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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. Binomial distribution
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Independently and Identically Distributed
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!)
Peaks over threshold - Collects dataset in excess of some threshold

2. Standard variable for non - normal distributions
Variance(x)
More than one random variable
Z = (Y - meany)/(stddev(y)/sqrt(n))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

3. GARCH
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 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)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
We accept a hypothesis that should have been rejected

4. Discrete random variable
Sample mean will near the population mean as the sample size increases
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

5. Standard error for Monte Carlo replications
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
Transformed to a unit variable - Mean = 0 Variance = 1
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Concerned with a single random variable (ex. Roll of a die)

6. Continuous representation of the GBM
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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)

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

8. Least squares estimator(m)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
For n>30 - sample mean is approximately normal

9. Adjusted R^2
Z = (Y - meany)/(stddev(y)/sqrt(n))
Rxy = Sxy/(Sx*Sy)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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

10. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

11. LAD
Mean = np - Variance = npq - Std dev = sqrt(npq)
Least absolute deviations estimator - used when extreme outliers are not uncommon
For n>30 - sample mean is approximately normal
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

12. Logistic distribution
We reject a hypothesis that is actually true
Has heavy tails
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)

13. Exact significance level
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
P - value
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

14. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Contains variables not explicit in model - Accounts for randomness

15. Type I error
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
We reject a hypothesis that is actually true
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

16. POT
When one regressor is a perfect linear function of the other regressors
Special type of pooled data in which the cross sectional unit is surveyed over time
Summation((xi - mean)^k)/n
Peaks over threshold - Collects dataset in excess of some threshold

17. Regime - switching volatility model
Sample mean will near the population mean as the sample size increases
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
When the sample size is large - the uncertainty about the value of the sample is very small
Does not depend on a prior event or information

18. Marginal unconditional probability function
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Does not depend on a prior event or information
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

19. Shortcomings of implied volatility
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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!)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Model dependent - Options with the same underlying assets may trade at different volatilities

20. Cholesky factorization (decomposition)
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
Independently and Identically Distributed
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Z = (Y - meany)/(stddev(y)/sqrt(n))

21. Inverse transform method
Regression can be non - linear in variables but must be linear in parameters
Returns over time for an individual asset
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

22. Joint probability functions
Based on an equation - P(A) = # of A/total outcomes
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Concerned with a single random variable (ex. Roll of a die)
Probability that the random variables take on certain values simultaneously

23. Sample mean
95% = 1.65 99% = 2.33 For one - tailed tests
Expected value of the sample mean is the population mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

24. Homoskedastic
Normal - Student's T - Chi - square - F distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(y)/n = variance of sample Y
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

25. Economical(elegant)
Based on a dataset
Only requires two parameters = mean and variance
Random walk (usually acceptable) - Constant volatility (unlikely)
P - value

26. Difference between population and sample variance
Has heavy tails
Population denominator = n - Sample denominator = n - 1
Model dependent - Options with the same underlying assets may trade at different volatilities
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

27. Multivariate probability
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
More than one random variable
Least absolute deviations estimator - used when extreme outliers are not uncommon

28. 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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Does not depend on a prior event or information
Easy to manipulate

29. Sample variance
Variance = (1/m) summation(u<n - i>^2)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

30. Variance of X+b
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(x)
Var(X) + Var(Y)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

31. Significance =1
Confidence level
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

32. What does the OLS minimize?
SSR
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Among all unbiased estimators - estimator with the smallest variance is efficient
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

33. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

34. Pooled data
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Summation((xi - mean)^k)/n
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Special type of pooled data in which the cross sectional unit is surveyed over time

35. Covariance calculations using weight sums (lambda)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Model dependent - Options with the same underlying assets may trade at different volatilities
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

36. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Var(X) + Var(Y)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

37. Sample covariance
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Has heavy tails
Low Frequency - High Severity events

38. F distribution
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)
Probability that the random variables take on certain values simultaneously
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

39. Expected future variance rate (t periods forward)
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
E(mean) = mean
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

40. Sample correlation
Confidence set for two coefficients - two dimensional analog for the confidence interval
Mean = np - Variance = npq - Std dev = sqrt(npq)
Rxy = Sxy/(Sx*Sy)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

41. Tractable
Easy to manipulate
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)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

42. WLS
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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

43. EWMA
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Based on an equation - P(A) = # of A/total outcomes
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

44. Potential reasons for fat tails in return distributions
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Confidence level

45. Monte Carlo Simulations
For n>30 - sample mean is approximately normal
Statement of the error or precision of an estimate
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

46. Weibul distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
We accept a hypothesis that should have been rejected
P(X=x - Y=y) = P(X=x) * P(Y=y)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

47. Simulation models
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

48. Critical z values
Sample mean will near the population mean as the sample size increases
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Rxy = Sxy/(Sx*Sy)
95% = 1.65 99% = 2.33 For one - tailed tests

49. Econometrics
We reject a hypothesis that is actually true
Expected value of the sample mean is the population mean
When one regressor is a perfect linear function of the other regressors
Application of mathematical statistics to economic data to lend empirical support to models

50. Mean reversion in asset dynamics
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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
Population denominator = n - Sample denominator = n - 1
Price/return tends to run towards a long - run level