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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. Perfect multicollinearity
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)
When one regressor is a perfect linear function of the other regressors
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Z = (Y - meany)/(stddev(y)/sqrt(n))

2. Simplified standard (un - weighted) variance
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)
Confidence level
Variance = (1/m) summation(u<n - i>^2)

3. SER
Population denominator = n - Sample denominator = n - 1
Based on an equation - P(A) = # of A/total outcomes
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

4. Exact significance level
Combine to form distribution with leptokurtosis (heavy tails)
95% = 1.65 99% = 2.33 For one - tailed tests
E(mean) = mean
P - value

5. Joint probability functions
Yi = B0 + B1Xi + ui
Probability that the random variables take on certain values simultaneously
Statement of the error or precision of an estimate
More than one random variable

6. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Variance(X) + Variance(Y) - 2*covariance(XY)
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!)
Expected value of the sample mean is the population mean

7. Weibul distribution
Peaks over threshold - Collects dataset in excess of some threshold
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Has heavy tails
Special type of pooled data in which the cross sectional unit is surveyed over time

8. Variance of sampling distribution of means when n<N
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

9. Mean reversion
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Sampling distribution of sample means tend to be normal
Average return across assets on a given day

10. Logistic distribution
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Has heavy tails
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

11. Normal distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance(X) + Variance(Y) - 2*covariance(XY)
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
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

12. Empirical frequency
Var(X) + Var(Y)
Easy to manipulate
Based on a dataset
When one regressor is a perfect linear function of the other regressors

13. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Model dependent - Options with the same underlying assets may trade at different volatilities
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

14. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Confidence level

15. Time series data
When one regressor is a perfect linear function of the other regressors
Combine to form distribution with leptokurtosis (heavy tails)
Independently and Identically Distributed
Returns over time for an individual asset

16. Two requirements of OVB
Variance(y)/n = variance of sample Y
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
If variance of the conditional distribution of u(i) is not constant
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

17. Confidence interval for sample mean
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Does not depend on a prior event or information
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

18. Econometrics
Average return across assets on a given day
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Nonlinearity
Application of mathematical statistics to economic data to lend empirical support to models

19. Cross - sectional
SSR
If variance of the conditional distribution of u(i) is not constant
Average return across assets on a given day
P(X=x - Y=y) = P(X=x) * P(Y=y)

20. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Var(X) + Var(Y)

21. Test for unbiasedness
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
We reject a hypothesis that is actually true
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E(mean) = mean

22. Statistical (or empirical) model
SSR
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Yi = B0 + B1Xi + ui
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

23. GARCH
Special type of pooled data in which the cross sectional unit is surveyed over time
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.
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)

24. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(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
Sample mean +/ - t*(stddev(s)/sqrt(n))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

25. Significance =1
Confidence level
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E(XY) - E(X)E(Y)
Confidence set for two coefficients - two dimensional analog for the confidence interval

26. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(x)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

27. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Random walk (usually acceptable) - Constant volatility (unlikely)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance(X) + Variance(Y) - 2*covariance(XY)

28. Persistence
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
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

29. Binomial distribution equations for mean variance and std dev
Returns over time for an individual asset
We accept a hypothesis that should have been rejected
Mean = np - Variance = npq - Std dev = sqrt(npq)
Confidence set for two coefficients - two dimensional analog for the confidence interval

30. What does the OLS minimize?
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)
SSR
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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

31. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

32. Pooled data
Variance(x) + Variance(Y) + 2*covariance(XY)
Price/return tends to run towards a long - run level
Returns over time for a combination of assets (combination of time series and cross - sectional data)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

33. Reliability
Random walk (usually acceptable) - Constant volatility (unlikely)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Statement of the error or precision of an estimate

34. Potential reasons for fat tails in return distributions
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Yi = B0 + B1Xi + ui
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

35. Type II Error
Independently and Identically Distributed
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
We accept a hypothesis that should have been rejected
Statement of the error or precision of an estimate

36. Block maxima
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

37. Adjusted R^2
Based on an equation - P(A) = # of A/total outcomes
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
Based on a dataset
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

38. Unbiased
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
Mean of sampling distribution is the population mean
Summation((xi - mean)^k)/n
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

39. Overall F - statistic
Among all unbiased estimators - estimator with the smallest variance is efficient
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
When the sample size is large - the uncertainty about the value of the sample is very small

40. Conditional probability functions
Random walk (usually acceptable) - Constant volatility (unlikely)
Confidence level
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

41. Sample correlation
Based on an equation - P(A) = # of A/total outcomes
Returns over time for an individual asset
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Rxy = Sxy/(Sx*Sy)

42. Consistent
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
When the sample size is large - the uncertainty about the value of the sample is very small
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

43. Inverse transform method
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Confidence set for two coefficients - two dimensional analog for the confidence interval

44. Poisson Distribution
Summation((xi - mean)^k)/n
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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

45. Multivariate Density Estimation (MDE)
Variance reverts to a long run 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)
95% = 1.65 99% = 2.33 For one - tailed tests
E(mean) = mean

46. Panel data (longitudinal or micropanel)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
SSR
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

47. Law of Large Numbers
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Sample mean will near the population mean as the sample size increases
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

48. Two drawbacks of moving average series
Summation((xi - mean)^k)/n
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
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

49. Central Limit Theorem
Returns over time for an individual asset
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
For n>30 - sample mean is approximately normal

50. F distribution
Low Frequency - High Severity events
Rxy = Sxy/(Sx*Sy)
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))