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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. Discrete random variable
Expected value of the sample mean is the population mean
Combine to form distribution with leptokurtosis (heavy tails)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Choose parameters that maximize the likelihood of what observations occurring

2. Heteroskedastic
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
If variance of the conditional distribution of u(i) is not constant

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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

4. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

5. Discrete representation of the GBM
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

6. Gamma distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Transformed to a unit variable - Mean = 0 Variance = 1
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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

7. Adjusted R^2
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
When one regressor is a perfect linear function of the other regressors

8. 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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E(mean) = mean
Var(X) + Var(Y)

9. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Choose parameters that maximize the likelihood of what observations occurring
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

10. WLS
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
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

11. Inverse transform method
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Independently and Identically Distributed
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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!)

12. Historical std dev
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Among all unbiased estimators - estimator with the smallest variance is efficient

13. Confidence interval for sample mean
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

14. Hybrid method for conditional volatility
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
SSR
Use historical simulation approach but use the EWMA weighting system
Returns over time for an individual asset

15. BLUE
P(X=x - Y=y) = P(X=x) * P(Y=y)
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)
Statement of the error or precision of an estimate

16. Skewness
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
Distribution with only two possible outcomes
Independently and Identically Distributed
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

17. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Mean of sampling distribution is the population mean
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Returns over time for an individual asset

18. Variance of X - Y assuming dependence
Contains variables not explicit in model - Accounts for randomness
Variance(X) + Variance(Y) - 2*covariance(XY)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

19. SER
(a^2)(variance(x)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

20. GARCH
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)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

21. T distribution
When one regressor is a perfect linear function of the other regressors
E(mean) = mean
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

22. LFHS
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
Low Frequency - High Severity events
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

23. Hazard rate of exponentially distributed random variable
Var(X) + Var(Y)
When the sample size is large - the uncertainty about the value of the sample is very small
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

24. Variance(discrete)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Mean of sampling distribution is the population mean
Variance reverts to a long run level

25. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
i = ln(Si/Si - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

26. Economical(elegant)
Random walk (usually acceptable) - Constant volatility (unlikely)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Only requires two parameters = mean and variance
SSR

27. Test for unbiasedness
E(mean) = mean
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Easy to manipulate
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

28. Exact significance level
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
P - value
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

29. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Special type of pooled data in which the cross sectional unit is surveyed over time
Choose parameters that maximize the likelihood of what observations occurring

30. Standard normal distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon
Transformed to a unit variable - Mean = 0 Variance = 1
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

31. Perfect multicollinearity
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Sampling distribution of sample means tend to be normal
When one regressor is a perfect linear function of the other regressors

32. Variance of X+b
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
Variance(x)
Population denominator = n - Sample denominator = n - 1
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

33. Variance of X+Y assuming dependence
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)
Average return across assets on a given day
Variance reverts to a long run level

34. Mean reversion
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

35. Marginal unconditional probability function
Does not depend on a prior event or information
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Application of mathematical statistics to economic data to lend empirical support to models
P(X=x - Y=y) = P(X=x) * P(Y=y)

36. Standard error
Peaks over threshold - Collects dataset in excess of some threshold
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

37. Sample covariance
[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
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

38. Central Limit Theorem(CLT)
Choose parameters that maximize the likelihood of what observations occurring
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)
P(Z>t)

39. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Population denominator = n - Sample denominator = n - 1
Contains variables not explicit in model - Accounts for randomness
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

40. Deterministic Simulation
Special type of pooled data in which the cross sectional unit is surveyed over time
i = ln(Si/Si - 1)
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

41. Logistic distribution
We reject a hypothesis that is actually true
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Based on a dataset
Has heavy tails

42. Cross - sectional
Average return across assets on a given day
Random walk (usually acceptable) - Constant volatility (unlikely)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Among all unbiased estimators - estimator with the smallest variance is efficient

43. Variance of X+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
Returns over time for an individual asset
Var(X) + Var(Y)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

44. Sample mean
Nonlinearity
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Expected value of the sample mean is the population mean

45. Shortcomings of implied volatility
Concerned with a single random variable (ex. Roll of a die)
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Sample mean +/ - t*(stddev(s)/sqrt(n))

46. Central Limit Theorem
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
For n>30 - sample mean is approximately normal
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

47. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
SSR
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
Regression can be non - linear in variables but must be linear in parameters

48. Continuous representation of the GBM
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Use historical simulation approach but use the EWMA weighting system
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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)

49. Key properties of linear regression
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
Regression can be non - linear in variables but must be linear in parameters
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
95% = 1.65 99% = 2.33 For one - tailed tests

50. i.i.d.
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Independently and Identically Distributed