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FRM Foundations Of Risk Management Quantitative Methods

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1. Deterministic Simulation
Mean = np - Variance = npq - Std dev = sqrt(npq)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Variance(y)/n = variance of sample Y

2. Monte Carlo Simulations
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
If variance of the conditional distribution of u(i) is not constant
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

3. Type I error
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Confidence set for two coefficients - two dimensional analog for the confidence interval
We reject a hypothesis that is actually true
Var(X) + Var(Y)

4. Two drawbacks of moving average series
Expected value of the sample mean is the population mean
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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

5. Type II Error
Easy to manipulate
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
We accept a hypothesis that should have been rejected
Confidence level

6. Perfect multicollinearity
Yi = B0 + B1Xi + ui
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
When one regressor is a perfect linear function of the other regressors
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

7. Reliability
Variance(x)
Transformed to a unit variable - Mean = 0 Variance = 1
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Statement of the error or precision of an estimate

8. Econometrics
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Application of mathematical statistics to economic data to lend empirical support to models
Variance reverts to a long run level
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

9. Homoskedastic
Regression can be non - linear in variables but must be linear in parameters
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
Peaks over threshold - Collects dataset in excess of some threshold
Use historical simulation approach but use the EWMA weighting system

10. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

11. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Z = (Y - meany)/(stddev(y)/sqrt(n))

12. Extreme Value Theory
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
We reject a hypothesis that is actually true
Special type of pooled data in which the cross sectional unit is surveyed over time
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

13. Difference between population and sample variance
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Population denominator = n - Sample denominator = n - 1
Probability that the random variables take on certain values simultaneously

14. i.i.d.
Sampling distribution of sample means tend to be normal
Independently and Identically Distributed
Choose parameters that maximize the likelihood of what observations occurring
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

15. 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
Regression can be non - linear in variables but must be linear in parameters
Independently and Identically Distributed
Confidence level

16. Economical(elegant)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Only requires two parameters = mean and variance
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

17. Variance of X - Y assuming dependence
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Only requires two parameters = mean and variance
Variance(X) + Variance(Y) - 2*covariance(XY)

18. Sample correlation
For n>30 - sample mean is approximately normal
Rxy = Sxy/(Sx*Sy)
Variance(X) + Variance(Y) - 2*covariance(XY)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

19. 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
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
Has heavy tails
Easy to manipulate

20. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Use historical simulation approach but use the EWMA weighting system
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
Transformed to a unit variable - Mean = 0 Variance = 1

21. Test for unbiasedness
Returns over time for a combination of assets (combination of time series and cross - sectional data)
For n>30 - sample mean is approximately normal
E(mean) = mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

22. Priori (classical) probability
Special type of pooled data in which the cross sectional unit is surveyed over time
Based on an equation - P(A) = # of A/total outcomes
Choose parameters that maximize the likelihood of what observations occurring
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

23. Logistic distribution
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
i = ln(Si/Si - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Has heavy tails

24. Poisson Distribution
Nonlinearity
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
When one regressor is a perfect linear function of the other regressors
Variance(x) + Variance(Y) + 2*covariance(XY)

25. Weibul distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

26. Standard normal distribution
Low Frequency - High Severity events
Returns over time for an individual asset
Transformed to a unit variable - Mean = 0 Variance = 1
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

27. Marginal unconditional probability function
Does not depend on a prior event or information
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Has heavy 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

28. Hazard rate of exponentially distributed random variable
Normal - Student's T - Chi - square - F distribution
95% = 1.65 99% = 2.33 For one - tailed tests
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
When one regressor is a perfect linear function of the other regressors

29. Antithetic variable technique
Regression can be non - linear in variables but must be linear in parameters
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

30. Central Limit Theorem
For n>30 - sample mean is approximately normal
We reject a hypothesis that is actually true
Normal - Student's T - Chi - square - F distribution
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

31. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
When one regressor is a perfect linear function of the other regressors
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Distribution with only two possible outcomes

32. Inverse transform method
Combine to form distribution with leptokurtosis (heavy tails)
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

33. Mean reversion in variance
Probability that the random variables take on certain values simultaneously
Variance reverts to a long run level
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

34. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Distribution with only two possible outcomes

35. Regime - switching volatility model
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Based on an equation - P(A) = # of A/total outcomes
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

36. Cross - sectional
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Average return across assets on a given day

37. Variance of X+b
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance(x)
Model dependent - Options with the same underlying assets may trade at different volatilities

38. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P - value
Var(X) + Var(Y)
Model dependent - Options with the same underlying assets may trade at different volatilities

39. Variance of aX
Contains variables not explicit in model - Accounts for randomness
Among all unbiased estimators - estimator with the smallest variance is efficient
E(XY) - E(X)E(Y)
(a^2)(variance(x)

40. Bootstrap method
Variance = (1/m) summation(u<n - i>^2)
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

41. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
We accept a hypothesis that should have been rejected
Yi = B0 + B1Xi + ui

42. Hybrid method for conditional volatility
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Use historical simulation approach but use the EWMA weighting system
SSR

43. Covariance
E(XY) - E(X)E(Y)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
For n>30 - sample mean is approximately normal

44. Confidence interval (from t)
Summation((xi - mean)^k)/n
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Sample mean +/ - t*(stddev(s)/sqrt(n))

45. Time series data
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Returns over time for an individual asset
Variance(X) + Variance(Y) - 2*covariance(XY)

46. Unbiased
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
P(Z>t)
Mean of sampling distribution is the population mean
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

47. Importance sampling technique
Attempts to sample along more important paths
Choose parameters that maximize the likelihood of what observations occurring
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

48. Discrete representation of the GBM
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

49. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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)
Returns over time for an individual asset
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

50. Tractable
Easy to manipulate
Returns over time for an individual asset
Special type of pooled data in which the cross sectional unit is surveyed over 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)

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FRM Foundations Of Risk Management Quantitative Methods

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