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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. Standard error
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

2. Variance of X+b
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 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
Sample mean will near the population mean as the sample size increases
Variance(x)

3. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
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)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

4. Unstable return distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Based on a dataset

5. Variance of weighted scheme
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

6. Regime - switching volatility model
Peaks over threshold - Collects dataset in excess of some threshold
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance(x) + Variance(Y) + 2*covariance(XY)

7. Pooled data
Rxy = Sxy/(Sx*Sy)
Concerned with a single random variable (ex. Roll of a die)
(a^2)(variance(x)) + (b^2)(variance(y))
Returns over time for a combination of assets (combination of time series and cross - sectional data)

8. Historical std dev
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Z = (Y - meany)/(stddev(y)/sqrt(n))

9. Multivariate Density Estimation (MDE)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Choose parameters that maximize the likelihood of what observations occurring
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Model dependent - Options with the same underlying assets may trade at different volatilities

10. Beta distribution
Statement of the error or precision of an estimate
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

11. Key properties of linear regression
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)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Statement of the error or precision of an estimate
Regression can be non - linear in variables but must be linear in parameters

12. Central Limit Theorem
For n>30 - sample mean is approximately normal
Transformed to a unit variable - Mean = 0 Variance = 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
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

13. Skewness
We accept a hypothesis that should have been rejected
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
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
Var(X) + Var(Y)

14. Binomial distribution
Random walk (usually acceptable) - Constant volatility (unlikely)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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!)

15. Implications of homoscedasticity
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

16. LAD
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

17. F distribution
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
Mean of sampling distribution is the population mean

18. Weibul distribution
Low Frequency - High Severity events
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Sampling distribution of sample means tend to be normal

19. R^2
Variance reverts to a long run level
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Model dependent - Options with the same underlying assets may trade at different volatilities
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

20. Control variates technique
E(mean) = mean
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

21. Homoskedastic only F - stat
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)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

22. Mean reversion in variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance reverts to a long run level
Mean of sampling distribution is the population mean
If variance of the conditional distribution of u(i) is not constant

23. What does the OLS minimize?
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Does not depend on a prior event or information
SSR

24. SER
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

25. Consistent
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
For n>30 - sample mean is approximately normal
When the sample size is large - the uncertainty about the value of the sample is very small

26. Hybrid method for conditional volatility
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Use historical simulation approach but use the EWMA weighting system
When the sample size is large - the uncertainty about the value of the sample is very small

27. Two ways to calculate historical volatility
Z = (Y - meany)/(stddev(y)/sqrt(n))
E(mean) = mean
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Var(X) + Var(Y)

28. Maximum likelihood method
Variance(X) + Variance(Y) - 2*covariance(XY)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance = (1/m) summation(u<n - i>^2)
Choose parameters that maximize the likelihood of what observations occurring

29. Priori (classical) probability
When one regressor is a perfect linear function of the other regressors
Based on an equation - P(A) = # of A/total outcomes
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

30. Variance of aX
Sample mean +/ - t*(stddev(s)/sqrt(n))
Nonlinearity
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
(a^2)(variance(x)

31. Tractable
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Easy to manipulate

32. Overall F - statistic
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Least absolute deviations estimator - used when extreme outliers are not uncommon

33. Joint probability functions
Probability that the random variables take on certain values simultaneously
Variance = (1/m) summation(u<n - i>^2)
For n>30 - sample mean is approximately normal
Peaks over threshold - Collects dataset in excess of some threshold

34. Antithetic variable technique
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample mean will near the population mean as the sample size increases
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many 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

35. Bootstrap method
More than one random variable
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
95% = 1.65 99% = 2.33 For one - tailed tests
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

36. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Summation((xi - mean)^k)/n
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

37. Covariance calculations using weight sums (lambda)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

38. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Based on an equation - P(A) = # of A/total outcomes
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(X) + Variance(Y) - 2*covariance(XY)

39. P - value
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
P(Z>t)
If variance of the conditional distribution of u(i) is not constant
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

40. Reliability
Var(X) + Var(Y)
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)
Statement of the error or precision of an estimate

41. Covariance
E(XY) - E(X)E(Y)
Probability that the random variables take on certain values simultaneously
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Yi = B0 + B1Xi + ui

42. Non - parametric vs parametric calculation of VaR
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Among all unbiased estimators - estimator with the smallest variance is efficient
95% = 1.65 99% = 2.33 For one - tailed tests

43. Persistence
Contains variables not explicit in model - Accounts for randomness
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

44. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Among all unbiased estimators - estimator with the smallest variance is efficient
We reject a hypothesis that is actually true
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

45. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Sample mean will near the population mean as the sample size increases
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean of sampling distribution is the population mean

46. Inverse transform method
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
Variance(y)/n = variance of sample Y
Average return across assets on a given day
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

47. Potential reasons for fat tails in return distributions
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
We reject a hypothesis that is actually true
Variance(y)/n = variance of sample Y

48. ESS
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(x) + Variance(Y) + 2*covariance(XY)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

49. Poisson Distribution
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)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
i = ln(Si/Si - 1)

50. Confidence interval for sample mean
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
SSR
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric