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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. Standard variable for non - normal distributions
Special type of pooled data in which the cross sectional unit is surveyed over time
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Mean = np - Variance = npq - Std dev = sqrt(npq)

2. Unconditional vs conditional distributions
E(mean) = mean
Sampling distribution of sample means tend to be normal
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

3. Four sampling distributions

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4. Expected future variance rate (t periods forward)
Choose parameters that maximize the likelihood of what observations occurring
Peaks over threshold - Collects dataset in excess of some threshold
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

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

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6. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Easy to manipulate
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

7. Critical z values
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
More than one random variable
i = ln(Si/Si - 1)
95% = 1.65 99% = 2.33 For one - tailed tests

8. T distribution
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)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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

9. K - th moment
Summation((xi - mean)^k)/n
Model dependent - Options with the same underlying assets may trade at different volatilities
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)
(a^2)(variance(x)) + (b^2)(variance(y))

10. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

11. Pooled data
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Returns over time for a combination of assets (combination of time series and cross - sectional data)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

12. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
P(X=x - Y=y) = P(X=x) * P(Y=y)
For n>30 - sample mean is approximately normal
Confidence set for two coefficients - two dimensional analog for the confidence interval

13. Variance of aX
i = ln(Si/Si - 1)
Combine to form distribution with leptokurtosis (heavy tails)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
(a^2)(variance(x)

14. Multivariate Density Estimation (MDE)
Independently and Identically Distributed
Average return across assets on a given day
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)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

15. Variance(discrete)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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)
Based on a dataset

16. Lognormal
More than one random variable
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
Sample mean +/ - t*(stddev(s)/sqrt(n))

17. Empirical frequency
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Based on a dataset
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

18. Type II Error
We accept a hypothesis that should have been rejected
(a^2)(variance(x)) + (b^2)(variance(y))
Concerned with a single random variable (ex. Roll of a die)
Z = (Y - meany)/(stddev(y)/sqrt(n))

19. 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)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Choose parameters that maximize the likelihood of what observations occurring

20. Normal distribution
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
Statement of the error or precision of an estimate
Transformed to a unit variable - Mean = 0 Variance = 1
SSR

21. Simulation models
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
Normal - Student's T - Chi - square - F distribution
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. Type I error
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
We reject a hypothesis that is actually true

23. Control variates technique
More than one random variable
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

24. Consistent
Among all unbiased estimators - estimator with the smallest variance is efficient
SSR
Average return across assets on a given day
When the sample size is large - the uncertainty about the value of the sample is very small

25. Conditional probability functions
If variance of the conditional distribution of u(i) is not constant
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(y)/n = variance of sample Y
Mean = np - Variance = npq - Std dev = sqrt(npq)

26. Poisson distribution equations for mean variance and std deviation
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

27. Variance of aX + bY
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
(a^2)(variance(x)) + (b^2)(variance(y))
When the sample size is large - the uncertainty about the value of the sample is very small
Only requires two parameters = mean and variance

28. Logistic distribution
Based on a dataset
Has heavy tails
Statement of the error or precision of an estimate
Z = (Y - meany)/(stddev(y)/sqrt(n))

29. Sample mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Expected value of the sample mean is the population mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

30. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
We accept a hypothesis that should have been rejected
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Confidence set for two coefficients - two dimensional analog for the confidence interval

31. Mean reversion
Returns over time for an individual asset
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

32. Confidence interval (from t)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Sampling distribution of sample means tend to be normal
Sample mean +/ - t*(stddev(s)/sqrt(n))

33. Unbiased
Variance(x) + Variance(Y) + 2*covariance(XY)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Mean of sampling distribution is the population mean
Sample mean will near the population mean as the sample size increases

34. Variance of sample mean
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(y)/n = variance of sample Y
Contains variables not explicit in model - Accounts for randomness
For n>30 - sample mean is approximately normal

35. Cholesky factorization (decomposition)
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
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
More than one random variable

36. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Use historical simulation approach but use the EWMA weighting system
95% = 1.65 99% = 2.33 For one - tailed tests

37. Sample correlation
Transformed to a unit variable - Mean = 0 Variance = 1
Rxy = Sxy/(Sx*Sy)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

38. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Mean = np - Variance = npq - Std dev = sqrt(npq)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
i = ln(Si/Si - 1)

39. Biggest (and only real) drawback of GARCH mode
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Nonlinearity
Normal - Student's T - Chi - square - F distribution

40. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
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)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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. Kurtosis
Variance = (1/m) summation(u<n - i>^2)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

42. Maximum likelihood method
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Peaks over threshold - Collects dataset in excess of some threshold
SSR
Choose parameters that maximize the likelihood of what observations occurring

43. Beta distribution
P - value
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

44. Priori (classical) probability
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 an equation - P(A) = # of A/total outcomes
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

45. Difference between population and sample variance
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
Sampling distribution of sample means tend to be normal
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Population denominator = n - Sample denominator = n - 1

46. Covariance calculations using weight sums (lambda)
E(XY) - E(X)E(Y)
Concerned with a single random variable (ex. Roll of a die)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

47. Variance of weighted scheme
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Normal - Student's T - Chi - square - F distribution
Only requires two parameters = mean and variance
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

48. Implications of homoscedasticity
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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

49. GARCH
SSR
Probability that the random variables take on certain values simultaneously
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)
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

50. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Regression can be non - linear in variables but must be linear in parameters
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for