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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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1. Four sampling distributions

2. R^2
Model dependent - Options with the same underlying assets may trade at different volatilities
When one regressor is a perfect linear function of the other regressors
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)

3. Kurtosis
P - value
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Among all unbiased estimators - estimator with the smallest variance is efficient
Does not depend on a prior event or information

4. SER
Confidence level
When the sample size is large - the uncertainty about the value of the sample is very small
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

5. Beta distribution
Peaks over threshold - Collects dataset in excess of some threshold
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

6. Mean reversion in variance
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
Variance reverts to a long run level
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Attempts to sample along more important paths

7. Consistent
Variance = (1/m) summation(u<n - i>^2)
When the sample size is large - the uncertainty about the value of the sample is very small
Attempts to sample along more important paths
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

8. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Mean of sampling distribution is the population mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E(XY) - E(X)E(Y)

9. Deterministic Simulation
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Expected value of the sample mean is the population mean

10. Marginal unconditional probability function
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Does not depend on a prior event or information
Returns over time for an individual asset
When one regressor is a perfect linear function of the other regressors

11. Covariance calculations using weight sums (lambda)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
i = ln(Si/Si - 1)
Has heavy tails
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

12. BLUE
Concerned with a single random variable (ex. Roll of a die)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Sample mean will near the population mean as the sample size increases
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

13. Expected future variance rate (t periods forward)
Based on an equation - P(A) = # of A/total outcomes
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Transformed to a unit variable - Mean = 0 Variance = 1
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

14. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
We accept a hypothesis that should have been rejected

15. Law of Large Numbers
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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
Sample mean will near the population mean as the sample size increases

16. SER
Transformed to a unit variable - Mean = 0 Variance = 1
We accept a hypothesis that should have been rejected
Choose parameters that maximize the likelihood of what observations occurring
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

17. Statistical (or empirical) model
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Yi = B0 + B1Xi + ui
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

18. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Attempts to sample along more important paths
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

19. K - th moment
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Summation((xi - mean)^k)/n
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

20. Confidence interval (from t)
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
Contains variables not explicit in model - Accounts for randomness
Statement of the error or precision of an estimate

21. Difference between population and sample variance
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Population denominator = n - Sample denominator = n - 1

22. Variance of X+b
Variance(x)
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
95% = 1.65 99% = 2.33 For one - tailed tests
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

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

24. Biggest (and only real) drawback of GARCH mode
Nonlinearity
We reject a hypothesis that is actually true
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Choose parameters that maximize the likelihood of what observations occurring

25. Weibul distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Application of mathematical statistics to economic data to lend empirical support to models
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
We accept a hypothesis that should have been rejected

26. Logistic distribution
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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)
Has heavy tails
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

27. Homoskedastic only F - stat
Application of mathematical statistics to economic data to lend empirical support to models
Nonlinearity
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

28. Poisson Distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

29. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Statement of the error or precision of an estimate
P(X=x - Y=y) = P(X=x) * P(Y=y)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

30. Time series data
Based on a dataset
Variance reverts to a long run level
Returns over time for an individual asset
Transformed to a unit variable - Mean = 0 Variance = 1

31. Variance - covariance approach for VaR of a portfolio
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Contains variables not explicit in model - Accounts for randomness
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Peaks over threshold - Collects dataset in excess of some threshold

32. 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)
Independently and Identically Distributed
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
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

33. Sample correlation
Rxy = Sxy/(Sx*Sy)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

34. Maximum likelihood method
Yi = B0 + B1Xi + ui
Sample mean will near the population mean as the sample size increases
Choose parameters that maximize the likelihood of what observations occurring
Normal - Student's T - Chi - square - F distribution

35. GEV
Only requires two parameters = mean and variance
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
When one regressor is a perfect linear function of the other regressors

36. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Concerned with a single random variable (ex. Roll of a die)
Variance(x)
Mean of sampling distribution is the population mean

37. Two drawbacks of moving average series
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)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

38. Sample variance
Independently and Identically Distributed
Application of mathematical statistics to economic data to lend empirical support to models
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

39. Sample covariance
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance = (1/m) summation(u<n - i>^2)

40. EWMA
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Normal - Student's T - Chi - square - F distribution

41. Tractable
Variance(x)
Easy to manipulate
Choose parameters that maximize the likelihood of what observations occurring
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

42. Econometrics
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(X=x - Y=y) = P(X=x) * P(Y=y)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Application of mathematical statistics to economic data to lend empirical support to models

43. Poisson distribution equations for mean variance and std deviation
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
i = ln(Si/Si - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

44. Joint probability functions
Confidence level
95% = 1.65 99% = 2.33 For one - tailed tests
Nonlinearity
Probability that the random variables take on certain values simultaneously

45. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
P(X=x - Y=y) = P(X=x) * P(Y=y)
More than one random variable

46. Importance sampling technique
Contains variables not explicit in model - Accounts for randomness
Attempts to sample along more important paths
Population denominator = n - Sample denominator = n - 1
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

47. Extreme Value Theory
If variance of the conditional distribution of u(i) is not constant
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

48. WLS
Price/return tends to run towards a long - run level
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

49. Adjusted R^2
Contains variables not explicit in model - Accounts for randomness
Special type of pooled data in which the cross sectional unit is surveyed over time
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
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

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