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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. Poisson Distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance = (1/m) summation(u<n - i>^2)
Returns over time for an individual asset

2. Beta distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

3. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Rxy = Sxy/(Sx*Sy)
Has heavy tails
Based on an equation - P(A) = # of A/total outcomes

4. P - value
P(Z>t)
Returns over time for an individual asset
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

5. Perfect multicollinearity
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
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
Easy to manipulate
When one regressor is a perfect linear function of the other regressors

6. Marginal unconditional probability function
Random walk (usually acceptable) - Constant volatility (unlikely)
Does not depend on a prior event or information
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

7. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
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
Rxy = Sxy/(Sx*Sy)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

8. Non - parametric vs parametric calculation of VaR
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Average return across assets on a given day
Variance(X) + Variance(Y) - 2*covariance(XY)

9. SER
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!)
Does not depend on a prior event or information
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

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

11. 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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

12. Type II Error
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
P - value
We accept a hypothesis that should have been rejected
Low Frequency - High Severity events

13. BLUE
Statement of the error or precision of an estimate
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

14. Simplified standard (un - weighted) variance
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance = (1/m) summation(u<n - i>^2)
Variance(sample y) = (variance(y)/n)*(N - n/N - 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!)

15. Conditional probability functions
Var(X) + Var(Y)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Attempts to sample along more important paths
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

16. Extending the HS approach for computing value of a portfolio

17. Continuous representation of the GBM
Least absolute deviations estimator - used when extreme outliers are not uncommon
Low Frequency - High Severity events
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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)

18. Lognormal
More than one random variable
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Combine to form distribution with leptokurtosis (heavy tails)

19. Mean reversion in asset dynamics
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

20. GPD
When the sample size is large - the uncertainty about the value of the sample is very small
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

21. Chi - squared distribution
Sampling distribution of sample means tend to be normal
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

22. WLS
Expected value of the sample mean is the population mean
Contains variables not explicit in model - Accounts for randomness
Does not depend on a prior event or information
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

23. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Choose parameters that maximize the likelihood of what observations occurring
Attempts to sample along more important paths
Sampling distribution of sample means tend to be normal

24. POT
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Peaks over threshold - Collects dataset in excess of some threshold
Z = (Y - meany)/(stddev(y)/sqrt(n))
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

25. Cholesky factorization (decomposition)
Independently and Identically Distributed
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
When one regressor is a perfect linear function of the other regressors

26. Econometrics
Mean = np - Variance = npq - Std dev = sqrt(npq)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Application of mathematical statistics to economic data to lend empirical support to models
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

27. Sample covariance
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Distribution with only two possible outcomes
[1/(n - 1)]*summation((Xi - X)(Yi - 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!)

28. GEV
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)
Nonlinearity
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

29. Empirical frequency
Based on a dataset
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

30. Variance of X - Y assuming dependence
Has heavy tails
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(X) + Variance(Y) - 2*covariance(XY)

31. Least squares estimator(m)
i = ln(Si/Si - 1)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Concerned with a single random variable (ex. Roll of a die)

32. Variance of X+b
Variance(x)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Nonlinearity
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

33. Standard normal distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
Transformed to a unit variable - Mean = 0 Variance = 1
For n>30 - sample mean is approximately 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

34. Homoskedastic
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

35. GARCH
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance(x)
Distribution with only two possible outcomes
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)

36. Continuous random variable
P(X=x - Y=y) = P(X=x) * P(Y=y)
Easy to manipulate
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Summation((xi - mean)^k)/n

37. EWMA
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance = (1/m) summation(u<n - i>^2)
Probability that the random variables take on certain values simultaneously
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

38. Sample variance
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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

39. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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)

40. ESS
Does not depend on a prior event or information
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

41. K - th moment
Summation((xi - mean)^k)/n
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)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

42. Mean(expected value)
Has heavy tails
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Returns over time for an individual asset
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

43. LFHS
Sampling distribution of sample means tend to be normal
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Low Frequency - High Severity events

44. Mean reversion in variance
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
SSR
Variance reverts to a long run level
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

45. Key properties of linear regression
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Regression can be non - linear in variables but must be linear in parameters
Contains variables not explicit in model - Accounts for randomness
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

46. Variance - covariance approach for VaR of a portfolio
SSR
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Mean of sampling distribution is the population mean
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

47. F distribution
P - value
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
Nonlinearity
Expected value of the sample mean is the population mean

48. Type I error
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
We reject a hypothesis that is actually true
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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

49. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
(a^2)(variance(x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Sample mean will near the population mean as the sample size increases

50. Covariance calculations using weight sums (lambda)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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)