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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Four sampling distributions

2. Continuous random variable
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

3. WLS
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Use historical simulation approach but use the EWMA weighting system
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

4. Persistence
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

5. Conditional probability functions
Special type of pooled data in which the cross sectional unit is surveyed over time
Confidence level
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

6. Monte Carlo Simulations
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)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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)

7. Empirical frequency
Based on a dataset
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
We accept a hypothesis that should have been rejected
Sampling distribution of sample means tend to be normal

8. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
We accept a hypothesis that should have been rejected
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

9. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Independently and Identically Distributed
(a^2)(variance(x)) + (b^2)(variance(y))
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

10. Economical(elegant)
We reject a hypothesis that is actually true
Rxy = Sxy/(Sx*Sy)
Only requires two parameters = mean and variance
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

11. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
We reject a hypothesis that is actually true
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

12. Sample mean
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Expected value of the sample mean is the population mean

13. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Probability that the random variables take on certain values simultaneously
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Random walk (usually acceptable) - Constant volatility (unlikely)

14. LAD
Variance(x) + Variance(Y) + 2*covariance(XY)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

15. 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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

16. Overall F - statistic
Easy to manipulate
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

17. Marginal unconditional probability function
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance = (1/m) summation(u<n - i>^2)
Does not depend on a prior event or information

18. What does the OLS minimize?
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
Sampling distribution of sample means tend to be normal
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
SSR

19. Standard variable for non - normal distributions
Has heavy tails
We reject a hypothesis that is actually true
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(x)

20. Hazard rate of exponentially distributed random variable
Easy to manipulate
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Independently and Identically Distributed
When one regressor is a perfect linear function of the other regressors

21. GPD
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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

22. Variance of aX + bY
Yi = B0 + B1Xi + ui
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
(a^2)(variance(x)) + (b^2)(variance(y))

23. GEV
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Based on a dataset
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

24. Binomial distribution
i = ln(Si/Si - 1)
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!)
When one regressor is a perfect linear function of the other regressors
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

25. Adjusted R^2
Combine to form distribution with leptokurtosis (heavy tails)
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
Average return across assets on a given day
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

26. Limitations of R^2 (what an increase doesn't necessarily imply)

27. Homoskedastic only F - stat
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance(y)/n = variance of sample Y
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

28. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
For n>30 - sample mean is approximately normal
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

29. Type I error
Var(X) + Var(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 = (1/m) summation(u<n - i>^2)
We reject a hypothesis that is actually true

30. Discrete random variable
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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

31. Cross - sectional
Population denominator = n - Sample denominator = n - 1
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Average return across assets on a given day
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

32. Two drawbacks of moving average series
Combine to form distribution with leptokurtosis (heavy tails)
Based on a dataset
We reject a hypothesis that is actually true
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

33. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
(a^2)(variance(x)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

34. Historical std dev
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
(a^2)(variance(x)) + (b^2)(variance(y))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

35. Control variates technique
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Population denominator = n - Sample denominator = n - 1
Normal - Student's T - Chi - square - F distribution

36. Tractable
Variance(X) + Variance(Y) - 2*covariance(XY)
Combine to form distribution with leptokurtosis (heavy tails)
If variance of the conditional distribution of u(i) is not constant
Easy to manipulate

37. Test for statistical independence
Does not depend on a prior event or information
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Mean = np - Variance = npq - Std dev = sqrt(npq)
P(X=x - Y=y) = P(X=x) * P(Y=y)

38. Priori (classical) probability
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Special type of pooled data in which the cross sectional unit is surveyed over time
Based on an equation - P(A) = # of A/total outcomes
Attempts to sample along more important paths

39. Pooled data
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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!)

40. Time series data
Returns over time for an individual asset
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

41. Gamma distribution
Expected value of the sample mean is the population mean
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
Based on a dataset
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

42. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
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
Combine to form distribution with leptokurtosis (heavy tails)
Summation((xi - mean)^k)/n

43. Confidence interval (from t)
Easy to manipulate
Distribution with only two possible outcomes
Variance(X) + Variance(Y) - 2*covariance(XY)
Sample mean +/ - t*(stddev(s)/sqrt(n))

44. Shortcomings of implied volatility
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Model dependent - Options with the same underlying assets may trade at different volatilities

45. Maximum likelihood method
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Choose parameters that maximize the likelihood of what observations occurring
Sampling distribution of sample means tend to be normal
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

46. POT
We reject a hypothesis that is actually true
Peaks over threshold - Collects dataset in excess of some threshold
Easy to manipulate
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

47. Bernouli Distribution
Distribution with only two possible outcomes
Price/return tends to run towards a long - run level
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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

48. Test for unbiasedness
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
i = ln(Si/Si - 1)
E(mean) = mean
Special type of pooled data in which the cross sectional unit is surveyed over time

49. Standard normal distribution
Nonlinearity
Transformed to a unit variable - Mean = 0 Variance = 1
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
Variance reverts to a long run level

50. Unconditional vs conditional distributions
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)
When one regressor is a perfect linear function of the other regressors
Average return across assets on a given day