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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. Continuous random variable
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Returns over time for a combination of assets (combination of time series and cross - sectional data)

2. Poisson Distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

3. Importance sampling technique
(a^2)(variance(x)
Contains variables not explicit in model - Accounts for randomness
Attempts to sample along more important paths
Distribution with only two possible outcomes

4. SER
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
95% = 1.65 99% = 2.33 For one - tailed tests
Var(X) + Var(Y)

5. Biggest (and only real) drawback of GARCH mode
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
We reject a hypothesis that is actually true
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Nonlinearity

6. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Sampling distribution of sample means tend to be normal

7. Logistic distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Has heavy tails
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

8. Lognormal
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Distribution with only two possible outcomes
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

9. Time series data
Returns over time for an individual asset
Special type of pooled data in which the cross sectional unit is surveyed over time
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

10. Adjusted R^2
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
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!)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(y)/n = variance of sample Y

11. Conditional probability functions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Expected value of the sample mean is the population mean
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

12. LFHS
More than one random variable
Low Frequency - High Severity events
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

13. Discrete representation of the GBM
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
Peaks over threshold - Collects dataset in excess of some threshold
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

14. Type II Error
(a^2)(variance(x)) + (b^2)(variance(y))
Rxy = Sxy/(Sx*Sy)
We accept a hypothesis that should have been rejected
More than one random variable

15. Central Limit Theorem
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
More than one random variable
Mean = np - Variance = npq - Std dev = sqrt(npq)
For n>30 - sample mean is approximately normal

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

17. Inverse transform method
P(Z>t)
(a^2)(variance(x)) + (b^2)(variance(y))
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Probability that the random variables take on certain values simultaneously

18. Hybrid method for conditional volatility
Summation((xi - mean)^k)/n
Use historical simulation approach but use the EWMA weighting system
Only requires two parameters = mean and variance
Nonlinearity

19. Implied standard deviation for options
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

20. Historical std dev
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)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance = (1/m) summation(u<n - i>^2)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

21. What does the OLS minimize?
Variance reverts to a long run level
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
SSR
Easy to manipulate

22. Variance - covariance approach for VaR of a portfolio
Use historical simulation approach but use the EWMA weighting system
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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

23. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
P - value
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

24. Non - parametric vs parametric calculation of VaR
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

25. 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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

26. Variance of X+Y
Price/return tends to run towards a long - run level
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Var(X) + Var(Y)
If variance of the conditional distribution of u(i) is not constant

27. Gamma distribution
E(XY) - E(X)E(Y)
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
Normal - Student's T - Chi - square - F distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

28. Marginal unconditional probability function
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Easy to manipulate
Sampling distribution of sample means tend to be normal
Does not depend on a prior event or information

29. Economical(elegant)
Only requires two parameters = mean and variance
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Confidence set for two coefficients - two dimensional analog for the confidence interval

30. Stochastic error term
Does not depend on a prior event or information
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Contains variables not explicit in model - Accounts for randomness

31. Simulating for VaR
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

32. Consistent
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
When the sample size is large - the uncertainty about the value of the sample is very small
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Based on a dataset

33. LAD
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
SSR
Confidence level

34. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance(X) + Variance(Y) - 2*covariance(XY)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance(x)

35. Covariance calculations using weight sums (lambda)
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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

36. Homoskedastic only F - stat
Variance(X) + Variance(Y) - 2*covariance(XY)
P - value
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

37. Two drawbacks of moving average series
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance(x)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Among all unbiased estimators - estimator with the smallest variance is efficient

38. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Independently and Identically Distributed
Confidence set for two coefficients - two dimensional analog for the confidence interval
Has heavy tails

39. Simulation models
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
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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

40. Test for unbiasedness
Independently and Identically Distributed
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E(mean) = mean
Least absolute deviations estimator - used when extreme outliers are not uncommon

41. Mean reversion
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

42. T distribution
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Normal - Student's T - Chi - square - F distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

43. Sample covariance
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Special type of pooled data in which the cross sectional unit is surveyed over time
When one regressor is a perfect linear function of the other regressors

44. Maximum likelihood method
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Confidence level
Choose parameters that maximize the likelihood of what observations occurring
Mean of sampling distribution is the population mean

45. GPD
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Returns over time for an individual asset
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

46. Bernouli Distribution
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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)

47. Mean(expected value)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
i = ln(Si/Si - 1)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Confidence level

48. Multivariate Density Estimation (MDE)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Z = (Y - meany)/(stddev(y)/sqrt(n))

49. Hazard rate of exponentially distributed random variable
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance(y)/n = variance of sample Y
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Population denominator = n - Sample denominator = n - 1

50. Homoskedastic
(a^2)(variance(x)) + (b^2)(variance(y))
Combine to form distribution with leptokurtosis (heavy tails)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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