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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Chi - squared distribution
Population denominator = n - Sample denominator = n - 1
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
Mean = np - Variance = npq - Std dev = sqrt(npq)

2. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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

3. Single variable (univariate) probability
Normal - Student's T - Chi - square - F distribution
Population denominator = n - Sample denominator = n - 1
Concerned with a single random variable (ex. Roll of a die)
P(X=x - Y=y) = P(X=x) * P(Y=y)

4. Confidence interval (from t)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sample mean +/ - t*(stddev(s)/sqrt(n))

5. Hybrid method for conditional volatility
Has heavy tails
Only requires two parameters = mean and variance
Use historical simulation approach but use the EWMA weighting system
Sample mean will near the population mean as the sample size increases

6. Consistent
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
We accept a hypothesis that should have been rejected
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
When the sample size is large - the uncertainty about the value of the sample is very small

7. Unbiased
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(x) = (1/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared
Mean of sampling distribution is the population 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!)

8. LFHS
Low Frequency - High Severity events
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Independently and Identically Distributed

9. Unstable return distribution
Easy to manipulate
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
If variance of the conditional distribution of u(i) is not constant

10. Bootstrap method
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Use historical simulation approach but use the EWMA weighting system
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

11. Law of Large Numbers
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sample mean will near the population mean as the sample size increases
Combine to form distribution with leptokurtosis (heavy tails)
Z = (Y - meany)/(stddev(y)/sqrt(n))

12. Variance of X - Y assuming dependence
Easy to manipulate
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance(X) + Variance(Y) - 2*covariance(XY)

13. GPD
Random walk (usually acceptable) - Constant volatility (unlikely)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

14. Mean(expected value)
Regression can be non - linear in variables but must be linear in parameters
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)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

15. F distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

16. Control variates technique
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

17. Variance of aX
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
(a^2)(variance(x)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

18. Historical std dev
Choose parameters that maximize the likelihood of what observations occurring
Contains variables not explicit in model - Accounts for randomness
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

19. Sample correlation
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Rxy = Sxy/(Sx*Sy)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Sample mean will near the population mean as the sample size increases

20. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Contains variables not explicit in model - Accounts for randomness
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

21. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Special type of pooled data in which the cross sectional unit is surveyed over time
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

22. Key properties of linear regression
Variance(x) + Variance(Y) + 2*covariance(XY)
Regression can be non - linear in variables but must be linear in parameters
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

23. SER
Returns over time for an individual asset
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance reverts to a long run level

24. Direction of OVB
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Confidence level
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

25. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Regression can be non - linear in variables but must be linear in parameters
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

26. Multivariate probability
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
More than one random variable
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(y)/n = variance of sample Y

27. Continuous representation of the GBM
E(XY) - E(X)E(Y)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
We reject a hypothesis that is actually true
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)

28. Homoskedastic only F - stat
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance = (1/m) summation(u<n - i>^2)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

29. GARCH
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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 = np - Variance = npq - Std dev = sqrt(npq)

30. Importance sampling technique
Attempts to sample along more important paths
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!)
Only requires two parameters = mean and variance
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

31. Mean reversion in variance
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance reverts to a long run level
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

32. Priori (classical) probability
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Based on an equation - P(A) = # of A/total outcomes
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

33. Stochastic error term
Variance(x)
Contains variables not explicit in model - Accounts for randomness
Probability that the random variables take on certain values simultaneously
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

34. Pooled data
E(XY) - E(X)E(Y)
Model dependent - Options with the same underlying assets may trade at different volatilities
Rxy = Sxy/(Sx*Sy)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

35. Homoskedastic
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
Model dependent - Options with the same underlying assets may trade at different volatilities
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 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

36. Poisson Distribution
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
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

37. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(x) + Variance(Y) + 2*covariance(XY)
Combine to form distribution with leptokurtosis (heavy tails)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

38. Sample mean
Expected value of the sample mean is the population mean
Use historical simulation approach but use the EWMA weighting system
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

39. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Special type of pooled data in which the cross sectional unit is surveyed over time
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
More than one random variable

40. Unconditional vs conditional distributions
P(X=x - Y=y) = P(X=x) * P(Y=y)
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

41. Mean reversion in asset dynamics
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
Price/return tends to run towards 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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

42. GEV
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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

43. Two assumptions of square root rule
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Random walk (usually acceptable) - Constant volatility (unlikely)
Contains variables not explicit in model - Accounts for randomness
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

44. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^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
For n>30 - sample mean is approximately normal
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

45. Implications of homoscedasticity
Sample mean will near the population mean as the sample size increases
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
Variance = (1/m) summation(u<n - i>^2)
Confidence level

46. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Sample mean +/ - t*(stddev(s)/sqrt(n))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

47. What does the OLS minimize?
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
SSR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

48. Adjusted R^2
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

49. Variance of aX + bY
When the sample size is large - the uncertainty about the value of the sample is very small
(a^2)(variance(x)) + (b^2)(variance(y))
Var(X) + Var(Y)
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

50. Cholesky factorization (decomposition)
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
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Combine to form distribution with leptokurtosis (heavy tails)