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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. Non - parametric vs parametric calculation of VaR
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Transformed to a unit variable - Mean = 0 Variance = 1
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

2. Economical(elegant)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Only requires two parameters = mean and variance
Based on an equation - P(A) = # of A/total outcomes

3. T distribution
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)
Sample mean will near the population mean as the sample size increases
Transformed to a unit variable - Mean = 0 Variance = 1
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

4. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

5. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Among all unbiased estimators - estimator with the smallest variance is efficient
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(sample y) = (variance(y)/n)*(N - n/N - 1)

6. Weibul distribution
Random walk (usually acceptable) - Constant volatility (unlikely)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance(X) + Variance(Y) - 2*covariance(XY)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

7. Persistence
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
For n>30 - sample mean is approximately normal
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
When one regressor is a perfect linear function of the other regressors

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

9. Confidence ellipse
Variance reverts to a long run level
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

10. Inverse transform method
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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

11. Confidence interval for sample mean
Nonlinearity
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Only requires two parameters = mean and variance
Statement of the error or precision of an estimate

12. Shortcomings of implied volatility
Does not depend on a prior event or information
Concerned with a single random variable (ex. Roll of a die)
Model dependent - Options with the same underlying assets may trade at different volatilities
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

13. Type I error
Variance(x)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
We reject a hypothesis that is actually true
Rxy = Sxy/(Sx*Sy)

14. Continuous random variable
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Statement of the error or precision of an estimate
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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)

15. Simulation models
E(XY) - E(X)E(Y)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

16. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Concerned with a single random variable (ex. Roll of a die)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
P(X=x - Y=y) = P(X=x) * P(Y=y)

17. Two ways to calculate historical volatility
Application of mathematical statistics to economic data to lend empirical support to models
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
P(Z>t)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

18. Mean(expected value)
Mean of sampling distribution is the population mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

19. Pooled data
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

20. Lognormal
Nonlinearity
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
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
Has heavy tails

21. Mean reversion in asset dynamics
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
Regression can be non - linear in variables but must be linear in parameters
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

22. Homoskedastic only F - stat
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
E(XY) - E(X)E(Y)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Confidence level

23. F distribution
Var(X) + Var(Y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
When the sample size is large - the uncertainty about the value of the sample is very small
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

24. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance(x) + Variance(Y) + 2*covariance(XY)
Regression can be non - linear in variables but must be linear in parameters

25. Deterministic Simulation
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
E(XY) - E(X)E(Y)
Contains variables not explicit in model - Accounts for randomness
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

26. Historical std dev
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Returns over time for an individual asset

27. Consistent
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Only requires two parameters = mean and variance
Population denominator = n - Sample denominator = n - 1
When the sample size is large - the uncertainty about the value of the sample is very small

28. Exact significance level
Random walk (usually acceptable) - Constant volatility (unlikely)
P - value
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

29. Covariance
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
E(XY) - E(X)E(Y)

30. Central Limit Theorem
(a^2)(variance(x)) + (b^2)(variance(y))
Based on an equation - P(A) = # of A/total outcomes
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
For n>30 - sample mean is approximately normal

31. Test for statistical independence
Based on a dataset
P(X=x - Y=y) = P(X=x) * P(Y=y)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
SSR

32. GARCH
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
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)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

33. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
We reject a hypothesis that is actually true
Special type of pooled data in which the cross sectional unit is surveyed over time
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

34. Continuous representation of the GBM
P - value
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)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

35. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Summation((xi - mean)^k)/n
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
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

36. Single variable (univariate) probability
E(mean) = mean
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Concerned with a single random variable (ex. Roll of a die)

37. Difference between population and sample variance
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Population denominator = n - Sample denominator = n - 1
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

38. Panel data (longitudinal or micropanel)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Expected value of the sample mean is the population mean
Special type of pooled data in which the cross sectional unit is surveyed over time

39. Result of combination of two normal with same means
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Among all unbiased estimators - estimator with the smallest variance is efficient
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Combine to form distribution with leptokurtosis (heavy tails)

40. Perfect multicollinearity
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
When one regressor is a perfect linear function of the other regressors
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

41. Two assumptions of square root rule
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Transformed to a unit variable - Mean = 0 Variance = 1
Random walk (usually acceptable) - Constant volatility (unlikely)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

42. Sample mean
Use historical simulation approach but use the EWMA weighting system
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
Expected value of the sample mean is the population mean

43. Sample correlation
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
Rxy = Sxy/(Sx*Sy)
Application of mathematical statistics to economic data to lend empirical support to models
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

44. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Mean = np - Variance = npq - Std dev = sqrt(npq)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

45. Discrete representation of the GBM
Contains variables not explicit in model - Accounts for randomness
Has heavy tails
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

46. Variance of sample mean
Variance(x) + Variance(Y) + 2*covariance(XY)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
(a^2)(variance(x)
Variance(y)/n = variance of sample Y

47. Unbiased
Mean of sampling distribution is the population mean
P(Z>t)
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 of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

48. Monte Carlo Simulations
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

49. Expected future variance rate (t periods forward)
(a^2)(variance(x)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

50. Extreme Value Theory
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Var(X) + Var(Y)