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FRM Foundations Of Risk Management Quantitative Methods

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1. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
We reject a hypothesis that is actually true
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
SSR

2. Covariance calculations using weight sums (lambda)
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

3. Time series data
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Returns over time for an individual asset
Low Frequency - High Severity events
Sampling distribution of sample means tend to be normal

4. Economical(elegant)
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Only requires two parameters = mean and variance

5. Logistic distribution
Has heavy tails
Confidence level
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance reverts to a long run level

6. Significance =1
Independently and Identically Distributed
Confidence level
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Regression can be non - linear in variables but must be linear in parameters

7. Mean reversion in variance
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance reverts to a long run level
P(X=x - Y=y) = P(X=x) * P(Y=y)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

8. Gamma distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

9. Variance(discrete)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Mean of sampling distribution is the population mean
Returns over time for an individual asset

10. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

11. Sample variance
Mean of sampling distribution is the population mean
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Normal - Student's T - Chi - square - F distribution
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

12. Unbiased
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Does not depend on a prior event or information
Var(X) + Var(Y)
Mean of sampling distribution is the population mean

13. Consistent
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
When the sample size is large - the uncertainty about the value of the sample is very small
Use historical simulation approach but use the EWMA weighting system

14. EWMA
Low Frequency - High Severity events
Sample variance = (1/(k - 1))Summation(Yi - mean)^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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

15. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Mean of sampling distribution is the population mean

16. Variance of X+Y
Yi = B0 + B1Xi + ui
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Var(X) + Var(Y)
P(Z>t)

17. Efficiency
Model dependent - Options with the same underlying assets may trade at different volatilities
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Summation((xi - mean)^k)/n
Among all unbiased estimators - estimator with the smallest variance is efficient

18. Sample correlation
Mean = np - Variance = npq - Std dev = sqrt(npq)
Rxy = Sxy/(Sx*Sy)
(a^2)(variance(x)) + (b^2)(variance(y))
SSR

19. Expected future variance rate (t periods forward)
When one regressor is a perfect linear function of the other regressors
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Attempts to sample along more important paths

20. Priori (classical) probability
Does not depend on a prior event or information
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Based on an equation - P(A) = # of A/total outcomes
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

21. Mean(expected value)
Mean of sampling distribution is the population mean
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Application of mathematical statistics to economic data to lend empirical support to models

22. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Use historical simulation approach but use the EWMA weighting system
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(y)/n = variance of sample Y

23. Variance of X+b
For n>30 - sample mean is approximately normal
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(x)
We reject a hypothesis that is actually true

24. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
P(Z>t)
Variance(X) + Variance(Y) - 2*covariance(XY)

25. Bootstrap method
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(X) + Variance(Y) - 2*covariance(XY)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

26. POT
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Peaks over threshold - Collects dataset in excess of some threshold
Rxy = Sxy/(Sx*Sy)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

27. Hazard rate of exponentially distributed random variable
Nonlinearity
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Only requires two parameters = mean and variance

28. Variance of aX
(a^2)(variance(x)
P(Z>t)
(a^2)(variance(x)) + (b^2)(variance(y))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

29. Implied standard deviation for options
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Statement of the error or precision of an estimate
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

30. Perfect multicollinearity
Variance reverts to a long run level
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
When one regressor is a perfect linear function of the other regressors
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

31. Extreme Value Theory
Among all unbiased estimators - estimator with the smallest variance is efficient
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

32. Poisson Distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
P - value
Regression can be non - linear in variables but must be linear in parameters
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

33. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
SSR
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

35. Variance of aX + bY
Sample mean +/ - t*(stddev(s)/sqrt(n))
(a^2)(variance(x)) + (b^2)(variance(y))
Yi = B0 + B1Xi + ui
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

36. Multivariate Density Estimation (MDE)
Population denominator = n - Sample denominator = n - 1
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Var(X) + Var(Y)

37. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance reverts to a long run level
Has heavy tails

38. Variance of sample mean
Variance(y)/n = variance of sample Y
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

39. GEV
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Transformed to a unit variable - Mean = 0 Variance = 1

40. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

41. Variance of X - Y assuming dependence
Attempts to sample along more important paths
Variance(x) + Variance(Y) + 2*covariance(XY)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance(X) + Variance(Y) - 2*covariance(XY)

42. Type II Error
Sample mean will near the population mean as the sample size increases
95% = 1.65 99% = 2.33 For one - tailed tests
We accept a hypothesis that should have been rejected
Least absolute deviations estimator - used when extreme outliers are not uncommon

43. Confidence interval for sample mean
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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

44. Joint probability functions
Var(X) + Var(Y)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Probability that the random variables take on certain values simultaneously
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

45. Regime - switching volatility model
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 an equation - P(A) = # of A/total outcomes
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

46. Empirical frequency
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Sample mean will near the population mean as the sample size increases
Probability that the random variables take on certain values simultaneously
Based on a dataset

47. Deterministic Simulation
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
P - value
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
Average return across assets on a given day

48. Confidence ellipse
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Confidence set for two coefficients - two dimensional analog for the confidence interval
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!)
Variance(x)

49. Central Limit Theorem(CLT)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Sampling distribution of sample means tend to be normal
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Special type of pooled data in which the cross sectional unit is surveyed over time

50. Monte Carlo Simulations
Returns over time for an individual asset
Has heavy tails
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Z = (Y - meany)/(stddev(y)/sqrt(n))

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