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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. K - th moment
Peaks over threshold - Collects dataset in excess of some threshold
Transformed to a unit variable - Mean = 0 Variance = 1
E(mean) = mean
Summation((xi - mean)^k)/n

2. Simulating for VaR
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Normal - Student's T - Chi - square - F distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

3. Law of Large Numbers
Model dependent - Options with the same underlying assets may trade at different volatilities
Least absolute deviations estimator - used when extreme outliers are not uncommon
Sample mean will near the population mean as the sample size increases
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

4. SER
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Contains variables not explicit in model - Accounts for randomness
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
(a^2)(variance(x)

5. Bernouli Distribution
Distribution with only two possible outcomes
Confidence level
Rxy = Sxy/(Sx*Sy)
(a^2)(variance(x)) + (b^2)(variance(y))

6. Mean(expected value)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

7. Time series data
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
Returns over time for an individual asset
Variance(x) + Variance(Y) + 2*covariance(XY)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

8. Covariance
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Application of mathematical statistics to economic data to lend empirical support to models
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
E(XY) - E(X)E(Y)

9. Marginal unconditional probability function
Summation((xi - mean)^k)/n
Does not depend on a prior event or information
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Special type of pooled data in which the cross sectional unit is surveyed over time

10. Logistic distribution
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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.
Has heavy tails

11. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Expected value of the sample mean is the population mean
Returns over time for an individual asset
Model dependent - Options with the same underlying assets may trade at different volatilities

12. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
Probability that the random variables take on certain values simultaneously
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

13. Type II Error
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
We accept a hypothesis that should have been rejected
Low Frequency - High Severity events

14. Confidence interval (from t)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Does not depend on a prior event or information
Sample mean +/ - t*(stddev(s)/sqrt(n))
Attempts to sample along more important paths

15. Block maxima
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
More than one random variable
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

16. Statistical (or empirical) model
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance = (1/m) summation(u<n - i>^2)
Variance(x)
Yi = B0 + B1Xi + ui

17. Homoskedastic only F - stat
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Population denominator = n - Sample denominator = n - 1

18. Non - parametric vs parametric calculation of VaR
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

19. Monte Carlo Simulations
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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!)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
(a^2)(variance(x)

20. Two ways to calculate historical volatility
Easy to manipulate
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!)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

21. Four sampling distributions

22. Historical std dev
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

23. Exact significance level
P - value
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))

24. Homoskedastic
Combine to form distribution with leptokurtosis (heavy tails)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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. Antithetic variable technique
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Returns over time for an individual asset
We accept a hypothesis that should have been rejected

26. Variance of aX
Variance(x) + Variance(Y) + 2*covariance(XY)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
(a^2)(variance(x)
Use historical simulation approach but use the EWMA weighting system

27. Simplified standard (un - weighted) variance
Nonlinearity
Variance = (1/m) summation(u<n - i>^2)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance reverts to a long run level

28. Regime - switching volatility model
P(X=x - Y=y) = P(X=x) * P(Y=y)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Use historical simulation approach but use the EWMA weighting system

29. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

30. GARCH
Choose parameters that maximize the likelihood of what observations occurring
When the sample size is large - the uncertainty about the value of the sample is very small
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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)

31. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
P - value
When one regressor is a perfect linear function of the other regressors
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

32. Multivariate probability
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
More than one random variable
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

33. 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!)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more 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

34. Extending the HS approach for computing value of a portfolio

35. POT
Peaks over threshold - Collects dataset in excess of some threshold
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
Yi = B0 + B1Xi + ui
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

36. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
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)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

37. Variance(discrete)
Z = (Y - meany)/(stddev(y)/sqrt(n))
(a^2)(variance(x)) + (b^2)(variance(y))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

38. Square root rule
Easy to manipulate
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Based on an equation - P(A) = # of A/total outcomes

39. Cholesky factorization (decomposition)
P - value
Choose parameters that maximize the likelihood of what observations occurring
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
Mean = np - Variance = npq - Std dev = sqrt(npq)

40. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many 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

41. Hazard rate of exponentially distributed random variable
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Transformed to a unit variable - Mean = 0 Variance = 1
Only requires two parameters = mean and variance
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

42. Exponential distribution
Contains variables not explicit in model - Accounts for randomness
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Sample mean will near the population mean as the sample size increases

43. Continuously compounded return equation
i = ln(Si/Si - 1)
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)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

44. Poisson distribution equations for mean variance and std deviation
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
We reject a hypothesis that is actually true
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

45. Empirical frequency
Based on a dataset
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Returns over time for an individual asset
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

46. Perfect multicollinearity
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
Normal - Student's T - Chi - square - F distribution
When one regressor is a perfect linear function of the other regressors
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

47. Mean reversion in variance
More than one random variable
Variance(x)
We reject a hypothesis that is actually true
Variance reverts to a long run level

48. Importance sampling technique
Application of mathematical statistics to economic data to lend empirical support to models
Attempts to sample along more important paths
Peaks over threshold - Collects dataset in excess of some threshold
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

49. GEV
Returns over time for an individual asset
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
SSR

50. Central Limit Theorem
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
For n>30 - sample mean is approximately normal
Combine to form distribution with leptokurtosis (heavy tails)
Distribution with only two possible outcomes