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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. Efficiency
Variance(y)/n = variance of sample Y
Among all unbiased estimators - estimator with the smallest variance is efficient
P(X=x - Y=y) = P(X=x) * P(Y=y)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

2. ESS
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
P(X=x - Y=y) = P(X=x) * P(Y=y)

3. Least squares estimator(m)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

4. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance(X) + Variance(Y) - 2*covariance(XY)

5. Mean reversion
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)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Statement of the error or precision of an estimate

6. SER
For n>30 - sample mean is approximately normal
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
Independently and Identically Distributed
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

7. Beta distribution
Application of mathematical statistics to economic data to lend empirical support to models
We reject a hypothesis that is actually true
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

8. Control variates technique
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

9. Variance of X+b
Average return across assets on a given day
Independently and Identically Distributed
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(x)

10. Cholesky factorization (decomposition)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Expected value of the sample mean is the population mean
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
Var(X) + Var(Y)

11. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Random walk (usually acceptable) - Constant volatility (unlikely)
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 = (1/m) summation(u<n - i>^2)

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

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13. Reliability
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!)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Statement of the error or precision of an estimate
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

14. R^2
Yi = B0 + B1Xi + ui
Sample mean will near the population mean as the sample size increases
SSR
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

15. Sample mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Expected value of the sample mean is the population mean
Attempts to sample along more important paths
Returns over time for an individual asset

16. Homoskedastic
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 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
SSR
When one regressor is a perfect linear function of the other regressors

17. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
SSR
Among all unbiased estimators - estimator with the smallest variance is efficient

18. Econometrics
Combine to form distribution with leptokurtosis (heavy tails)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Application of mathematical statistics to economic data to lend empirical support to models
Transformed to a unit variable - Mean = 0 Variance = 1

19. EWMA
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

20. Extreme Value Theory
Transformed to a unit variable - Mean = 0 Variance = 1
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Independently and Identically Distributed
Returns over time for a combination of assets (combination of time series and cross - sectional data)

21. Chi - squared distribution
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
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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

22. Implications of homoscedasticity
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
i = ln(Si/Si - 1)
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

23. Stochastic error term
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
(a^2)(variance(x)) + (b^2)(variance(y))

24. Bernouli Distribution
Distribution with only two possible outcomes
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

25. Joint probability functions
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Variance = (1/m) summation(u<n - i>^2)
Probability that the random variables take on certain values simultaneously

26. Difference between population and sample variance
Probability that the random variables take on certain values simultaneously
Population denominator = n - Sample denominator = n - 1
E(mean) = mean
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

27. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
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

28. Unconditional vs conditional distributions
95% = 1.65 99% = 2.33 For one - tailed tests
When the sample size is large - the uncertainty about the value of the sample is very small
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Easy to manipulate

29. Persistence
Sampling distribution of sample means tend to be normal
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
Rxy = Sxy/(Sx*Sy)
If variance of the conditional distribution of u(i) is not constant

30. Binomial distribution equations for mean variance and std dev
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Mean = np - Variance = npq - Std dev = sqrt(npq)
Mean of sampling distribution is the population mean
When the sample size is large - the uncertainty about the value of the sample is very small

31. Weibul distribution
Variance = (1/m) summation(u<n - i>^2)
Special type of pooled data in which the cross sectional unit is surveyed over time
Peaks over threshold - Collects dataset in excess of some threshold
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

32. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

33. Sample variance
Summation((xi - mean)^k)/n
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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

34. Variance(discrete)
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
P(Z>t)
Returns over time for an individual asset

35. Panel data (longitudinal or micropanel)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Special type of pooled data in which the cross sectional unit is surveyed over time
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Use historical simulation approach but use the EWMA weighting system

36. GARCH
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
We accept a hypothesis that should have been rejected
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)

37. Confidence interval for sample mean
Peaks over threshold - Collects dataset in excess of some threshold
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Yi = B0 + B1Xi + ui
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

38. Covariance calculations using weight sums (lambda)
When one regressor is a perfect linear function of the other regressors
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Among all unbiased estimators - estimator with the smallest variance is efficient

39. T distribution
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)
Returns over time for an individual asset
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

40. Variance of X+Y
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance = (1/m) summation(u<n - i>^2)
Var(X) + Var(Y)

41. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Probability that the random variables take on certain values simultaneously
Nonlinearity
Sample mean +/ - t*(stddev(s)/sqrt(n))

42. Multivariate Density Estimation (MDE)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
i = ln(Si/Si - 1)

43. Test for statistical independence
Based on a dataset
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)
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

44. Maximum likelihood method
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Choose parameters that maximize the likelihood of what observations occurring
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Attempts to sample along more important paths

45. Test for unbiasedness
E(mean) = mean
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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

46. Result of combination of two normal with same means
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Combine to form distribution with leptokurtosis (heavy tails)

47. P - value
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
P(Z>t)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

48. Two drawbacks of moving average series
Sample mean +/ - t*(stddev(s)/sqrt(n))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Based on a dataset
Random walk (usually acceptable) - Constant volatility (unlikely)

49. Variance of X+Y assuming dependence
Only requires two parameters = mean and variance
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(x) + Variance(Y) + 2*covariance(XY)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

50. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Has heavy tails
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
We reject a hypothesis that is actually true