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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. Joint probability functions
Probability that the random variables take on certain values simultaneously
SSR
More than one random variable
Based on an equation - P(A) = # of A/total outcomes

2. Binomial distribution equations for mean variance and std dev
Variance(x) + Variance(Y) + 2*covariance(XY)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Mean = np - Variance = npq - Std dev = sqrt(npq)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

3. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Use historical simulation approach but use the EWMA weighting system
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

4. Skewness
P(Z>t)
P - value
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

5. Law of Large Numbers
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Sample mean will near the population mean as the sample size increases
E(XY) - E(X)E(Y)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

6. Unconditional vs conditional distributions
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Low Frequency - High Severity events

7. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Transformed to a unit variable - Mean = 0 Variance = 1
Returns over time for a combination of assets (combination of time series and cross - sectional data)

8. Perfect multicollinearity
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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!)
Model dependent - Options with the same underlying assets may trade at different volatilities
When one regressor is a perfect linear function of the other regressors

9. Time series data
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
Returns over time for an individual asset
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Concerned with a single random variable (ex. Roll of a die)

10. i.i.d.
Independently and Identically Distributed
Based on a dataset
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!)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

11. Panel data (longitudinal or micropanel)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Price/return tends to run towards a long - run level
Based on a dataset
Special type of pooled data in which the cross sectional unit is surveyed over time

12. Marginal unconditional probability function
Does not depend on a prior event or information
Z = (Y - meany)/(stddev(y)/sqrt(n))
Returns over time for an individual asset
Independently and Identically Distributed

13. Continuous random variable
Transformed to a unit variable - Mean = 0 Variance = 1
More than one random variable
Z = (Y - meany)/(stddev(y)/sqrt(n))
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

14. Discrete representation of the GBM
When the sample size is large - the uncertainty about the value of the sample is very small
Use historical simulation approach but use the EWMA weighting system
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

15. Continuously compounded return equation
Model dependent - Options with the same underlying assets may trade at different volatilities
i = ln(Si/Si - 1)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Peaks over threshold - Collects dataset in excess of some threshold

16. Regime - switching volatility model
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)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance(x) + Variance(Y) + 2*covariance(XY)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

17. Variance of sampling distribution of means when n<N
Contains variables not explicit in model - Accounts for randomness
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance reverts to a long run level
When the sample size is large - the uncertainty about the value of the sample is very small

18. Pooled data
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Price/return tends to run towards a long - run level
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Returns over time for a combination of assets (combination of time series and cross - sectional data)

19. Simulation models
Attempts to sample along more important paths
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of 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
Among all unbiased estimators - estimator with the smallest variance is efficient

20. Implied standard deviation for options
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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 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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

21. Cholesky factorization (decomposition)
Based on an equation - P(A) = # of A/total outcomes
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
Contains variables not explicit in model - Accounts for randomness
Independently and Identically Distributed

22. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
i = ln(Si/Si - 1)
Independently and Identically Distributed
Easy to manipulate

23. POT
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Peaks over threshold - Collects dataset in excess of some threshold
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
E(mean) = mean

24. Two drawbacks of moving average series
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

25. GARCH
For n>30 - sample mean is approximately normal
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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)
Yi = B0 + B1Xi + ui

26. Unbiased
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
E(XY) - E(X)E(Y)
Statement of the error or precision of an estimate
Mean of sampling distribution is the population mean

27. Control variates technique
Use historical simulation approach but use the EWMA weighting system
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Regression can be non - linear in variables but must be linear in parameters

28. Heteroskedastic
Least absolute deviations estimator - used when extreme outliers are not uncommon
If variance of the conditional distribution of u(i) is not constant
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

29. Confidence interval for sample mean
Peaks over threshold - Collects dataset in excess of some threshold
Regression can be non - linear in variables but must be linear in parameters
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

30. Importance sampling technique
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Choose parameters that maximize the likelihood of what observations occurring
Attempts to sample along more important paths
Rxy = Sxy/(Sx*Sy)

31. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance(y)/n = variance of sample Y

32. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
i = ln(Si/Si - 1)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

33. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
95% = 1.65 99% = 2.33 For one - tailed tests
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

34. Expected future variance rate (t periods forward)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
If variance of the conditional distribution of u(i) is not constant
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

35. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Combine to form distribution with leptokurtosis (heavy tails)
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

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

37. Continuous representation of the GBM
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Summation((xi - mean)^k)/n
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)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

38. What does the OLS minimize?
Distribution with only two possible outcomes
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
SSR

39. Bernouli Distribution
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Distribution with only two possible outcomes

40. Variance - covariance approach for VaR of a portfolio
Model dependent - Options with the same underlying assets may trade at different volatilities
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Attempts to sample along more important paths
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

41. Maximum likelihood method
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Choose parameters that maximize the likelihood of what observations occurring
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

42. GEV
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Confidence level
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

43. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
SSR
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

44. Efficiency
SSR
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Among all unbiased estimators - estimator with the smallest variance is efficient

45. R^2
Population denominator = n - Sample denominator = n - 1
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Expected value of the sample mean is the population mean

46. Reliability
Statement of the error or precision of an estimate
Attempts to sample along more important paths
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)

47. Cross - sectional
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
Average return across assets on a given day
Does not depend on a prior event or information

48. Variance of aX + bY
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
(a^2)(variance(x)) + (b^2)(variance(y))
Summation((xi - mean)^k)/n

49. Kurtosis
Easy to manipulate
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Var(X) + Var(Y)

50. Variance of aX
(a^2)(variance(x)
Normal - Student's T - Chi - square - F distribution
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
P(X=x - Y=y) = P(X=x) * P(Y=y)