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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. Kurtosis
Nonlinearity
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

2. Hazard rate of exponentially distributed random variable
Variance = (1/m) summation(u<n - i>^2)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Distribution with only two possible outcomes

3. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Variance(X) + Variance(Y) - 2*covariance(XY)

4. T distribution
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

5. Reliability
We reject a hypothesis that is actually true
We accept a hypothesis that should have been rejected
Mean of sampling distribution is the population mean
Statement of the error or precision of an estimate

6. Discrete representation of the GBM
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

7. Non - parametric vs parametric calculation of VaR
Expected value of the sample mean is the population mean
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Does not depend on a prior event or information
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

8. Tractable
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Easy to manipulate
Peaks over threshold - Collects dataset in excess of some threshold
Attempts to sample along more important paths

9. Variance of aX + bY
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
(a^2)(variance(x)) + (b^2)(variance(y))
Mean = np - Variance = npq - Std dev = sqrt(npq)

10. P - value
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
P(Z>t)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

11. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Mean of sampling distribution is the population mean
Concerned with a single random variable (ex. Roll of a die)

12. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
For n>30 - sample mean is approximately normal
Variance(X) + Variance(Y) - 2*covariance(XY)
When the sample size is large - the uncertainty about the value of the sample is very small

13. LAD
Regression can be non - linear in variables but must be linear in parameters
If variance of the conditional distribution of u(i) is not constant
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Least absolute deviations estimator - used when extreme outliers are not uncommon

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

15. Continuously compounded return equation
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)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
i = ln(Si/Si - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

16. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Distribution with only two possible outcomes
Based on an equation - P(A) = # of A/total outcomes
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!)

17. Two drawbacks of moving average series
Easy to manipulate
Random walk (usually acceptable) - Constant volatility (unlikely)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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)

18. Exact significance level
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
P - value

19. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Variance(x) + Variance(Y) + 2*covariance(XY)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Combine to form distribution with leptokurtosis (heavy tails)

20. GEV
P(Z>t)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
P(X=x - Y=y) = P(X=x) * P(Y=y)
Least absolute deviations estimator - used when extreme outliers are not uncommon

21. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Based on a dataset
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

22. Homoskedastic
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Sample mean will near the population mean as the sample size increases

23. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(x)
Transformed to a unit variable - Mean = 0 Variance = 1
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

24. Variance of weighted scheme
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Contains variables not explicit in model - Accounts for randomness
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

25. Normal distribution
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
Independently and Identically Distributed
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
Regression can be non - linear in variables but must be linear in parameters

26. Monte Carlo Simulations
Only requires two parameters = mean and variance
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Mean of sampling distribution is the population mean

27. Homoskedastic only F - stat
Average return across assets on a given day
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

28. Expected future variance rate (t periods forward)
Probability that the random variables take on certain values simultaneously
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

29. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

30. Difference between population and sample variance
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Population denominator = n - Sample denominator = n - 1
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Mean = np - Variance = npq - Std dev = sqrt(npq)

31. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Attempts to sample along more important paths

32. Antithetic variable technique
Combine to form distribution with leptokurtosis (heavy tails)
When one regressor is a perfect linear function of the other regressors
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Independently and Identically Distributed

33. Variance(discrete)
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Concerned with a single random variable (ex. Roll of a die)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

34. Binomial distribution
Normal - Student's T - Chi - square - F 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!)
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
Does not depend on a prior event or information

35. Least squares estimator(m)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Application of mathematical statistics to economic data to lend empirical support to models

36. Binomial distribution equations for mean variance and std dev
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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)
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)

37. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

38. Empirical frequency
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Use historical simulation approach but use the EWMA weighting system
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Based on a dataset

39. Hybrid method for conditional volatility
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Use historical simulation approach but use the EWMA weighting system
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
Special type of pooled data in which the cross sectional unit is surveyed over time

40. Weibul distribution
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

41. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Rxy = Sxy/(Sx*Sy)
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)
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

42. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Least absolute deviations estimator - used when extreme outliers are not uncommon

43. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

44. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Attempts to sample along more important paths

45. Mean reversion in variance
Variance reverts to a long run level
Choose parameters that maximize the likelihood of what observations occurring
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

46. Importance sampling technique
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Attempts to sample along more important paths
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

47. ESS
Use historical simulation approach but use the EWMA weighting system
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
When the sample size is large - the uncertainty about the value of the sample is very small

48. Bootstrap method
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

49. Sample mean
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Expected value of the sample mean is the population mean

50. Key properties of linear regression
P - value
Sample mean +/ - t*(stddev(s)/sqrt(n))
Sampling distribution of sample means tend to be normal
Regression can be non - linear in variables but must be linear in parameters