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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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Match each statement with the correct term.
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1. Discrete random variable
Population denominator = n - Sample denominator = n - 1
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)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

2. Sample covariance
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

3. Efficiency
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Among all unbiased estimators - estimator with the smallest variance is efficient
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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

4. Weibul distribution
i = ln(Si/Si - 1)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

5. Economical(elegant)
Only requires two parameters = mean and variance
Model dependent - Options with the same underlying assets may trade at different volatilities
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

6. Potential reasons for fat tails in return distributions
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Nonlinearity
Expected value of the sample mean is the population mean

7. Joint probability functions
Yi = B0 + B1Xi + ui
Variance(X) + Variance(Y) - 2*covariance(XY)
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
Probability that the random variables take on certain values simultaneously

8. Variance - covariance approach for VaR of a portfolio
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Normal - Student's T - Chi - square - F distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

9. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

10. Square root rule
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Based on a dataset
Normal - Student's T - Chi - square - F distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

11. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
We accept a hypothesis that should have been rejected
For n>30 - sample mean is approximately normal

12. Type II Error
We accept a hypothesis that should have been rejected
Variance = (1/m) summation(u<n - i>^2)
Variance reverts to a long run level
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

13. Historical std dev
SSR
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
i = ln(Si/Si - 1)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

14. Confidence interval (from t)
i = ln(Si/Si - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))
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)
We reject a hypothesis that is actually true

15. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Price/return tends to run towards a long - run level
Population denominator = n - Sample denominator = n - 1
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

16. Mean reversion in asset dynamics
Regression can be non - linear in variables but must be linear in parameters
Price/return tends to run towards a long - run level
Variance = (1/m) summation(u<n - i>^2)
95% = 1.65 99% = 2.33 For one - tailed tests

17. Variance(discrete)
Variance(x)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(x) + Variance(Y) + 2*covariance(XY)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

18. Skewness
Easy to manipulate
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

19. Two ways to calculate historical volatility
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
We accept a hypothesis that should have been rejected
E(XY) - E(X)E(Y)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

20. Lognormal
Statement of the error or precision of an estimate
When the sample size is large - the uncertainty about the value of the sample is very small
SSR
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

21. Homoskedastic
Combine to form distribution with leptokurtosis (heavy tails)
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
Sampling distribution of sample means tend to be normal
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

22. Chi - squared distribution
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

23. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Low Frequency - High Severity events
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

24. Variance of sampling distribution of means when n<N
Choose parameters that maximize the likelihood of what observations occurring
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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)

25. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
P - value
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

26. Priori (classical) probability
Combine to form distribution with leptokurtosis (heavy tails)
Returns over time for an individual asset
Based on an equation - P(A) = # of A/total outcomes
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

27. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Model dependent - Options with the same underlying assets may trade at different volatilities
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
More than one random variable

28. Type I error
We reject a hypothesis that is actually true
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))
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

29. Two assumptions of square root rule
Easy to manipulate
Random walk (usually acceptable) - Constant volatility (unlikely)
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)

30. Variance of X+Y assuming dependence
P(Z>t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance(x) + Variance(Y) + 2*covariance(XY)

31. What does the OLS minimize?
Probability that the random variables take on certain values simultaneously
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
SSR
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

32. 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
Price/return tends to run towards a long - run level
When the sample size is large - the uncertainty about the value of the sample is very small
Statement of the error or precision of an estimate

33. Logistic distribution
Has heavy tails
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(x) + Variance(Y) + 2*covariance(XY)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

34. GARCH
Distribution with only two possible outcomes
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
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)

35. Persistence
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Expected value of the sample mean is the population mean

36. Mean(expected value)
Based on an equation - P(A) = # of A/total outcomes
Only requires two parameters = mean and variance
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E(mean) = mean

37. Multivariate Density Estimation (MDE)
For n>30 - sample mean is approximately normal
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

38. Kurtosis
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

39. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Statement of the error or precision of an estimate
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Variance(y)/n = variance of sample Y

40. P - value
P(Z>t)
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
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

41. Variance of aX
P(Z>t)
(a^2)(variance(x)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

42. Beta distribution
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

43. Variance of X+b
Expected value of the sample mean is the population mean
Variance(x)
Easy to manipulate
Var(X) + Var(Y)

44. Standard variable for non - normal distributions
Special type of pooled data in which the cross sectional unit is surveyed over time
Z = (Y - meany)/(stddev(y)/sqrt(n))
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

45. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
P - value
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

46. ESS
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

47. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Probability that the random variables take on certain values simultaneously
Rxy = Sxy/(Sx*Sy)
Based on a dataset

48. POT
We accept a hypothesis that should have been rejected
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Peaks over threshold - Collects dataset in excess of some threshold
Based on a dataset

49. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Application of mathematical statistics to economic data to lend empirical support to models
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
(a^2)(variance(x)

50. Block maxima
Sampling distribution of sample means tend to be normal
Independently and Identically Distributed
Population denominator = n - Sample denominator = n - 1
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.