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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. Variance of X+Y
Peaks over threshold - Collects dataset in excess of some threshold
Summation((xi - mean)^k)/n
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Var(X) + Var(Y)

2. Type II Error
Yi = B0 + B1Xi + ui
Confidence set for two coefficients - two dimensional analog for the confidence interval
Independently and Identically Distributed
We accept a hypothesis that should have been rejected

3. Binomial distribution
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!)
Concerned with a single random variable (ex. Roll of a die)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Use historical simulation approach but use the EWMA weighting system

4. Confidence interval for sample mean
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(y)/n = variance of sample Y
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

5. Poisson Distribution
More than one random 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
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

6. Standard error for Monte Carlo replications
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

7. Least squares estimator(m)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

8. Joint probability functions
Probability that the random variables take on certain values simultaneously
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Summation((xi - mean)^k)/n
Transformed to a unit variable - Mean = 0 Variance = 1

9. Single variable (univariate) probability
Transformed to a unit variable - Mean = 0 Variance = 1
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Concerned with a single random variable (ex. Roll of a die)
Confidence set for two coefficients - two dimensional analog for the confidence interval

10. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Choose parameters that maximize the likelihood of what observations occurring
Regression can be non - linear in variables but must be linear in parameters
Expected value of the sample mean is the population mean

11. Potential reasons for fat tails in return distributions
Average return across assets on a given day
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Yi = B0 + B1Xi + ui
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

12. Bernouli Distribution
For n>30 - sample mean is approximately normal
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
Distribution with only two possible outcomes
Var(X) + Var(Y)

13. Mean reversion in asset dynamics
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
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
Average return across assets on a given day

14. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Easy to manipulate
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Model dependent - Options with the same underlying assets may trade at different volatilities

15. Covariance calculations using weight sums (lambda)
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
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
Concerned with a single random variable (ex. Roll of a die)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

16. Multivariate probability
Nonlinearity
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
Rxy = Sxy/(Sx*Sy)
More than one random variable

17. Biggest (and only real) drawback of GARCH mode
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
P(Z>t)
Nonlinearity
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!)

18. Variance(discrete)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Confidence level

19. Lognormal
Confidence level
Mean = np - Variance = npq - Std dev = sqrt(npq)
Var(X) + Var(Y)
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

20. Continuous random variable
Summation((xi - mean)^k)/n
Special type of pooled data in which the cross sectional unit is surveyed over time
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

21. Unbiased
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Mean of sampling distribution is the population mean
Special type of pooled data in which the cross sectional unit is surveyed over time
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

22. Conditional probability functions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

23. Two assumptions of square root rule
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Random walk (usually acceptable) - Constant volatility (unlikely)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

24. Time series data
Returns over time for an individual asset
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

25. 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
Combine to form distribution with leptokurtosis (heavy tails)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

26. ESS
Normal - Student's T - Chi - square - F distribution
Use historical simulation approach but use the EWMA weighting system
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Probability that the random variables take on certain values simultaneously

27. Variance - covariance approach for VaR of a portfolio
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Population denominator = n - Sample denominator = n - 1

28. Perfect multicollinearity
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
(a^2)(variance(x)) + (b^2)(variance(y))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
When one regressor is a perfect linear function of the other regressors

29. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Yi = B0 + B1Xi + ui
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

30. Two drawbacks of moving average series
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Concerned with a single random variable (ex. Roll of a die)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

31. Monte Carlo Simulations
Independently and Identically Distributed
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Attempts to sample along more important paths

32. What does the OLS minimize?
Confidence set for two coefficients - two dimensional analog for the confidence interval
SSR
P(Z>t)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

33. Economical(elegant)
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
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
Only requires two parameters = mean and variance
Based on a dataset

34. Variance of weighted scheme
We accept a hypothesis that should have been rejected
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

35. Mean reversion in variance
Variance(x) + Variance(Y) + 2*covariance(XY)
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
Sampling distribution of sample means tend to be normal
Variance reverts to a long run level

36. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Transformed to a unit variable - Mean = 0 Variance = 1
Contains variables not explicit in model - Accounts for randomness
Sampling distribution of sample means tend to be normal

37. Test for statistical independence
Expected value of the sample mean is the population mean
P(X=x - Y=y) = P(X=x) * P(Y=y)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(y)/n = variance of sample Y

38. Four sampling distributions

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39. Sample variance
Model dependent - Options with the same underlying assets may trade at different volatilities
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Price/return tends to run towards a long - run level
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

40. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
SSR
Summation((xi - mean)^k)/n

41. Continuous representation of the GBM
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)
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Choose parameters that maximize the likelihood of what observations occurring

42. R^2
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!)
Summation((xi - mean)^k)/n
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

43. Variance of sample mean
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(y)/n = variance of sample Y
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
For n>30 - sample mean is approximately normal

44. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Nonlinearity

45. Standard normal 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
Transformed to a unit variable - Mean = 0 Variance = 1
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Application of mathematical statistics to economic data to lend empirical support to models

46. Normal distribution
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
(a^2)(variance(x)) + (b^2)(variance(y))
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

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

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48. 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
Random walk (usually acceptable) - Constant volatility (unlikely)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

49. Square root rule
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Special type of pooled data in which the cross sectional unit is surveyed over time

50. Empirical frequency
Based on a dataset
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients