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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. Marginal unconditional probability function
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Does not depend on a prior event or information
We accept a hypothesis that should have been rejected
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

2. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

3. Standard normal distribution
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
Transformed to a unit variable - Mean = 0 Variance = 1
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
SSR

4. Simplified standard (un - weighted) variance
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance = (1/m) summation(u<n - i>^2)
Normal - Student's T - Chi - square - F distribution

5. Multivariate probability
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
More than one random variable
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

6. LFHS
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Low Frequency - High Severity events
Yi = B0 + B1Xi + ui

7. Biggest (and only real) drawback of GARCH mode
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!)
Nonlinearity
SSR
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

8. Mean(expected value)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Use historical simulation approach but use the EWMA weighting system
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

9. SER
When one regressor is a perfect linear function of the other regressors
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

10. Control variates technique
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
We accept a hypothesis that should have been rejected
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

11. Potential reasons for fat tails in return distributions
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Independently and Identically Distributed

12. Single variable (univariate) probability
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Normal - Student's T - Chi - square - F distribution
Confidence level
Concerned with a single random variable (ex. Roll of a die)

13. Skewness
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
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

14. Simulation models
Probability that the random variables take on certain values simultaneously
Application of mathematical statistics to economic data to lend empirical support to 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
Special type of pooled data in which the cross sectional unit is surveyed over time

15. Sample correlation
Expected value of the sample mean is the population mean
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
Rxy = Sxy/(Sx*Sy)

16. Variance of aX + bY
i = ln(Si/Si - 1)
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
(a^2)(variance(x)) + (b^2)(variance(y))
Returns over time for a combination of assets (combination of time series and cross - sectional data)

17. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
E(mean) = mean

18. Sample covariance
P(X=x - Y=y) = P(X=x) * P(Y=y)
Confidence set for two coefficients - two dimensional analog for the confidence interval
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
i = ln(Si/Si - 1)

19. Beta distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

20. POT
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
Yi = B0 + B1Xi + ui
Sample mean will near the population mean as the sample size increases
Peaks over threshold - Collects dataset in excess of some threshold

21. GEV
Nonlinearity
SSR
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

22. Two requirements of OVB
Contains variables not explicit in model - Accounts for randomness
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

23. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Confidence level
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

24. Covariance
E(XY) - E(X)E(Y)
Based on a dataset
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

25. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
(a^2)(variance(x)) + (b^2)(variance(y))
We accept a hypothesis that should have been rejected

26. Exact significance level
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
P - value

27. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
More than one random variable
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

28. Continuous representation of the GBM
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E(mean) = mean
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)

29. Economical(elegant)
Only requires two parameters = mean and variance
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - 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
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

30. Unconditional vs conditional distributions
Price/return tends to run towards a long - run level
Model dependent - Options with the same underlying assets may trade at different volatilities
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

31. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Sample mean +/ - t*(stddev(s)/sqrt(n))
Confidence set for two coefficients - two dimensional analog for the confidence interval

32. Discrete random variable
Confidence set for two coefficients - two dimensional analog for the confidence interval
Easy to manipulate
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

33. K - th moment
We accept a hypothesis that should have been rejected
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Summation((xi - mean)^k)/n
Special type of pooled data in which the cross sectional unit is surveyed over time

34. Significance =1
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Summation((xi - mean)^k)/n
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Confidence level

35. EWMA
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Concerned with a single random variable (ex. Roll of a die)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

36. Cross - sectional
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Average return across assets on a given day

37. Bootstrap method
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

38. Sample variance
Application of mathematical statistics to economic data to lend empirical support to models
Probability that the random variables take on certain values simultaneously
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

39. Two drawbacks of moving average series
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

40. Result of combination of two normal with same means
Variance(X) + Variance(Y) - 2*covariance(XY)
Combine to form distribution with leptokurtosis (heavy tails)
Transformed to a unit variable - Mean = 0 Variance = 1
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. Maximum likelihood method
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Choose parameters that maximize the likelihood of what observations occurring
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)
Statement of the error or precision of an estimate

42. WLS
SSR
Sample mean +/ - t*(stddev(s)/sqrt(n))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

43. Standard error
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
More than one random variable
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

44. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Easy to manipulate
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Sample mean will near the population mean as the sample size increases

45. Bernouli Distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Summation((xi - mean)^k)/n
Distribution with only two possible outcomes

46. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Does not depend on a prior event or information
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

47. Type I error
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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

48. Variance of X+b
Variance(x)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Average return across assets on a given day
i = ln(Si/Si - 1)

49. Variance of X+Y assuming dependence
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!)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance(x) + Variance(Y) + 2*covariance(XY)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

50. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())