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FRM Foundations Of Risk Management Quantitative Methods

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1. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Peaks over threshold - Collects dataset in excess of some threshold
(a^2)(variance(x)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

2. Binomial distribution equations for mean variance and std dev
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Regression can be non - linear in variables but must be linear in parameters
Mean = np - Variance = npq - Std dev = sqrt(npq)
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

3. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
When the sample size is large - the uncertainty about the value of the sample is very small
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

4. Simplified standard (un - weighted) variance
Independently and Identically Distributed
Returns over time for an individual asset
Variance = (1/m) summation(u<n - i>^2)
Use historical simulation approach but use the EWMA weighting system

5. Mean(expected value)
Variance(X) + Variance(Y) - 2*covariance(XY)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Independently and Identically Distributed
When one regressor is a perfect linear function of the other regressors

6. Type II Error
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
When the sample size is large - the uncertainty about the value of the sample is very small
Variance(X) + Variance(Y) - 2*covariance(XY)
We accept a hypothesis that should have been rejected

7. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Peaks over threshold - Collects dataset in excess of some threshold

8. Lognormal
Returns over time for an individual asset
Summation((xi - mean)^k)/n
Application of mathematical statistics to economic data to lend empirical support to models
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

9. Variance of aX + bY
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Returns over time for a combination of assets (combination of time series and cross - sectional data)
(a^2)(variance(x)) + (b^2)(variance(y))
95% = 1.65 99% = 2.33 For one - tailed tests

10. Variance of X+Y assuming dependence
Among all unbiased estimators - estimator with the smallest variance is efficient
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Low Frequency - High Severity events
Variance(x) + Variance(Y) + 2*covariance(XY)

11. Historical std dev
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)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Refers to whether distribution is symmetrical - Sigma^3 = E[(x - mean)^3]/sigma^3 - Positive skew = mean>median>mode - Negative skew = mean<median<mode - if zero - all are equal - Function of the third moment
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

12. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance reverts to a long run level

13. Cross - sectional
When the sample size is large - the uncertainty about the value of the sample is very small
Average return across assets on a given day
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Distribution with only two possible outcomes

14. Unconditional vs conditional distributions
P - value
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Regression can be non - linear in variables but must be linear in parameters
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

15. K - th moment
Sample mean +/ - t*(stddev(s)/sqrt(n))
Special type of pooled data in which the cross sectional unit is surveyed over time
Summation((xi - mean)^k)/n
Contains variables not explicit in model - Accounts for randomness

16. Direction of OVB
Independently and Identically Distributed
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
For n>30 - sample mean is approximately normal

17. Sample covariance
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)
Mean of sampling distribution is the population mean
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

18. Result of combination of two normal with same means
SSR
Combine to form distribution with leptokurtosis (heavy tails)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Among all unbiased estimators - estimator with the smallest variance is efficient

19. Importance sampling technique
95% = 1.65 99% = 2.33 For one - tailed tests
Attempts to sample along more important paths
Summation((xi - mean)^k)/n
Yi = B0 + B1Xi + ui

20. Key properties of linear regression
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Regression can be non - linear in variables but must be linear in parameters
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

21. Statistical (or empirical) model
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Easy to manipulate
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Yi = B0 + B1Xi + ui

22. Deterministic Simulation
Instead of independent samples - systematically fills space left by previous numbers in the series - Std error shrinks at 1/k instead of 1/sqrt(k) but accuracy determination is hard since variables are not independent
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
Contains variables not explicit in model - Accounts for randomness

23. Mean reversion in asset dynamics
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)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Price/return tends to run towards a long - run level
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

24. Simulation models
Combine to form distribution with leptokurtosis (heavy tails)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
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

25. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Attempts to sample along more important paths
Returns over time for an individual asset

26. Skewness
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Refers to whether distribution is symmetrical - Sigma^3 = E[(x - mean)^3]/sigma^3 - Positive skew = mean>median>mode - Negative skew = mean<median<mode - if zero - all are equal - Function of the third moment
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance(y)/n = variance of sample Y

27. Logistic distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon
Has heavy tails
Population denominator = n - Sample denominator = n - 1
Yi = B0 + B1Xi + ui

28. Sample mean
We accept a hypothesis that should have been rejected
Special type of pooled data in which the cross sectional unit is surveyed over time
Expected value of the sample mean is the population mean
Returns over time for a combination of assets (combination of time series and cross - sectional data)

29. What does the OLS minimize?
Easy to manipulate
SSR
Choose parameters that maximize the likelihood of what observations occurring
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

30. Standard normal distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Combine to form distribution with leptokurtosis (heavy tails)
Transformed to a unit variable - Mean = 0 Variance = 1

31. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

32. Law of Large Numbers
Variance(x) + Variance(Y) + 2*covariance(XY)
Rxy = Sxy/(Sx*Sy)
Sample mean will near the population mean as the sample size increases
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

33. Unstable return distribution
P - value
Sample mean +/ - t*(stddev(s)/sqrt(n))
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

34. Hazard rate of exponentially distributed random variable
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Sample mean +/ - t*(stddev(s)/sqrt(n))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

35. Variance of sample mean
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(y)/n = variance of sample Y
Probability that the random variables take on certain values simultaneously
Independently and Identically Distributed

36. Cholesky factorization (decomposition)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Concerned with a single random variable (ex. Roll of a die)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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

37. Multivariate probability
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)
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
More than one random variable

38. Mean reversion in variance
Variance reverts to a long run level
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
(a^2)(variance(x)) + (b^2)(variance(y))

39. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(X) + Variance(Y) - 2*covariance(XY)
Use historical simulation approach but use the EWMA weighting system

40. Continuous random variable
Variance(x)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance = (1/m) summation(u<n - i>^2)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

41. GEV
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Expected value of the sample mean is the population mean
When one regressor is a perfect linear function of the other regressors
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

42. Biggest (and only real) drawback of GARCH mode
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)
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
Nonlinearity
Variance reverts to a long run level

43. Homoskedastic only F - stat
Among all unbiased estimators - estimator with the smallest variance is efficient
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

44. GARCH
Based on a dataset
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

45. Implications of homoscedasticity
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Distribution with only two possible outcomes

46. Extreme Value Theory
Sampling distribution of sample means tend to be normal
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
When one regressor is a perfect linear function of the other regressors
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

47. Sample variance
F(x) = (1/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

48. Heteroskedastic
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
95% = 1.65 99% = 2.33 For one - tailed tests
If variance of the conditional distribution of u(i) is not constant

49. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Has heavy tails
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E(XY) - E(X)E(Y)

50. Continuous representation of the GBM
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)
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
Variance(X) + Variance(Y) - 2*covariance(XY)

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