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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. Homoskedastic
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Sampling distribution of sample means tend to be normal
E(mean) = mean
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

2. 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!)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
95% = 1.65 99% = 2.33 For one - tailed tests

3. Priori (classical) probability
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Among all unbiased estimators - estimator with the smallest variance is efficient
Based on an equation - P(A) = # of A/total outcomes
Model dependent - Options with the same underlying assets may trade at different volatilities

4. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance(y)/n = variance of sample Y
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!)
Expected value of the sample mean is the population mean

5. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Based on a dataset
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

6. Unstable return distribution
Rxy = Sxy/(Sx*Sy)
Normal - Student's T - Chi - square - F distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
95% = 1.65 99% = 2.33 For one - tailed tests

7. Unconditional vs conditional distributions
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Returns over time for an individual asset
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

8. Poisson Distribution
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))
Average return across assets on a given day
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

9. Lognormal
When the sample size is large - the uncertainty about the value of the sample is very small
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
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
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)

10. POT
Peaks over threshold - Collects dataset in excess of some threshold
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Distribution with only two possible outcomes
Returns over time for a combination of assets (combination of time series and cross - sectional data)

11. Reliability
Variance reverts to a long run level
Statement of the error or precision of an estimate
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Normal - Student's T - Chi - square - F distribution

12. R^2
Variance(x)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Has heavy tails
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

13. Persistence
Transformed to a unit variable - Mean = 0 Variance = 1
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

14. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

15. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

16. Discrete random variable
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

17. Two ways to calculate historical volatility
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(x)
Sample mean will near the population mean as the sample size increases

18. Marginal unconditional probability function
Rxy = Sxy/(Sx*Sy)
Does not depend on a prior event or information
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 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

19. Direction of OVB
Probability that the random variables take on certain values simultaneously
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
When the sample size is large - the uncertainty about the value of the sample is very small
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

20. Variance of aX + bY
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
(a^2)(variance(x)) + (b^2)(variance(y))

21. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
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
Nonlinearity
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

22. Overall F - statistic
Expected value of the sample mean is the population mean
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Based on a dataset
Independently and Identically Distributed

23. Sample mean
P - value
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Has heavy tails
Expected value of the sample mean is the population mean

24. Confidence interval (from t)
Low Frequency - High Severity events
Random walk (usually acceptable) - Constant volatility (unlikely)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Sample mean +/ - t*(stddev(s)/sqrt(n))

25. Variance of X - Y assuming dependence
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance(X) + Variance(Y) - 2*covariance(XY)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

26. LFHS
Confidence level
Peaks over threshold - Collects dataset in excess of some threshold
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!)
Low Frequency - High Severity events

27. Central Limit Theorem
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
For n>30 - sample mean is approximately normal
i = ln(Si/Si - 1)
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

28. Four sampling distributions

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29. Joint probability functions
Sampling distribution of sample means tend to be normal
Use historical simulation approach but use the EWMA weighting system
Probability that the random variables take on certain values simultaneously
Variance = (1/m) summation(u<n - i>^2)

30. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Confidence level
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

31. Critical z values
Only requires two parameters = mean and variance
Mean of sampling distribution is the population mean
95% = 1.65 99% = 2.33 For one - tailed tests
Easy to manipulate

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

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33. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance(y)/n = variance of sample Y
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Sampling distribution of sample means tend to be normal

34. SER
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
We reject a hypothesis that is actually true

35. Mean reversion in asset dynamics
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Price/return tends to run towards a long - run level

36. Unbiased
Mean of sampling distribution is the population mean
Contains variables not explicit in model - Accounts for randomness
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

37. Variance of weighted scheme
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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

38. Type I error
P - value
We reject a hypothesis that is actually true
Confidence set for two coefficients - two dimensional analog for the confidence interval
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

39. Sample variance
We accept a hypothesis that should have been rejected
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Yi = B0 + B1Xi + ui
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

40. Non - parametric vs parametric calculation of VaR
More than one random variable
When the sample size is large - the uncertainty about the value of the sample is very small
For n>30 - sample mean is approximately normal
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

41. Key properties of linear regression
Application of mathematical statistics to economic data to lend empirical support to models
Regression can be non - linear in variables but must be linear in parameters
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Sample mean will near the population mean as the sample size increases

42. Covariance calculations using weight sums (lambda)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
(a^2)(variance(x)

43. Consistent
Confidence set for two coefficients - two dimensional analog for the confidence interval
When the sample size is large - the uncertainty about the value of the sample is very small
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

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

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45. Variance of X+b
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Low Frequency - High Severity events
(a^2)(variance(x)
Variance(x)

46. Variance of aX
(a^2)(variance(x)
SSR
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Random walk (usually acceptable) - Constant volatility (unlikely)

47. Continuous representation of the GBM
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)
Confidence level
Combine to form distribution with leptokurtosis (heavy tails)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

48. Expected future variance rate (t periods forward)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(X) + Variance(Y) - 2*covariance(XY)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

49. Cross - sectional
Average return across assets on a given day
Use historical simulation approach but use the EWMA weighting system
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance(y)/n = variance of sample Y

50. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
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!)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution