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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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1. Overall F - statistic
Var(X) + Var(Y)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Nonlinearity
Price/return tends to run towards a long - run level

2. Variance of X+Y assuming dependence
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(x) + Variance(Y) + 2*covariance(XY)

3. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Returns over time for an individual asset
Summation((xi - mean)^k)/n
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

4. Implications of homoscedasticity
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Choose parameters that maximize the likelihood of what observations occurring
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

5. Time series data
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Returns over time for an individual asset

6. Variance of sampling distribution of means when n<N
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
We accept a hypothesis that should have been rejected
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

7. BLUE
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
We reject a hypothesis that is actually true
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

8. Confidence interval (from t)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E(XY) - E(X)E(Y)

9. Tractable
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
Easy to manipulate
E(XY) - E(X)E(Y)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

10. Variance of X - Y assuming dependence
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
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)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

11. Empirical frequency
When the sample size is large - the uncertainty about the value of the sample is very small
Choose parameters that maximize the likelihood of what observations occurring
Based on a dataset
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

12. Biggest (and only real) drawback of GARCH mode
Nonlinearity
If variance of the conditional distribution of u(i) is not constant
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

13. i.i.d.
When one regressor is a perfect linear function of the other regressors
Independently and Identically Distributed
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
We accept a hypothesis that should have been rejected

14. Variance of X+Y
Confidence set for two coefficients - two dimensional analog for the confidence interval
Var(X) + Var(Y)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

15. Exact significance level
Random walk (usually acceptable) - Constant volatility (unlikely)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
P - value
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

16. What does the OLS minimize?
Peaks over threshold - Collects dataset in excess of some threshold
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Variance(y)/n = variance of sample Y
SSR

17. Standard error for Monte Carlo replications
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance reverts to a long run level

18. Multivariate probability
Combine to form distribution with leptokurtosis (heavy tails)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
More than one random variable

19. LFHS
Variance(y)/n = variance of sample Y
Low Frequency - High Severity events
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

20. Bernouli Distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Distribution with only two possible outcomes

21. SER
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

22. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

23. Multivariate Density Estimation (MDE)
95% = 1.65 99% = 2.33 For one - tailed tests
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

24. WLS
Variance(X) + Variance(Y) - 2*covariance(XY)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
P(X=x - Y=y) = P(X=x) * P(Y=y)

25. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

26. Confidence interval for sample mean
Rxy = Sxy/(Sx*Sy)
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

27. Kurtosis
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

28. Reliability
Confidence level
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance = (1/m) summation(u<n - i>^2)
Statement of the error or precision of an estimate

29. Unconditional vs conditional distributions
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
We reject a hypothesis that is actually true
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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

30. Variance - covariance approach for VaR of a portfolio
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
Transformed to a unit variable - Mean = 0 Variance = 1
Low Frequency - High Severity events
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

31. Non - parametric vs parametric calculation of VaR
Statement of the error or precision of an estimate
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
E(mean) = mean
Average return across assets on a given day

32. Unstable return distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
P - value

33. Law of Large Numbers
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean will near the population mean as the sample size increases
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)
Low Frequency - High Severity events

34. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Var(X) + Var(Y)
Choose parameters that maximize the likelihood of what observations occurring
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

35. Cross - sectional
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Average return across assets on a given day
Independently and Identically Distributed

36. Economical(elegant)
Based on an equation - P(A) = # of A/total outcomes
Only requires two parameters = mean and variance
Variance(x) + Variance(Y) + 2*covariance(XY)
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

37. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Application of mathematical statistics to economic data to lend empirical support to models
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

38. Sample correlation
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Regression can be non - linear in variables but must be linear in parameters
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Rxy = Sxy/(Sx*Sy)

39. Standard error
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Has heavy tails
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

40. Homoskedastic only F - stat
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Has heavy tails
Transformed to a unit variable - Mean = 0 Variance = 1

41. Econometrics
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Application of mathematical statistics to economic data to lend empirical support to models

42. Variance of weighted scheme
Normal - Student's T - Chi - square - F distribution
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Based on an equation - P(A) = # of A/total outcomes

43. Consistent
Confidence level
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
When the sample size is large - the uncertainty about the value of the sample is very small

44. Simplified standard (un - weighted) variance
Low Frequency - High Severity events
Population denominator = n - Sample denominator = n - 1
Price/return tends to run towards a long - run level
Variance = (1/m) summation(u<n - i>^2)

45. Joint probability functions
Concerned with a single random variable (ex. Roll of a die)
P - value
Attempts to sample along more important paths
Probability that the random variables take on certain values simultaneously

46. Variance of sample mean
Peaks over threshold - Collects dataset in excess of some threshold
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance(y)/n = variance of sample Y
Sampling distribution of sample means tend to be normal

47. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
(a^2)(variance(x)) + (b^2)(variance(y))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Contains variables not explicit in model - Accounts for randomness

48. Binomial distribution equations for mean variance and std dev
Independently and Identically Distributed
Mean = np - Variance = npq - Std dev = sqrt(npq)
Only requires two parameters = mean and variance
When one regressor is a perfect linear function of the other regressors

49. Pooled data
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(y)/n = variance of sample Y
Returns over time for a combination of assets (combination of time series and cross - sectional data)
We reject a hypothesis that is actually true

50. Two ways to calculate historical volatility
Choose parameters that maximize the likelihood of what observations occurring
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Based on an equation - P(A) = # of A/total outcomes
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric