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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. Test for unbiasedness
Confidence level
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E(mean) = mean

2. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P(Z>t)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

3. Two drawbacks of moving average series
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
P(Z>t)

4. Central Limit Theorem(CLT)
Use historical simulation approach but use the EWMA weighting system
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Sampling distribution of sample means tend to be normal

5. Reliability
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Statement of the error or precision of an estimate
Summation((xi - mean)^k)/n
Special type of pooled data in which the cross sectional unit is surveyed over time

6. Unstable return distribution
P(Z>t)
Model dependent - Options with the same underlying assets may trade at different volatilities
Random walk (usually acceptable) - Constant volatility (unlikely)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

7. Mean(expected value)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Does not depend on a prior event or information

8. Beta distribution
E(XY) - E(X)E(Y)
Low Frequency - High Severity events
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
We accept a hypothesis that should have been rejected

9. Standard error for Monte Carlo replications
Nonlinearity
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

10. Difference between population and sample variance
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Population denominator = n - Sample denominator = n - 1
E(XY) - E(X)E(Y)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

11. Time series data
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Returns over time for an individual asset
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

12. F distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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

13. Variance - covariance approach for VaR of a portfolio
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
We reject a hypothesis that is actually true
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)

14. P - value
P(Z>t)
More than one random variable
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

15. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
(a^2)(variance(x)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

16. Statistical (or empirical) model
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)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Yi = B0 + B1Xi + ui
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

17. Continuously compounded return equation
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
i = ln(Si/Si - 1)
95% = 1.65 99% = 2.33 For one - tailed tests
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

18. Unbiased
Mean of sampling distribution is the population mean
Regression can be non - linear in variables but must be linear in parameters
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

19. Result of combination of two normal with same means
Only requires two parameters = mean and variance
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Among all unbiased estimators - estimator with the smallest variance is efficient
Combine to form distribution with leptokurtosis (heavy tails)

20. Maximum likelihood method
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)
Choose parameters that maximize the likelihood of what observations occurring
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

21. Key properties of linear regression
Low Frequency - High Severity events
Sampling distribution of sample means tend to be normal
Regression can be non - linear in variables but must be linear in parameters
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

22. Variance of aX
(a^2)(variance(x)
We reject a hypothesis that is actually true
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
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)

23. R^2
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Does not depend on a prior event or information
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

24. Economical(elegant)
Does not depend on a prior event or information
Only requires two parameters = mean and variance
Use historical simulation approach but use the EWMA weighting system
Contains variables not explicit in model - Accounts for randomness

25. T distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

26. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Concerned with a single random variable (ex. Roll of a die)
Variance(x) + Variance(Y) + 2*covariance(XY)

27. Homoskedastic only F - stat
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

28. Historical std dev
Special type of pooled data in which the cross sectional unit is surveyed over time
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
Variance = (1/m) summation(u<n - i>^2)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

29. WLS
Choose parameters that maximize the likelihood of what observations occurring
SSR
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

30. Variance of sampling distribution of means when n<N
(a^2)(variance(x)
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample mean will near the population mean as the sample size increases

31. Kurtosis
(a^2)(variance(x)) + (b^2)(variance(y))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Probability that the random variables take on certain values simultaneously

32. Two requirements of OVB
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Price/return tends to run towards a long - run level

33. Least squares estimator(m)
Sample mean +/ - t*(stddev(s)/sqrt(n))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

34. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
E(XY) - E(X)E(Y)
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
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!)

35. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance(x) + Variance(Y) + 2*covariance(XY)
We reject a hypothesis that is actually true
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

36. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Mean of sampling distribution is the population mean
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
When the sample size is large - the uncertainty about the value of the sample is very small

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

38. Control variates technique
Combine to form distribution with leptokurtosis (heavy tails)
Population denominator = n - Sample denominator = n - 1
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

39. Law of Large Numbers
E(XY) - E(X)E(Y)
Sample mean will near the population mean as the sample size increases
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

40. Chi - squared distribution
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
P(Z>t)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

41. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
95% = 1.65 99% = 2.33 For one - tailed tests

42. Persistence
Model dependent - Options with the same underlying assets may trade at different volatilities
When the sample size is large - the uncertainty about the value of the sample is very small
Does not depend on a prior event or information
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

43. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
For n>30 - sample mean is approximately normal
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
Transformed to a unit variable - Mean = 0 Variance = 1

44. Binomial distribution
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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)
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!)

45. Covariance calculations using weight sums (lambda)
P(Z>t)
E(XY) - E(X)E(Y)
Use historical simulation approach but use the EWMA weighting system
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

46. K - th moment
Sample mean will near the population mean as the sample size increases
i = ln(Si/Si - 1)
Summation((xi - mean)^k)/n
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

47. ESS
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Variance(x)

48. Variance of X - Y assuming dependence
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(X) + Variance(Y) - 2*covariance(XY)
When one regressor is a perfect linear function of the other regressors
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

49. Homoskedastic
Concerned with a single random variable (ex. Roll of a die)
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

50. POT
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Peaks over threshold - Collects dataset in excess of some threshold
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates