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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. Maximum likelihood method
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 an equation - P(A) = # of A/total outcomes
Choose parameters that maximize the likelihood of what observations occurring
Summation((xi - mean)^k)/n

2. Inverse transform method
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
SSR
Sample mean will near the population mean as the sample size increases

3. Square root rule
Use historical simulation approach but use the EWMA weighting system
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

4. Block maxima
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Contains variables not explicit in model - Accounts for randomness
Returns over time for a combination of assets (combination of time series and cross - sectional data)

5. Variance of aX + bY
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Contains variables not explicit in model - Accounts for randomness
(a^2)(variance(x)) + (b^2)(variance(y))
Confidence level

6. Simulating for VaR
Model dependent - Options with the same underlying assets may trade at different volatilities
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Variance(x)
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

7. Standard error for Monte Carlo replications
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Variance(y)/n = variance of sample Y
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

8. Poisson Distribution
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Z = (Y - meany)/(stddev(y)/sqrt(n))

9. Antithetic variable technique
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

10. Variance of X+b
Summation((xi - mean)^k)/n
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Based on a dataset
Variance(x)

11. Sample correlation
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Rxy = Sxy/(Sx*Sy)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

12. Tractable
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
SSR
Easy to manipulate
Confidence level

13. WLS
Variance(x) + Variance(Y) + 2*covariance(XY)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Use historical simulation approach but use the EWMA weighting system
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

14. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

15. Economical(elegant)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of 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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Only requires two parameters = mean and variance

16. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
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)
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

17. Standard normal distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance reverts to a long run level
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Transformed to a unit variable - Mean = 0 Variance = 1

18. Weibul distribution
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

19. P - value
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Confidence set for two coefficients - two dimensional analog for the confidence interval
P(Z>t)
More than one random variable

20. Econometrics
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
For n>30 - sample mean is approximately normal
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Application of mathematical statistics to economic data to lend empirical support to models

21. What does the OLS minimize?
SSR
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)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

22. Critical z values
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Attempts to sample along more important paths
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
95% = 1.65 99% = 2.33 For one - tailed tests

23. Two drawbacks of moving average series
Contains variables not explicit in model - Accounts for randomness
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(X) + Variance(Y) - 2*covariance(XY)
Normal - Student's T - Chi - square - F distribution

24. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Confidence set for two coefficients - two dimensional analog for the confidence interval

25. Regime - switching volatility model
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
Only requires two parameters = mean and variance
We accept a hypothesis that should have been rejected

26. Discrete representation of the GBM
Confidence level
Independently and Identically Distributed
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

27. Kurtosis
Sampling distribution of sample means tend to be normal
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

28. Cholesky factorization (decomposition)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
For n>30 - sample mean is approximately normal
Sample mean +/ - t*(stddev(s)/sqrt(n))

29. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
[1/(n - 1)]*summation((Xi - X)(Yi - 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)

30. Overall F - statistic
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Choose parameters that maximize the likelihood of what observations occurring

31. Persistence
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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

32. Sample mean
Expected value of the sample mean is the population mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
Normal - Student's T - Chi - square - F distribution

33. Central Limit Theorem(CLT)
Random walk (usually acceptable) - Constant volatility (unlikely)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Sampling distribution of sample means tend to be normal
Returns over time for a combination of assets (combination of time series and cross - sectional data)

34. Bernouli Distribution
P(Z>t)
Distribution with only two possible outcomes
Independently and Identically Distributed
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)

35. Variance of sample mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(y)/n = variance of sample Y

36. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance = (1/m) summation(u<n - i>^2)

37. Variance of X - Y assuming dependence
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(X) + Variance(Y) - 2*covariance(XY)
(a^2)(variance(x)) + (b^2)(variance(y))
Statement of the error or precision of an estimate

38. Mean(expected value)
Yi = B0 + B1Xi + ui
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

39. Type II Error
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Confidence set for two coefficients - two dimensional analog for the confidence interval
We accept a hypothesis that should have been rejected
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

40. Shortcomings of implied volatility
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Model dependent - Options with the same underlying assets may trade at different volatilities
Use historical simulation approach but use the EWMA weighting system

41. Direction of OVB
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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

42. Result of combination of two normal with same means
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Combine to form distribution with leptokurtosis (heavy tails)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

43. Consistent
P(X=x - Y=y) = P(X=x) * P(Y=y)
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
Rxy = Sxy/(Sx*Sy)
When the sample size is large - the uncertainty about the value of the sample is very small

44. Extreme Value Theory
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Z = (Y - meany)/(stddev(y)/sqrt(n))
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Least absolute deviations estimator - used when extreme outliers are not uncommon

45. Conditional probability functions
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

46. Unconditional vs conditional distributions
i = ln(Si/Si - 1)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance(y)/n = variance of sample Y
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

47. Bootstrap method
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)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
SSR

48. Deterministic Simulation
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Combine to form distribution with leptokurtosis (heavy tails)
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

49. GEV
Summation((xi - mean)^k)/n
When the sample size is large - the uncertainty about the value of the sample is very small
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

50. Hybrid method for conditional volatility
Variance(x)
Use historical simulation approach but use the EWMA weighting system
Combine to form distribution with leptokurtosis (heavy tails)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)