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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. Stochastic error term
Returns over time for an individual asset
Contains variables not explicit in model - Accounts for randomness
Mean of sampling distribution is the population mean
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

2. Binomial distribution
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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!)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

3. Non - parametric vs parametric calculation of VaR
Does not depend on a prior event or information
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
i = ln(Si/Si - 1)
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

4. Continuously compounded return equation
i = ln(Si/Si - 1)
E(XY) - E(X)E(Y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

5. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Price/return tends to run towards a long - run level
Easy to manipulate
Random walk (usually acceptable) - Constant volatility (unlikely)

6. Tractable
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Confidence set for two coefficients - two dimensional analog for the confidence interval
Easy to manipulate
Least absolute deviations estimator - used when extreme outliers are not uncommon

7. Inverse transform method
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Least absolute deviations estimator - used when extreme outliers are not uncommon
Price/return tends to run towards a long - run level

8. Monte Carlo Simulations
Application of mathematical statistics to economic data to lend empirical support to models
Variance(X) + Variance(Y) - 2*covariance(XY)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Yi = B0 + B1Xi + ui

9. Extending the HS approach for computing value of a portfolio

10. Single variable (univariate) probability
Contains variables not explicit in model - Accounts for randomness
Sample mean will near the population mean as the sample size increases
Concerned with a single random variable (ex. Roll of a die)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

11. Test for unbiasedness
E(mean) = mean
Variance(y)/n = variance of sample Y
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

12. Standard error
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Only requires two parameters = mean and variance
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

13. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
For n>30 - sample mean is approximately normal
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

14. Variance of X+Y
Application of mathematical statistics to economic data to lend empirical support to models
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)
When the sample size is large - the uncertainty about the value of the sample is very small
Var(X) + Var(Y)

15. Variance - covariance approach for VaR of a portfolio
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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 a combination of assets (combination of time series and cross - sectional data)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

16. Panel data (longitudinal or micropanel)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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)
Sample mean will near the population mean as the sample size increases
Special type of pooled data in which the cross sectional unit is surveyed over time

17. Result of combination of two normal with same means
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance reverts to a long run level
Does not depend on a prior event or information
Combine to form distribution with leptokurtosis (heavy tails)

18. Binomial distribution equations for mean variance and std dev
Summation((xi - mean)^k)/n
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Does not depend on a prior event or information
Mean = np - Variance = npq - Std dev = sqrt(npq)

19. Skewness
Change in S = S<t - 1>(meanchange in time + stddev 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
Transformed to a unit variable - Mean = 0 Variance = 1
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

20. Statistical (or empirical) model
Confidence set for two coefficients - two dimensional analog for the confidence interval
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Probability that the random variables take on certain values simultaneously
Yi = B0 + B1Xi + ui

21. Chi - squared distribution
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
More than one random variable

22. Exponential 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
Confidence level
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

23. Maximum likelihood method
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
Choose parameters that maximize the likelihood of what observations occurring
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

24. Sample variance
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Nonlinearity
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

25. Reliability
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Yi = B0 + B1Xi + ui
Nonlinearity
Statement of the error or precision of an estimate

26. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Peaks over threshold - Collects dataset in excess of some threshold
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

27. Economical(elegant)
Transformed to a unit variable - Mean = 0 Variance = 1
Yi = B0 + B1Xi + ui
Only requires two parameters = mean and variance
E(XY) - E(X)E(Y)

28. Cholesky factorization (decomposition)
Independently and Identically Distributed
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
Random walk (usually acceptable) - Constant volatility (unlikely)
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

29. POT
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Summation((xi - mean)^k)/n
Z = (Y - meany)/(stddev(y)/sqrt(n))
Peaks over threshold - Collects dataset in excess of some threshold

30. Homoskedastic
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

31. Multivariate probability
We accept a hypothesis that should have been rejected
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
More than one random variable
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

32. 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
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

33. Adjusted R^2
Confidence level
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
SSR

34. What does the OLS minimize?
SSR
Var(X) + Var(Y)
i = ln(Si/Si - 1)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

35. Bootstrap method
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

36. Expected future variance rate (t periods forward)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
Contains variables not explicit in model - Accounts for randomness
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

37. Discrete random variable
Variance(X) + Variance(Y) - 2*covariance(XY)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Use historical simulation approach but use the EWMA weighting system
Based on a dataset

38. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Expected value of the sample mean is the population mean
Nonlinearity

39. Discrete representation of the GBM
Expected value of the sample mean is the population mean
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Combine to form distribution with leptokurtosis (heavy tails)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

40. Significance =1
Confidence level
Sample mean +/ - t*(stddev(s)/sqrt(n))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Returns over time for an individual asset

41. Biggest (and only real) drawback of GARCH mode
Peaks over threshold - Collects dataset in excess of some threshold
Nonlinearity
95% = 1.65 99% = 2.33 For one - tailed tests
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

42. GARCH
Concerned with a single random variable (ex. Roll of a die)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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)

43. Lognormal
Population denominator = n - Sample denominator = n - 1
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
Application of mathematical statistics to economic data to lend empirical support to models
Price/return tends to run towards a long - run level

44. Priori (classical) probability
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Based on an equation - P(A) = # of A/total outcomes

45. Deterministic Simulation
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
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
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
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

46. Difference between population and sample variance
Least absolute deviations estimator - used when extreme outliers are not uncommon
Population denominator = n - Sample denominator = n - 1
For n>30 - sample mean is approximately normal
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

47. Weibul distribution
Only requires two parameters = mean and variance
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

48. Marginal unconditional probability function
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance = (1/m) summation(u<n - i>^2)
Variance(X) + Variance(Y) - 2*covariance(XY)
Does not depend on a prior event or information

49. Sample covariance
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
Has heavy tails

50. Sample mean
P - value
Based on a dataset
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Expected value of the sample mean is the population mean