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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. Multivariate Density Estimation (MDE)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
We accept a hypothesis that should have been rejected
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

2. Conditional probability functions
Variance(y)/n = variance of sample Y
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
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

3. Variance of sample mean
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(y)/n = variance of sample Y
Has heavy tails
Returns over time for an individual asset

4. Overall F - statistic
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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

5. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Independently and Identically Distributed
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E(XY) - E(X)E(Y)

6. Type II Error
We accept a hypothesis that should have been rejected
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Rxy = Sxy/(Sx*Sy)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

7. Simulation models
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Nonlinearity
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

8. Multivariate probability
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
E(XY) - E(X)E(Y)
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
More than one random variable

9. i.i.d.
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Only requires two parameters = mean and variance
Independently and Identically Distributed
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

10. Standard error for Monte Carlo replications
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Random walk (usually acceptable) - Constant volatility (unlikely)

11. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Probability that the random variables take on certain values simultaneously

12. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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

13. Bootstrap method
Attempts to sample along more important paths
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
P(X=x - Y=y) = P(X=x) * P(Y=y)

14. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Expected value of the sample mean is the population mean
Z = (Y - meany)/(stddev(y)/sqrt(n))
Special type of pooled data in which the cross sectional unit is surveyed over time

15. Economical(elegant)
Sampling distribution of sample means tend to be normal
Variance(y)/n = variance of sample Y
Only requires two parameters = mean and variance
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

16. Law of Large Numbers
Confidence set for two coefficients - two dimensional analog for the confidence interval
Sample mean will near the population mean as the sample size increases
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Sample mean +/ - t*(stddev(s)/sqrt(n))

17. Two ways to calculate historical volatility
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)
Attempts to sample along more important paths
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

18. Variance(discrete)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Nonlinearity

19. Two assumptions of square root rule
Sampling distribution of sample means tend to be normal
Random walk (usually acceptable) - Constant volatility (unlikely)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

20. Normal distribution
Combine to form distribution with leptokurtosis (heavy tails)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Random walk (usually acceptable) - Constant volatility (unlikely)

21. Kurtosis
P(Z>t)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
(a^2)(variance(x)) + (b^2)(variance(y))
Average return across assets on a given day

22. Maximum likelihood method
Mean of sampling distribution is the population mean
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Choose parameters that maximize the likelihood of what observations occurring
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

23. K - th moment
Summation((xi - mean)^k)/n
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

24. Binomial distribution
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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!)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
We accept a hypothesis that should have been rejected

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

26. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Rxy = Sxy/(Sx*Sy)
Does not depend on a prior event or information
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

27. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
More than one random variable
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

28. Empirical frequency
Z = (Y - meany)/(stddev(y)/sqrt(n))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Based on a dataset
Yi = B0 + B1Xi + ui

29. Exponential distribution
Average return across assets on a given day
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Application of mathematical statistics to economic data to lend empirical support to models

30. Significance =1
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Confidence level
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

31. Hybrid method for conditional volatility
Yi = B0 + B1Xi + ui
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
Use historical simulation approach but use the EWMA weighting system
Concerned with a single random variable (ex. Roll of a die)

32. Statistical (or empirical) model
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Based on a dataset
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

33. Implied standard deviation for options
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Among all unbiased estimators - estimator with the smallest variance is efficient
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

34. GEV
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

35. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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!)

36. Homoskedastic only F - stat
E(XY) - E(X)E(Y)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sample mean will near the population mean as the sample size increases
We reject a hypothesis that is actually true

37. Central Limit Theorem
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Mean of sampling distribution is the population mean
For n>30 - sample mean is approximately normal
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

38. Skewness
If variance of the conditional distribution of u(i) is not constant
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 exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Returns over time for a combination of assets (combination of time series and cross - sectional data)

39. Variance of weighted scheme
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)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
(a^2)(variance(x)

40. Perfect multicollinearity
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When one regressor is a perfect linear function of the other regressors
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Summation((xi - mean)^k)/n

41. Covariance
E(XY) - E(X)E(Y)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
SSR

42. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Sampling distribution of sample means tend to be normal
Normal - Student's T - Chi - square - F distribution

43. P - value
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
P(Z>t)
Sampling distribution of sample means tend to be normal
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

44. Biggest (and only real) drawback of GARCH mode
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Nonlinearity
E(mean) = mean

45. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Low Frequency - High Severity events
Based on a dataset

46. Least squares estimator(m)
Based on an equation - P(A) = # of A/total outcomes
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

47. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Var(X) + Var(Y)

48. Variance of X+b
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Z = (Y - meany)/(stddev(y)/sqrt(n))
Returns over time for an individual asset
Variance(x)

49. Simplified standard (un - weighted) variance
Regression can be non - linear in variables but must be linear in parameters
Variance = (1/m) summation(u<n - i>^2)
We reject a hypothesis that is actually true
When one regressor is a perfect linear function of the other regressors

50. Sample variance
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
We reject a hypothesis that is actually true
Sample mean will near the population mean as the sample size increases