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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. Skewness
Sample mean +/ - t*(stddev(s)/sqrt(n))
Mean of sampling distribution is the population mean
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
Choose parameters that maximize the likelihood of what observations occurring

2. Time series data
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Combine to form distribution with leptokurtosis (heavy tails)
Returns over time for an individual asset
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

3. Direction of OVB
Sample mean will near the population mean as the sample size increases
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

4. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Confidence set for two coefficients - two dimensional analog for the confidence interval
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

5. Adjusted R^2
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
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
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 = np - Variance = npq - Std dev = sqrt(npq)

6. Discrete representation of the GBM
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

7. Significance =1
Mean = np - Variance = npq - Std dev = sqrt(npq)
Confidence level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

8. Square root rule
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Distribution with only two possible outcomes
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

9. Mean(expected value)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Does not depend on a prior event or information
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

10. Variance of aX
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
(a^2)(variance(x)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

11. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Application of mathematical statistics to economic data to lend empirical support to models
More than one random variable

12. Statistical (or empirical) model
Special type of pooled data in which the cross sectional unit is surveyed over time
P(Z>t)
Yi = B0 + B1Xi + ui
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

13. Poisson distribution equations for mean variance and std deviation
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

14. Bootstrap method
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Summation((xi - mean)^k)/n
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

15. Hybrid method for conditional volatility
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Use historical simulation approach but use the EWMA weighting system
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

16. Heteroskedastic
Attempts to sample along more important paths
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
If variance of the conditional distribution of u(i) is not constant
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

17. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
P - value
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!)
Concerned with a single random variable (ex. Roll of a die)

18. Standard variable for non - normal distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Use historical simulation approach but use the EWMA weighting system
Variance reverts to a long run level

19. Variance of sampling distribution of means when n<N
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Regression can be non - linear in variables but must be linear in parameters

20. Variance of weighted scheme
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

21. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance(x) + Variance(Y) + 2*covariance(XY)
If variance of the conditional distribution of u(i) is not constant
Regression can be non - linear in variables but must be linear in parameters

22. Variance of X+Y
Var(X) + Var(Y)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

23. Variance of sample mean
Variance(y)/n = variance of sample Y
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Special type of pooled data in which the cross sectional unit is surveyed over time
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

24. ESS
Easy to manipulate
Contains variables not explicit in model - Accounts for randomness
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

25. Economical(elegant)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Only requires two parameters = mean and variance

26. 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)
Price/return tends to run towards a long - run level
95% = 1.65 99% = 2.33 For one - tailed tests
Variance reverts to a long run level

27. Two assumptions of square root rule
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)
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)
Random walk (usually acceptable) - Constant volatility (unlikely)

28. Key properties of linear regression
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
E(mean) = mean
Regression can be non - linear in variables but must be linear in parameters
Sampling distribution of sample means tend to be normal

29. Consistent
Attempts to sample along more important paths
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
When the sample size is large - the uncertainty about the value of the sample is very small
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

30. Implied standard deviation for options
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
(a^2)(variance(x)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

31. Variance of aX + bY
Easy to manipulate
E(mean) = mean
(a^2)(variance(x)) + (b^2)(variance(y))
When the sample size is large - the uncertainty about the value of the sample is very small

32. Reliability
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Statement of the error or precision of an estimate
Peaks over threshold - Collects dataset in excess of some threshold
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

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

34. Persistence
We accept a hypothesis that should have been rejected
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
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
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

35. Normal distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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

36. Homoskedastic
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Var(X) + Var(Y)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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

37. Joint probability functions
If variance of the conditional distribution of u(i) is not constant
P(Z>t)
Probability that the random variables take on certain values simultaneously
Mean = np - Variance = npq - Std dev = sqrt(npq)

38. Confidence ellipse
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Combine to form distribution with leptokurtosis (heavy tails)
Confidence set for two coefficients - two dimensional analog for the confidence interval
More than one random variable

39. Homoskedastic only F - stat
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Population denominator = n - Sample denominator = n - 1

40. SER
E(mean) = mean
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

41. Non - parametric vs parametric calculation of VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Variance(x)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
More than one random variable

42. P - value
i = ln(Si/Si - 1)
Population denominator = n - Sample denominator = n - 1
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
P(Z>t)

43. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance reverts to a long run level
Summation((xi - mean)^k)/n
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

44. Discrete random variable
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
Variance(x)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

45. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Mean of sampling distribution is the population mean
Sample mean +/ - t*(stddev(s)/sqrt(n))
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. Kurtosis
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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)

47. Historical std dev
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

48. Continuous random variable
95% = 1.65 99% = 2.33 For one - tailed tests
E(mean) = mean
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Sample mean +/ - t*(stddev(s)/sqrt(n))

49. Simulating for VaR
Attempts to sample along more important paths
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

50. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Combine to form distribution with leptokurtosis (heavy tails)