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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Variance(discrete)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

2. Simulation models
Choose parameters that maximize the likelihood of what observations occurring
Random walk (usually acceptable) - Constant volatility (unlikely)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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

3. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Use historical simulation approach but use the EWMA weighting system

4. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Returns over time for an individual asset
Among all unbiased estimators - estimator with the smallest variance is efficient
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

5. Two requirements of OVB
Variance(x)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
P - value

6. Implications of homoscedasticity
When one regressor is a perfect linear function of the other regressors
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

7. Overall F - statistic
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Based on an equation - P(A) = # of A/total outcomes

8. Standard error
Population denominator = n - Sample denominator = n - 1
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
If variance of the conditional distribution of u(i) is not constant
Variance(X) + Variance(Y) - 2*covariance(XY)

9. Direction of OVB
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sample mean will near the population mean as the sample size increases
Z = (Y - meany)/(stddev(y)/sqrt(n))

10. Pooled data
Does not depend on a prior event or information
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Returns over time for a combination of assets (combination of time series and cross - sectional data)

11. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
95% = 1.65 99% = 2.33 For one - tailed tests
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

12. Variance of weighted scheme
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Average return across assets on a given day
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

13. Homoskedastic
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Choose parameters that maximize the likelihood of what observations occurring

14. Standard error for Monte Carlo replications
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
We reject a hypothesis that is actually true
Among all unbiased estimators - estimator with the smallest variance is efficient
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

15. Efficiency
Least absolute deviations estimator - used when extreme outliers are not uncommon
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

16. Difference between population and sample variance
Var(X) + Var(Y)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
SSR
Population denominator = n - Sample denominator = n - 1

17. Law of Large Numbers
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sample mean will near the population mean as the sample size increases
Concerned with a single random variable (ex. Roll of a die)
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. Non - parametric vs parametric calculation of VaR
Confidence set for two coefficients - two dimensional analog for the confidence interval
Nonlinearity
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

19. K - th moment
Based on an equation - P(A) = # of A/total outcomes
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Contains variables not explicit in model - Accounts for randomness
Summation((xi - mean)^k)/n

20. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

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

22. Hazard rate of exponentially distributed random variable
Variance(x) + Variance(Y) + 2*covariance(XY)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

23. Significance =1
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Confidence level
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 set for two coefficients - two dimensional analog for the confidence interval

24. Panel data (longitudinal or micropanel)
Average return across assets on a given day
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Confidence level
Special type of pooled data in which the cross sectional unit is surveyed over time

25. Variance of X+Y
(a^2)(variance(x)) + (b^2)(variance(y))
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Var(X) + Var(Y)

26. Conditional probability functions
Yi = B0 + B1Xi + ui
Among all unbiased estimators - estimator with the smallest variance is efficient
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

27. Sample mean
Expected value of the sample mean is the population mean
Variance reverts to a long run level
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Transformed to a unit variable - Mean = 0 Variance = 1

28. Variance of sample mean
Confidence level
Variance(y)/n = variance of sample Y
Sampling distribution of sample means tend to be normal
P - value

29. Statistical (or empirical) model
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Combine to form distribution with leptokurtosis (heavy tails)
Yi = B0 + B1Xi + ui

30. Chi - squared distribution
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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Rxy = Sxy/(Sx*Sy)
Variance(x) + Variance(Y) + 2*covariance(XY)

31. Persistence
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
Contains variables not explicit in model - Accounts for randomness
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)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

32. Square root rule
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
i = ln(Si/Si - 1)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

33. Discrete random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Price/return tends to run towards a long - run level
Concerned with a single random variable (ex. Roll of a die)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

34. Adjusted R^2
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

35. Gamma distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon
More than one random variable
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

36. Mean reversion in asset dynamics
Choose parameters that maximize the likelihood of what observations occurring
When one regressor is a perfect linear function of the other regressors
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
Price/return tends to run towards a long - run level

37. Covariance calculations using weight sums (lambda)
Sampling distribution of sample means tend to be normal
(a^2)(variance(x)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Regression can be non - linear in variables but must be linear in parameters

38. Variance of aX + bY
Average return across assets on a given day
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
(a^2)(variance(x)) + (b^2)(variance(y))

39. Variance of X+b
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Concerned with a single random variable (ex. Roll of a die)
Variance(x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

40. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Distribution with only two possible outcomes
Confidence level

41. R^2
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
i = ln(Si/Si - 1)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

42. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
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
Variance reverts to a long run level
Var(X) + Var(Y)

43. Kurtosis
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

44. i.i.d.
Contains variables not explicit in model - Accounts for randomness
Independently and Identically Distributed
Mean of sampling distribution is the population mean
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

45. Homoskedastic only F - stat
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

46. 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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(y)/n = variance of sample Y
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

47. Central Limit Theorem
Random walk (usually acceptable) - Constant volatility (unlikely)
For n>30 - sample mean is approximately normal
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

48. Bernouli Distribution
Distribution with only two possible outcomes
Returns over time for an individual asset
Concerned with a single random variable (ex. Roll of a die)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

49. Poisson Distribution
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
P - value

50. Mean(expected value)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Population denominator = n - Sample denominator = n - 1