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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Difference between population and sample variance
Application of mathematical statistics to economic data to lend empirical support to models
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!)
Sampling distribution of sample means tend to be normal
Population denominator = n - Sample denominator = n - 1

2. Result of combination of two normal with same means
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Combine to form distribution with leptokurtosis (heavy tails)

3. Variance of X+b
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Least absolute deviations estimator - used when extreme outliers are not uncommon
Variance(x)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

4. Type I error
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)
Transformed to a unit variable - Mean = 0 Variance = 1
We reject a hypothesis that is actually true
Based on a dataset

5. Cholesky factorization (decomposition)
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
Sample mean will near the population mean as the sample size increases
Application of mathematical statistics to economic data to lend empirical support to models
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

6. WLS
Low Frequency - High Severity events
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

7. Control variates technique
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
P(Z>t)

8. Discrete representation of the GBM
Variance(x)
Easy to manipulate
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
More than one random variable

9. Regime - switching volatility model
Contains variables not explicit in model - Accounts for randomness
Peaks over threshold - Collects dataset in excess of some threshold
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

10. Biggest (and only real) drawback of GARCH mode
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Nonlinearity
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

11. Empirical frequency
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Based on a dataset

12. Hazard rate of exponentially distributed random variable
Among all unbiased estimators - estimator with the smallest variance is efficient
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
i = ln(Si/Si - 1)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

13. Weibul distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Price/return tends to run towards a long - run level

14. Variance of sampling distribution of means when n<N
Special type of pooled data in which the cross sectional unit is surveyed over time
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample mean will near the population mean as the sample size increases

15. Simplified standard (un - weighted) variance
Sampling distribution of sample means tend to be normal
SSR
Variance = (1/m) summation(u<n - i>^2)
Does not depend on a prior event or information

16. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
If variance of the conditional distribution of u(i) is not constant
Probability that the random variables take on certain values simultaneously
E(XY) - E(X)E(Y)

17. Unconditional vs conditional distributions
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
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

18. Marginal unconditional probability function
Does not depend on a prior event or information
i = ln(Si/Si - 1)
Based on a dataset
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

19. Sample correlation
Low Frequency - High Severity events
Variance(x) + Variance(Y) + 2*covariance(XY)
Rxy = Sxy/(Sx*Sy)
Variance reverts to a long run level

20. Importance sampling technique
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Does not depend on a prior event or information
Attempts to sample along more important paths

21. Standard variable for non - normal distributions
Least absolute deviations estimator - used when extreme outliers are not uncommon
Yi = B0 + B1Xi + ui
Z = (Y - meany)/(stddev(y)/sqrt(n))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

22. 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
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
Summation((xi - mean)^k)/n
Variance(x) + Variance(Y) + 2*covariance(XY)

23. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
P(Z>t)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

24. Four sampling distributions

25. Kurtosis
We reject a hypothesis that is actually true
Var(X) + Var(Y)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

26. Bootstrap method
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
P(X=x - Y=y) = P(X=x) * P(Y=y)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

27. Exponential distribution
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Returns over time for an individual asset
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

28. Mean reversion
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

29. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Summation((xi - mean)^k)/n
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

30. Central Limit Theorem
For n>30 - sample mean is approximately normal
Yi = B0 + B1Xi + ui
We accept a hypothesis that should have been rejected
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

31. Critical z values
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
95% = 1.65 99% = 2.33 For one - tailed tests
Contains variables not explicit in model - Accounts for randomness
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

32. GPD
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

33. Inverse transform method
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

34. Square root rule
Transformed to a unit variable - Mean = 0 Variance = 1
P(X=x - Y=y) = P(X=x) * P(Y=y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

35. Tractable
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)
Easy to manipulate
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

36. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Contains variables not explicit in model - Accounts for randomness
Average return across assets on a given day

37. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
E(mean) = mean
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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)

38. Joint probability functions
Probability that the random variables take on certain values simultaneously
Returns over time for an individual asset
Summation((xi - mean)^k)/n
For n>30 - sample mean is approximately normal

39. Shortcomings of implied volatility
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(X) + Variance(Y) - 2*covariance(XY)
Model dependent - Options with the same underlying assets may trade at different volatilities
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)

40. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
When one regressor is a perfect linear function of the other regressors
Least absolute deviations estimator - used when extreme outliers are not uncommon
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

41. Deterministic Simulation
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Only requires two parameters = mean and variance
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
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

42. Homoskedastic only F - stat
Price/return tends to run towards a long - run level
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Returns over time for a combination of assets (combination of time series and cross - sectional data)

43. Confidence interval (from t)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Sample mean +/ - t*(stddev(s)/sqrt(n))

44. ESS
Based on an equation - P(A) = # of A/total outcomes
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

45. Multivariate Density Estimation (MDE)
Sampling distribution of sample means tend to be normal
Combine to form distribution with leptokurtosis (heavy tails)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
95% = 1.65 99% = 2.33 For one - tailed tests

46. Bernouli Distribution
SSR
Distribution with only two possible outcomes
Variance(x)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

47. Two drawbacks of moving average series
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Distribution with only two possible outcomes
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

48. Variance - covariance approach for VaR of a portfolio
Price/return tends to run towards a long - run level
Least absolute deviations estimator - used when extreme outliers are not uncommon
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Variance(y)/n = variance of sample Y

49. EWMA
More than one random variable
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

50. Simulating for VaR
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)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Random walk (usually acceptable) - Constant volatility (unlikely)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)