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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. GARCH
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
Statement of the error or precision of an estimate

2. Standard normal distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Transformed to a unit variable - Mean = 0 Variance = 1

3. Homoskedastic
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)
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

4. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Easy to manipulate

5. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

6. Unconditional vs conditional distributions
Mean of sampling distribution is the population mean
Variance(x)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

7. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
E(XY) - E(X)E(Y)
Summation((xi - mean)^k)/n
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

8. EWMA
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!)
i = ln(Si/Si - 1)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

9. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Confidence level
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Nonlinearity

10. Two requirements of OVB
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Confidence set for two coefficients - two dimensional analog for the confidence interval
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

11. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Mean of sampling distribution is the population mean
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

12. Variance of sample mean
We reject a hypothesis that is actually true
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance(y)/n = variance of sample Y
Returns over time for a combination of assets (combination of time series and cross - sectional data)

13. Variance - covariance approach for VaR of a portfolio
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
More than one random variable
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

14. Joint probability functions
Probability that the random variables take on certain values simultaneously
Easy to manipulate
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

15. What does the OLS minimize?
Model dependent - Options with the same underlying assets may trade at different volatilities
SSR
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Contains variables not explicit in model - Accounts for randomness

16. Gamma distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

17. Maximum likelihood method
SSR
Model dependent - Options with the same underlying assets may trade at different volatilities
We reject a hypothesis that is actually true
Choose parameters that maximize the likelihood of what observations occurring

18. Bootstrap method
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Based on a dataset
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

19. Variance(discrete)
Variance = (1/m) summation(u<n - i>^2)
Sample mean will near the population mean as the sample size increases
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^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

20. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Application of mathematical statistics to economic data to lend empirical support to models
Sample mean will near the population mean as the sample size increases
Peaks over threshold - Collects dataset in excess of some threshold

21. Panel data (longitudinal or micropanel)
Among all unbiased estimators - estimator with the smallest variance is efficient
Special type of pooled data in which the cross sectional unit is surveyed over time
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

22. Lognormal
When one regressor is a perfect linear function of the other regressors
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

23. Skewness
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
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)
E(mean) = mean

24. Covariance
E(XY) - E(X)E(Y)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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
(a^2)(variance(x)) + (b^2)(variance(y))

25. Mean(expected value)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

26. GEV
Application of mathematical statistics to economic data to lend empirical support to models
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E(XY) - E(X)E(Y)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

27. Antithetic variable technique
P - value
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Special type of pooled data in which the cross sectional unit is surveyed over time

28. Four sampling distributions

29. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance(x) + Variance(Y) + 2*covariance(XY)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

30. Cholesky factorization (decomposition)
Based on a dataset
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
Rxy = Sxy/(Sx*Sy)
Price/return tends to run towards a long - run level

31. Statistical (or empirical) model
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Low Frequency - High Severity events
Yi = B0 + B1Xi + ui
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)

32. Sample correlation
Rxy = Sxy/(Sx*Sy)
Application of mathematical statistics to economic data to lend empirical support to models
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
Among all unbiased estimators - estimator with the smallest variance is efficient

33. Importance sampling technique
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
For n>30 - sample mean is approximately normal
P(Z>t)
Attempts to sample along more important paths

34. Poisson Distribution
(a^2)(variance(x)
P - value
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

35. Marginal unconditional probability function
Does not depend on a prior event or information
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Contains variables not explicit in model - Accounts for randomness

36. Binomial distribution
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Regression can be non - linear in variables but must be linear in parameters
Transformed to a unit variable - Mean = 0 Variance = 1
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!)

37. Continuous random variable
Nonlinearity
Random walk (usually acceptable) - Constant volatility (unlikely)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(X) + Variance(Y) - 2*covariance(XY)

38. Variance of X+Y assuming dependence
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance(x) + Variance(Y) + 2*covariance(XY)

39. Time series data
Returns over time for an individual asset
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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

40. Variance of X - Y assuming dependence
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance(X) + Variance(Y) - 2*covariance(XY)
Peaks over threshold - Collects dataset in excess of some threshold

41. Expected future variance rate (t periods forward)
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

42. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Based on a dataset
Population denominator = n - Sample denominator = n - 1
Variance = (1/m) summation(u<n - i>^2)

43. Key properties of linear regression
Least absolute deviations estimator - used when extreme outliers are not uncommon
Regression can be non - linear in variables but must be linear in parameters
P(Z>t)
Contains variables not explicit in model - Accounts for randomness

44. Persistence
P(Z>t)
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Price/return tends to run towards a long - run level

45. Implied standard deviation for options
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Concerned with a single random variable (ex. Roll of a die)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

46. i.i.d.
Independently and Identically Distributed
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many 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)
Summation((xi - mean)^k)/n

47. Sample mean
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
Expected value of the sample mean is the population mean
Peaks over threshold - Collects dataset in excess of some threshold
Does not depend on a prior event or information

48. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
For n>30 - sample mean is approximately normal
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

49. F distribution
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
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Attempts to sample along more important paths

50. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
i = ln(Si/Si - 1)
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
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