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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. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
E(XY) - E(X)E(Y)
Sample mean +/ - t*(stddev(s)/sqrt(n))
(a^2)(variance(x)

2. Binomial distribution equations for mean variance and std dev
We reject a hypothesis that is actually true
Mean = np - Variance = npq - Std dev = sqrt(npq)
P(Z>t)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

3. Cholesky factorization (decomposition)
Among all unbiased estimators - estimator with the smallest variance is efficient
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
SSR
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

4. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Rxy = Sxy/(Sx*Sy)
If variance of the conditional distribution of u(i) is not constant

5. Lognormal
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
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
Mean = np - Variance = npq - Std dev = sqrt(npq)

6. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance(y)/n = variance of sample Y
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

7. Single variable (univariate) probability
Random walk (usually acceptable) - Constant volatility (unlikely)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Rxy = Sxy/(Sx*Sy)
Concerned with a single random variable (ex. Roll of a die)

8. Expected future variance rate (t periods forward)
Var(X) + Var(Y)
Combine to form distribution with leptokurtosis (heavy tails)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

9. Covariance
E(XY) - E(X)E(Y)
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
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

10. Chi - squared distribution
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Independently and Identically Distributed
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

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

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12. Continuous representation of the GBM
E(XY) - E(X)E(Y)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

13. Variance of aX + bY
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
(a^2)(variance(x)) + (b^2)(variance(y))
Does not depend on a prior event or information

14. Unbiased
Transformed to a unit variable - Mean = 0 Variance = 1
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Mean of sampling distribution is the population mean

15. Reliability
Statement of the error or precision of an estimate
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
P(Z>t)

16. Variance - covariance approach for VaR of a portfolio
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(mean) = mean
Combine to form distribution with leptokurtosis (heavy tails)

17. Hazard rate of exponentially distributed random variable
Population denominator = n - Sample denominator = n - 1
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)
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

18. Logistic distribution
(a^2)(variance(x)) + (b^2)(variance(y))
Has heavy tails
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Random walk (usually acceptable) - Constant volatility (unlikely)

19. Implications of homoscedasticity
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
For n>30 - sample mean is approximately normal
Mean = np - Variance = npq - Std dev = sqrt(npq)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

20. What does the OLS minimize?
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
Confidence level
Contains variables not explicit in model - Accounts for randomness
SSR

21. Conditional probability functions
Variance reverts to a long run level
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

22. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Transformed to a unit variable - Mean = 0 Variance = 1
We accept a hypothesis that should have been rejected
Variance = (1/m) summation(u<n - i>^2)

23. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
If variance of the conditional distribution of u(i) is not constant
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

24. LAD
Transformed to a unit variable - Mean = 0 Variance = 1
Easy to manipulate
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Least absolute deviations estimator - used when extreme outliers are not uncommon

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

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26. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance = (1/m) summation(u<n - i>^2)

27. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*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

28. Standard variable for non - normal distributions
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Z = (Y - meany)/(stddev(y)/sqrt(n))

29. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Model dependent - Options with the same underlying assets may trade at different volatilities
If variance of the conditional distribution of u(i) is not constant

30. Central Limit Theorem
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
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
For n>30 - sample mean is approximately normal

31. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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

32. Type I error
E(mean) = mean
We reject a hypothesis that is actually true
Confidence level
P - value

33. Result of combination of two normal with same means
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
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)
Combine to form distribution with leptokurtosis (heavy tails)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

34. Econometrics
Expected value of the sample mean is the population mean
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Peaks over threshold - Collects dataset in excess of some threshold
Application of mathematical statistics to economic data to lend empirical support to models

35. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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
E(XY) - E(X)E(Y)
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)

36. Variance of X+Y assuming dependence
Distribution with only two possible outcomes
Variance(x) + Variance(Y) + 2*covariance(XY)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

37. BLUE
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
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.
(a^2)(variance(x)) + (b^2)(variance(y))

38. Non - parametric vs parametric calculation of VaR
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Price/return tends to run towards a long - run level
Rxy = Sxy/(Sx*Sy)

39. Beta distribution
Use historical simulation approach but use the EWMA weighting system
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

40. P - value
Has heavy tails
P(Z>t)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

41. Variance of X+Y
Var(X) + Var(Y)
Mean = np - Variance = npq - Std dev = sqrt(npq)
If variance of the conditional distribution of u(i) is not constant
Sample mean will near the population mean as the sample size increases

42. R^2
SSR
Has heavy tails
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Probability that the random variables take on certain values simultaneously

43. T distribution
Transformed to a unit variable - Mean = 0 Variance = 1
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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

44. Discrete representation of the GBM
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Regression can be non - linear in variables but must be linear in parameters
Concerned with a single random variable (ex. Roll of a die)

45. Binomial distribution
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!)
Choose parameters that maximize the likelihood of what observations occurring
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

46. Tractable
Easy to manipulate
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Average return across assets on a given day

47. Consistent
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
When the sample size is large - the uncertainty about the value of the sample is very small

48. Square root rule
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Expected value of the sample mean is the population mean
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

49. Central Limit Theorem(CLT)
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sampling distribution of sample means tend to be normal
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

50. Heteroskedastic
Population denominator = n - Sample denominator = n - 1
If variance of the conditional distribution of u(i) is not constant
Sampling distribution of sample means tend to be normal
More than one random variable