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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. Unbiased
Mean of sampling distribution is the population mean
Sample mean will near the population mean as the sample size increases
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)
Sampling distribution of sample means tend to be normal

2. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
SSR
Z = (Y - meany)/(stddev(y)/sqrt(n))
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

3. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
For n>30 - sample mean is approximately normal
Variance(x)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

4. Importance sampling technique
Attempts to sample along more important paths
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
We accept a hypothesis that should have been rejected
Random walk (usually acceptable) - Constant volatility (unlikely)

5. Monte Carlo Simulations
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

6. Bernouli Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
i = ln(Si/Si - 1)
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
Distribution with only two possible outcomes

7. Standard normal distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Transformed to a unit variable - Mean = 0 Variance = 1
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Var(X) + Var(Y)

8. Covariance
P(Z>t)
E(XY) - E(X)E(Y)
For n>30 - sample mean is approximately normal
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

9. Test for statistical independence
Average return across assets on a given day
P(X=x - Y=y) = P(X=x) * P(Y=y)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
P - value

10. T distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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)

11. BLUE
(a^2)(variance(x)) + (b^2)(variance(y))
Sample mean +/ - t*(stddev(s)/sqrt(n))
Application of mathematical statistics to economic data to lend empirical support to models
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

12. Conditional probability functions
Distribution with only two possible outcomes
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Combine to form distribution with leptokurtosis (heavy tails)
Price/return tends to run towards a long - run level

13. Variance of X+Y
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Var(X) + Var(Y)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

14. Logistic distribution
Based on an equation - P(A) = # of A/total outcomes
Has heavy tails
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
Based on a dataset

15. Mean reversion in asset dynamics
When one regressor is a perfect linear function of the other regressors
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Price/return tends to run towards a long - run level
Population denominator = n - Sample denominator = n - 1

16. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
For n>30 - sample mean is approximately normal
Sample mean will near the population mean as the sample size increases

17. Four sampling distributions

18. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Mean = np - Variance = npq - Std dev = sqrt(npq)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
More than one random variable

19. Empirical frequency
Combine to form distribution with leptokurtosis (heavy tails)
Based on a dataset
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
Attempts to sample along more important paths

20. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
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
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!)
Easy to manipulate

21. Variance of aX
Attempts to sample along more important paths
(a^2)(variance(x)
Does not depend on a prior event or information
Expected value of the sample mean is the population mean

22. Variance of aX + bY
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
(a^2)(variance(x)) + (b^2)(variance(y))
Concerned with a single random variable (ex. Roll of a die)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

23. Confidence interval for sample mean
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Sample mean +/ - t*(stddev(s)/sqrt(n))

24. Key properties of linear regression
Confidence level
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
Regression can be non - linear in variables but must be linear in parameters
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

25. Tractable
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
P(Z>t)
Easy to manipulate
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)

26. Square root rule
SSR
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Least absolute deviations estimator - used when extreme outliers are not uncommon
When the sample size is large - the uncertainty about the value of the sample is very small

27. Two assumptions of square root rule
Variance = (1/m) summation(u<n - i>^2)
Random walk (usually acceptable) - Constant volatility (unlikely)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

28. Sample correlation
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
Rxy = Sxy/(Sx*Sy)
More than one random variable
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

29. Continuously compounded return equation
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
More than one random variable
i = ln(Si/Si - 1)

30. Type II Error
Normal - Student's T - Chi - square - F distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
We accept a hypothesis that should have been rejected
Returns over time for an individual asset

31. Simulating for VaR
Choose parameters that maximize the likelihood of what observations occurring
Returns over time for an individual asset
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

32. Joint probability functions
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Easy to manipulate
Probability that the random variables take on certain values simultaneously
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

33. R^2
Average return across assets on a given day
When the sample size is large - the uncertainty about the value of the sample is very small
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Only requires two parameters = mean and variance

34. Variance of sampling distribution of means when n<N
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance(sample y) = (variance(y)/n)*(N - n/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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

35. Standard error
Has heavy tails
When one regressor is a perfect linear function of the other regressors
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

36. Implied standard deviation for options
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
Easy to manipulate
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

37. Adjusted R^2
Independently and Identically Distributed
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
If variance of the conditional distribution of u(i) is not constant
Variance(y)/n = variance of sample Y

38. Homoskedastic
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Confidence level
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

39. LFHS
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Low Frequency - High Severity 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

40. Homoskedastic only F - stat
Sample mean +/ - t*(stddev(s)/sqrt(n))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 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
For n>30 - sample mean is approximately normal

41. Multivariate probability
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
More than one random variable
Special type of pooled data in which the cross sectional unit is surveyed over time
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

42. Direction of OVB
Sample mean will near the population mean as the sample size increases
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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)
Peaks over threshold - Collects dataset in excess of some threshold

43. Variance(discrete)
Mean of sampling distribution is the population mean
Low Frequency - High Severity events
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Var(X) + Var(Y)

44. Significance =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)
Confidence level
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Summation((xi - mean)^k)/n

45. Binomial distribution
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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!)

46. Poisson distribution equations for mean variance and std deviation
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)
Statement of the error or precision of an estimate
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

47. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Regression can be non - linear in variables but must be linear in parameters
Yi = B0 + B1Xi + ui

48. Central Limit Theorem
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
For n>30 - sample mean is approximately normal
Only requires two parameters = mean and variance
Confidence set for two coefficients - two dimensional analog for the confidence interval

49. Variance of sample mean
Variance(y)/n = variance of sample Y
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

50. Time series data
Returns over time for an individual asset
Variance = (1/m) summation(u<n - i>^2)
Variance(x)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2