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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. Significance =1
Returns over time for an individual asset
Confidence level
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

2. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
For n>30 - sample mean is approximately normal
Mean = np - Variance = npq - Std dev = sqrt(npq)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

3. Implied standard deviation for options
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Does not depend on a prior event or information
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

4. Deterministic Simulation
More than one random variable
Expected value of the sample mean is the population mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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

5. Hazard rate of exponentially distributed random variable
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for an individual asset

6. Two requirements of OVB
Based on an equation - P(A) = # of A/total outcomes
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Statement of the error or precision of an estimate
Sampling distribution of sample means tend to be normal

7. Time series data
Use historical simulation approach but use the EWMA weighting system
Returns over time for an individual asset
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Does not depend on a prior event or information

8. Type I error
E(XY) - E(X)E(Y)
We reject a hypothesis that is actually true
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

9. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
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)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

10. Multivariate probability
More than one random variable
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
We reject a hypothesis that is actually true
Normal - Student's T - Chi - square - F distribution

11. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Based on a dataset
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

12. Consistent
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
When the sample size is large - the uncertainty about the value of the sample is very small
Confidence level

13. Variance of X - Y assuming dependence
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Regression can be non - linear in variables but must be linear in parameters
Variance(X) + Variance(Y) - 2*covariance(XY)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

14. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Model dependent - Options with the same underlying assets may trade at different volatilities

15. Two drawbacks of moving average series
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

16. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
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
Contains variables not explicit in model - Accounts for randomness

17. Adjusted R^2
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!)
P(Z>t)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

18. Square root rule
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
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!)

19. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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

20. Homoskedastic
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Independently and Identically Distributed
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
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

21. Discrete random variable
Least absolute deviations estimator - used when extreme outliers are not uncommon
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

22. P - value
P(Z>t)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Concerned with a single random variable (ex. Roll of a die)
Variance(X) + Variance(Y) - 2*covariance(XY)

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

24. Discrete representation of the GBM
Does not depend on a prior event or information
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

25. Continuous random variable
Returns over time for an individual asset
We reject a hypothesis that is actually true
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Random walk (usually acceptable) - Constant volatility (unlikely)

26. Kurtosis
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
Price/return tends to run towards a long - run level
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Based on a dataset

27. Normal distribution
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
P(Z>t)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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

28. Unbiased
Mean of sampling distribution is the population mean
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Combine to form distribution with leptokurtosis (heavy tails)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

29. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
P(X=x - Y=y) = P(X=x) * P(Y=y)
Has heavy tails
Yi = B0 + B1Xi + ui

30. Panel data (longitudinal or micropanel)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Average return across assets on a given day
Independently and Identically Distributed
Special type of pooled data in which the cross sectional unit is surveyed over time

31. Cholesky factorization (decomposition)
Only requires two parameters = mean and variance
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
When one regressor is a perfect linear function of the other regressors

32. Homoskedastic only F - stat
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)
Summation((xi - mean)^k)/n
When one regressor is a perfect linear function of the other regressors

33. Standard error for Monte Carlo replications
Confidence set for two coefficients - two dimensional analog for the confidence interval
E(XY) - E(X)E(Y)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

34. Variance of X+Y
We accept a hypothesis that should have been rejected
Var(X) + Var(Y)
Low Frequency - High Severity events
Mean of sampling distribution is the population mean

35. Empirical frequency
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Based on a dataset
E(mean) = mean
Sampling distribution of sample means tend to be normal

36. Type II Error
We accept a hypothesis that should have been rejected
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

37. LAD
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Least absolute deviations estimator - used when extreme outliers are not uncommon

38. Poisson Distribution
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Summation((xi - mean)^k)/n
Combine to form distribution with leptokurtosis (heavy tails)

39. Standard error
Does not depend on a prior event or information
Variance(x) + Variance(Y) + 2*covariance(XY)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

40. Continuous representation of the GBM
Expected value of the sample mean is the population mean
SSR
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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)

41. Cross - sectional
Average return across assets on a given day
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

42. Simplified standard (un - weighted) variance
Has heavy tails
Variance = (1/m) summation(u<n - i>^2)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

43. Lognormal
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Statement of the error or precision of an estimate

44. Standard variable for non - normal distributions
Normal - Student's T - Chi - square - F distribution
Z = (Y - meany)/(stddev(y)/sqrt(n))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Yi = B0 + B1Xi + ui

45. Key properties of linear regression
(a^2)(variance(x)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Regression can be non - linear in variables but must be linear in parameters
E(mean) = mean

46. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Returns over time for an individual asset
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

47. Variance of sample mean
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
Variance(y)/n = variance of sample Y
Based on an equation - P(A) = # of A/total outcomes
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

48. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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!)
Population denominator = n - Sample denominator = n - 1

49. Heteroskedastic
Contains variables not explicit in model - Accounts for randomness
Independently and Identically Distributed
Application of mathematical statistics to economic data to lend empirical support to models
If variance of the conditional distribution of u(i) is not constant

50. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)