two parameters = mean and variance
Confidence level
Variance ratio distribution F = (variance(x)/variance(y)) - Greater sample variance is numerator - Nonnegative and skewed right - Approaches normal as df increases - Square of t - distribution has a F distribution with 1 -k df - M*F(m -n) = Chi - s
OLS estimators are unbiased - consistent - and normal regardless of homo or heterskedasticity - OLS estimates are efficient - Can use homoscedasticity - only variance formula - OLS is BLUE

18. Adjusted R^2
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Independently and Identically Distributed
Adjusted R^2 does not increase from addition of new independent variables -Adjusted R^2 = 1 - (n - 1)/(n - k - 1) * (SSR/TSS) = 1 - su^2/sy^2
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia

19. Variance of X+Y assuming dependence
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generalized Auto Regressive Conditional Heteroscedasticity model - GARCH(1 -1) is the weighted sum of a long term variance (weight=gamma) - the most recent squared return (weight=alpha) and the most recent variance (weight=beta)
Variance(x) + Variance(Y) + 2*covariance(XY)

20. Law of Large Numbers
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Sample mean will near the population mean as the sample size increases
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

21. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Model dependent - Options with the same underlying assets may trade at different volatilities
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

22. Confidence interval for sample mean
Variance(x) + Variance(Y) + 2*covariance(XY)
P(X=x - Y=y) = P(X=x) * P(Y=y)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

23. Standard error
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Sum of n i.i.d. Bernouli variables - Probability of k successes: (combination n over k)(p^k)(1 - p)^(n - k) - (n over k) = (n!)/((n - k)!k!)
Variance(x) + Variance(Y) + 2*covariance(XY)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

24. Overall F - statistic
Sum of n i.i.d. Bernouli variables - Probability of k successes: (combination n over k)(p^k)(1 - p)^(n - k) - (n over k) = (n!)/((n - k)!k!)
Yi = B0 + B1Xi + ui
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

25. Hazard rate of exponentially distributed random variable
Concerned with a single random variable (ex. Roll of a die)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Among all unbiased estimators - estimator with the smallest variance is efficient

26. Variance(discrete)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

27. Monte Carlo Simulations
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
(a^2)(variance(x)
Low Frequency - High Severity events

28. Variance of X+b
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Variance(x)
F(x) = (1/stddev(x)sqrt(2pi))e^ - (x - mean)^2/(2variance) - skew = 0 - Parsimony = only requires mean and variance - Summation stability = combination of two normal distributions is a normal distribution - Kurtosis = 3
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

29. Poisson distribution equations for mean variance and std deviation
Variance(X) + Variance(Y) - 2*covariance(XY)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Peaks over threshold - Collects dataset in excess of some threshold

30. Difference between population and sample variance
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Population denominator = n - Sample denominator = n - 1

31. SER
Confidence level
Normal - Student's T - Chi - square - F distribution
(a^2)(variance(x)) + (b^2)(variance(y))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

32. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance ratio distribution F = (variance(x)/variance(y)) - Greater sample variance is numerator - Nonnegative and skewed right - Approaches normal as df increases - Square of t - distribution has a F distribution with 1 -k df - M*F(m -n) = Chi - s
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Standard error of the regression - SER = sqrt(SSR/(n - 2)) = sqrt((ei^2)/(n - 2)) SSR - Sum of squared residuals - Summation[(Yi - predicted Yi)^2] - Summation of each squared deviation between the actual Y and the predicted Y - Directly related

33. Binomial distribution equations for mean variance and std dev
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Mean = np - Variance = npq - Std dev = sqrt(npq)
For n>30 - sample mean is approximately normal
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

34. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Confidence set for two coefficients - two dimensional analog for the confidence interval

35. Hybrid method for conditional volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Use historical simulation approach but use the EWMA weighting system
(a^2)(variance(x)) + (b^2)(variance(y))

36. Time series data
Z = (Y - meany)/(stddev(y)/sqrt(n))
Returns over time for an individual asset
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Based on an equation - P(A) = # of A/total outcomes

37. Simplified standard (un - weighted) variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance = (1/m) summation(u<n - i>^2)
When one regressor is a perfect linear function of the other regressors

38. Variance of sample mean
Only requires two parameters = mean and variance
Based on an equation - P(A) = # of A/total outcomes
Rxy = Sxy/(Sx*Sy)
Variance(y)/n = variance of sample Y

39. Confidence ellipse
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(X) + Variance(Y) - 2*covariance(XY)
Use historical simulation approach but use the EWMA weighting system

40. Type II Error
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
We accept a hypothesis that should have been rejected

41. Stochastic error term
Confidence level
Low Frequency - High Severity events
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Contains variables not explicit in model - Accounts for randomness

42. Shortcomings of implied volatility
When asset return(r) is normally distributed - the continuously compounded future asset price level is lognormal - Reverse is true - if a variable is lognormal - its natural log is normal
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test
Variance(y)/n = variance of sample Y
Model dependent - Options with the same underlying assets may trade at different volatilities

43. Economical(elegant)
Application of mathematical statistics to economic data to lend empirical support to models
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Based on a dataset
Only requires two parameters = mean and variance

44. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

45. Continuously compounded return equation
i = ln(Si/Si - 1)
Independently and Identically Distributed
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Standard error of the regression - SER = sqrt(SSR/(n - 2)) = sqrt((ei^2)/(n - 2)) SSR - Sum of squared residuals - Summation[(Yi - predicted Yi)^2] - Summation of each squared deviation between the actual Y and the predicted Y - Directly related

46. P - value
P(Z>t)
Mean = np - Variance = npq - Std dev = sqrt(npq)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
95% = 1.65 99% = 2.33 For one - tailed tests

47. Efficiency
Expected value of the sample mean is the population mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Among all unbiased estimators - estimator with the smallest variance is efficient

48. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Low Frequency - High Severity events
When the sample size is large - the uncertainty about the value of the sample is very small
Variance ratio distribution F = (variance(x)/variance(y)) - Greater sample variance is numerator - Nonnegative and skewed right - Approaches normal as df increases - Square of t - distribution has a F distribution with 1 -k df - M*F(m -n) = Chi - s

49. F distribution
Variance ratio distribution F = (variance(x)/variance(y)) - Greater sample variance is numerator - Nonnegative and skewed right - Approaches normal as df increases - Square of t - distribution has a F distribution with 1 -k df - M*F(m -n) = Chi - s
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Random walk (usually acceptable) - Constant volatility (unlikely)
Var(X) + Var(Y)

50. Mean reversion in variance
Variance reverts to a long run level
Flexible and postulate stochastic process or resample historical data - Full valuation on target date - More prone to model risk - Slow and loses precision due to sampling variation
We accept a hypothesis that should have been rejected
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)