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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. Mean(expected value)
Concerned with a single random variable (ex. Roll of a die)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Average return across assets on a given day
Returns over time for an individual asset

2. Difference between population and sample variance
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
We reject a hypothesis that is actually true
Population denominator = n - Sample denominator = n - 1
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

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

4. Two drawbacks of moving average series
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

5. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Rxy = Sxy/(Sx*Sy)
Peaks over threshold - Collects dataset in excess of some threshold

6. Binomial distribution
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!)
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Mean of sampling distribution is the population mean

7. Multivariate probability
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
More than one random variable
Confidence level
Var(X) + Var(Y)

8. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Sampling distribution of sample means tend to be normal
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

9. What does the OLS minimize?
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
SSR
Transformed to a unit variable - Mean = 0 Variance = 1
Expected value of the sample mean is the population mean

10. Block maxima
We accept a hypothesis that should have been rejected
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
(a^2)(variance(x)) + (b^2)(variance(y))
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

11. Potential reasons for fat tails in return distributions
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample mean will near the population mean as the sample size increases
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

12. Beta 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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
P(Z>t)
95% = 1.65 99% = 2.33 For one - tailed tests

13. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Among all unbiased estimators - estimator with the smallest variance is efficient
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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

14. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Confidence level
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
We accept a hypothesis that should have been rejected

15. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
If variance of the conditional distribution of u(i) is not constant
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

16. Pooled data
Independently and Identically Distributed
Low Frequency - High Severity events
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Combine to form distribution with leptokurtosis (heavy tails)

17. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Summation((xi - mean)^k)/n
Mean = np - Variance = npq - Std dev = sqrt(npq)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

18. Overall F - statistic
i = ln(Si/Si - 1)
Returns over time for an individual asset
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

19. Exact significance level
P - value
Only requires two parameters = mean and variance
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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

20. Continuously compounded return equation
Does not depend on a prior event or information
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
i = ln(Si/Si - 1)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

21. Conditional probability functions
Price/return tends to run towards a long - run level
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Regression can be non - linear in variables but must be linear in parameters
Rxy = Sxy/(Sx*Sy)

22. Standard variable for non - normal distributions
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Z = (Y - meany)/(stddev(y)/sqrt(n))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

23. Extreme Value Theory
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
P(Z>t)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

24. Statistical (or empirical) model
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!)
Yi = B0 + B1Xi + ui
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
Model dependent - Options with the same underlying assets may trade at different volatilities

25. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When the sample size is large - the uncertainty about the value of the sample is very small

26. Covariance calculations using weight sums (lambda)
(a^2)(variance(x)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Easy to manipulate
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

27. Confidence interval for sample mean
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Price/return tends to run towards a long - run level
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Returns over time for an individual asset

28. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Regression can be non - linear in variables but must be linear in parameters
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

29. Marginal unconditional probability function
E(mean) = mean
Does not depend on a prior event or information
Returns over time for an individual asset
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

30. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
For n>30 - sample mean is approximately normal
Peaks over threshold - Collects dataset in excess of some threshold
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

31. Gamma distribution
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
Regression can be non - linear in variables but must be linear in parameters
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
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

32. Mean reversion
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Population denominator = n - Sample denominator = n - 1
Statement of the error or precision of an estimate

33. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance(x) + Variance(Y) + 2*covariance(XY)
Probability that the random variables take on certain values simultaneously

34. Test for statistical independence
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Attempts to sample along more important paths
P(X=x - Y=y) = P(X=x) * P(Y=y)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

35. Multivariate Density Estimation (MDE)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Based on an equation - P(A) = # of A/total outcomes
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Average return across assets on a given day

36. Continuous random variable
Probability that the random variables take on certain values simultaneously
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

37. Sample mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Expected value of the sample mean is the population mean
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

38. R^2
Use historical simulation approach but use the EWMA weighting system
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Easy to manipulate

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

40. LAD
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

41. Result of combination of two normal with same means
E(XY) - E(X)E(Y)
(a^2)(variance(x)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Combine to form distribution with leptokurtosis (heavy tails)

42. WLS
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

43. Law of Large Numbers
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sample mean will near the population mean as the sample size increases
Variance = (1/m) summation(u<n - i>^2)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

44. SER
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Expected value of the sample mean is the population mean
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

45. Weibul distribution
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Yi = B0 + B1Xi + ui
Only requires two parameters = mean and variance

46. Variance of aX + bY
Easy to manipulate
(a^2)(variance(x)) + (b^2)(variance(y))
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

47. Square root rule
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Only requires two parameters = mean and variance
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

48. Simulating for VaR
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
P(Z>t)

49. Heteroskedastic
Choose parameters that maximize the likelihood of what observations occurring
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
If variance of the conditional distribution of u(i) is not constant
Mean of sampling distribution is the population mean

50. Economical(elegant)
Only requires two parameters = mean and variance
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
For n>30 - sample mean is approximately normal