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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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1. Biggest (and only real) drawback of GARCH mode
Average return across assets on a given day
Nonlinearity
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Sample mean will near the population mean as the sample size increases

2. GEV
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Distribution with only two possible outcomes
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Among all unbiased estimators - estimator with the smallest variance is efficient

3. R^2
Returns over time for an individual asset
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

4. Variance of X - Y assuming dependence
Average return across assets on a given day
Confidence level
Variance(X) + Variance(Y) - 2*covariance(XY)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

5. Skewness
Application of mathematical statistics to economic data to lend empirical support to models
(a^2)(variance(x)) + (b^2)(variance(y))
Yi = B0 + B1Xi + ui
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

6. Variance of X+Y assuming dependence
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance(x) + Variance(Y) + 2*covariance(XY)

7. Mean reversion in variance
Variance reverts to a long run level
Least absolute deviations estimator - used when extreme outliers are not uncommon
Special type of pooled data in which the cross sectional unit is surveyed over time
Var(X) + Var(Y)

8. Overall F - statistic
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sample mean will near the population mean as the sample size increases
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Statement of the error or precision of an estimate

9. POT
We accept a hypothesis that should have been rejected
Transformed to a unit variable - Mean = 0 Variance = 1
Peaks over threshold - Collects dataset in excess of some threshold
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

10. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Sample mean will near the population mean as the sample size increases
Rxy = Sxy/(Sx*Sy)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

11. Control variates technique
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Nonlinearity

12. 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
Use historical simulation approach but use the EWMA weighting system
Confidence set for two coefficients - two dimensional analog for the confidence interval
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

13. Stochastic error term
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Average return across assets on a given day
Contains variables not explicit in model - Accounts for randomness
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

14. 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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Expected value of the sample mean is the population mean

15. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Distribution with only two possible outcomes
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
i = ln(Si/Si - 1)

16. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

17. Pooled data
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Returns over time for a combination of assets (combination of time series and cross - sectional data)

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

19. Unconditional vs conditional distributions
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

20. Implied standard deviation for options
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Mean of sampling distribution is the population mean

21. Sample mean
SSR
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Expected value of the sample mean is the population mean
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

22. What does the OLS minimize?
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
SSR
Only requires two parameters = mean and variance
Yi = B0 + B1Xi + ui

23. Variance of aX
(a^2)(variance(x)
Distribution with only two possible outcomes
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

24. Economical(elegant)
Based on an equation - P(A) = # of A/total outcomes
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Only requires two parameters = mean and variance

25. Test for statistical independence
E(mean) = mean
P(X=x - Y=y) = P(X=x) * P(Y=y)
Var(X) + Var(Y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

26. Empirical frequency
Statement of the error or precision of an estimate
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Based on a dataset
If variance of the conditional distribution of u(i) is not constant

27. GPD
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Only requires two parameters = mean and variance
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Based on an equation - P(A) = # of A/total outcomes

28. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
(a^2)(variance(x)) + (b^2)(variance(y))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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)

29. Non - parametric vs parametric calculation of VaR
E(mean) = mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

30. i.i.d.
Independently and Identically Distributed
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Var(X) + Var(Y)

31. Type II Error
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
We accept a hypothesis that should have been rejected

32. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Application of mathematical statistics to economic data to lend empirical support to models
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

33. Central Limit Theorem
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
For n>30 - sample mean is approximately normal
(a^2)(variance(x)) + (b^2)(variance(y))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

34. Unbiased
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Z = (Y - meany)/(stddev(y)/sqrt(n))
Mean of sampling distribution is the population mean

35. Continuously compounded return equation
Rxy = Sxy/(Sx*Sy)
E(XY) - E(X)E(Y)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
i = ln(Si/Si - 1)

36. SER
Variance(x)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

37. Confidence interval for sample mean
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
We reject a hypothesis that is actually true
Only requires two parameters = mean and variance
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

38. LAD
Among all unbiased estimators - estimator with the smallest variance is efficient
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

39. Implications of homoscedasticity
We reject a hypothesis that is actually true
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Rxy = Sxy/(Sx*Sy)

40. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Nonlinearity
Normal - Student's T - Chi - square - F distribution
Choose parameters that maximize the likelihood of what observations occurring

41. Extreme Value Theory
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
We accept a hypothesis that should have been rejected
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

42. Kurtosis
Confidence set for two coefficients - two dimensional analog for the confidence interval
Transformed to a unit variable - Mean = 0 Variance = 1
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

43. Two drawbacks of moving average series
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)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

44. Mean reversion
P - value
P(X=x - Y=y) = P(X=x) * P(Y=y)
(a^2)(variance(x)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

45. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Independently and Identically Distributed
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

46. Critical z values
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
95% = 1.65 99% = 2.33 For one - tailed tests
Choose parameters that maximize the likelihood of what observations occurring
Confidence set for two coefficients - two dimensional analog for the confidence interval

47. Persistence
Random walk (usually acceptable) - Constant volatility (unlikely)
If variance of the conditional distribution of u(i) is not constant
Normal - Student's T - Chi - square - F distribution
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

48. Significance =1
Application of mathematical statistics to economic data to lend empirical support to models
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Confidence level
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

49. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Only requires two parameters = mean and variance
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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

50. Covariance
Summation((xi - mean)^k)/n
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Model dependent - Options with the same underlying assets may trade at different volatilities
E(XY) - E(X)E(Y)