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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. Econometrics
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Application of mathematical statistics to economic data to lend empirical support to models
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

2. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
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)
Average return across assets on a given day
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

3. Mean reversion
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

4. Tractable
Statement of the error or precision of an estimate
Easy to manipulate
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

5. Marginal unconditional probability function
Easy to manipulate
Does not depend on a prior event or information
Concerned with a single random variable (ex. Roll of a die)
Probability that the random variables take on certain values simultaneously

6. Stochastic error term
Price/return tends to run towards a long - run level
Contains variables not explicit in model - Accounts for randomness
E(mean) = mean
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

7. Least squares estimator(m)
SSR
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Model dependent - Options with the same underlying assets may trade at different volatilities

8. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(x)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

9. Deterministic Simulation
Has heavy tails
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
For n>30 - sample mean is approximately normal
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

10. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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)
Z = (Y - meany)/(stddev(y)/sqrt(n))
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

11. Test for unbiasedness
E(mean) = mean
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Returns over time for an individual asset
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

12. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

13. Variance of sample mean
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(y)/n = variance of sample Y
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors

14. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
95% = 1.65 99% = 2.33 For one - tailed tests
Attempts to sample along more important paths

15. Bernouli Distribution
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance(x) + Variance(Y) + 2*covariance(XY)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Distribution with only two possible outcomes

16. Type I error
Combine to form distribution with leptokurtosis (heavy tails)
We reject a hypothesis that is actually true
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
Special type of pooled data in which the cross sectional unit is surveyed over time

17. Covariance
E(XY) - E(X)E(Y)
Attempts to sample along more important paths
Confidence set for two coefficients - two dimensional analog for the confidence interval
We reject a hypothesis that is actually true

18. Discrete random variable
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance(x) + Variance(Y) + 2*covariance(XY)
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
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Combine to form distribution with leptokurtosis (heavy tails)
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

20. Logistic distribution
Has heavy tails
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
More than one random variable
Average return across assets on a given day

21. Confidence interval (from t)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Sample mean +/ - t*(stddev(s)/sqrt(n))
P - value
Variance(x) + Variance(Y) + 2*covariance(XY)

22. GPD
Contains variables not explicit in model - Accounts for randomness
When one regressor is a perfect linear function of the other regressors
Based on a dataset
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

23. Continuous representation of the GBM
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!)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

24. Variance of X+b
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Normal - Student's T - Chi - square - F distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(x)

25. Unbiased
Mean of sampling distribution is the population mean
Peaks over threshold - Collects dataset in excess of some threshold
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

26. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sampling distribution of sample means tend to be normal
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

27. SER
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
Transformed to a unit variable - Mean = 0 Variance = 1
E(XY) - E(X)E(Y)

28. Confidence interval for sample mean
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
Transformed to a unit variable - Mean = 0 Variance = 1
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

29. Sample mean
Expected value of the sample mean is the population mean
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)
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

30. Single variable (univariate) probability
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
Concerned with a single random variable (ex. Roll of a die)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Application of mathematical statistics to economic data to lend empirical support to models

31. Multivariate probability
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Has heavy tails
More than one random variable
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

32. Test for statistical independence
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
P(X=x - Y=y) = P(X=x) * P(Y=y)
Statement of the error or precision of an estimate
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

33. R^2
Expected value of the sample mean is the population mean
More than one random variable
Special type of pooled data in which the cross sectional unit is surveyed over time
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

34. Discrete representation of the GBM
(a^2)(variance(x)
Price/return tends to run towards a long - run level
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

35. Standard error for Monte Carlo replications
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Does not depend on a prior event or information
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

36. Mean reversion in variance
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance reverts to a long run level

37. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
(a^2)(variance(x)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

38. Implications of homoscedasticity
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
Variance(X) + Variance(Y) - 2*covariance(XY)

39. Perfect multicollinearity
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)
Low Frequency - High Severity events
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
When one regressor is a perfect linear function of the other regressors

40. Square root rule
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
We accept a hypothesis that should have been rejected
Price/return tends to run towards a long - run level

41. Antithetic variable technique
E(mean) = mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Normal - Student's T - Chi - square - F distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

42. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
When the sample size is large - the uncertainty about the value of the sample is very small
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

43. Direction of OVB
Sample mean +/ - t*(stddev(s)/sqrt(n))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

44. Time series data
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Returns over time for an individual asset
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Has heavy tails

45. F distribution
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
Population denominator = n - Sample denominator = n - 1
Does not depend on a prior event or information
Rxy = Sxy/(Sx*Sy)

46. Variance of sampling distribution of means when n<N
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Independently and Identically Distributed
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

47. Skewness
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
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
Statement of the error or precision of an estimate

48. Standard normal distribution
When the sample size is large - the uncertainty about the value of the sample is very small
Rxy = Sxy/(Sx*Sy)
Transformed to a unit variable - Mean = 0 Variance = 1
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

49. Sample correlation
Summation((xi - mean)^k)/n
Population denominator = n - Sample denominator = n - 1
Rxy = Sxy/(Sx*Sy)
When the sample size is large - the uncertainty about the value of the sample is very small

50. Central Limit Theorem
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
For n>30 - sample mean is approximately normal
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))