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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Binomial distribution equations for mean variance and std dev
95% = 1.65 99% = 2.33 For one - tailed tests
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Mean = np - Variance = npq - Std dev = sqrt(npq)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

2. Perfect multicollinearity
Confidence level
When one regressor is a perfect linear function of the other regressors
Contains variables not explicit in model - Accounts for randomness
Variance(x)

3. Panel data (longitudinal or micropanel)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Special type of pooled data in which the cross sectional unit is surveyed over time

4. Maximum likelihood method
Var(X) + Var(Y)
Contains variables not explicit in model - Accounts for randomness
Variance(x)
Choose parameters that maximize the likelihood of what observations occurring

5. Biggest (and only real) drawback of GARCH mode
Confidence set for two coefficients - two dimensional analog for the confidence interval
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Sample mean will near the population mean as the sample size increases
Nonlinearity

6. Variance of X+b
Variance(x)
Normal - Student's T - Chi - square - F distribution
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Concerned with a single random variable (ex. Roll of a die)

7. EWMA
Regression can be non - linear in variables but must be linear in parameters
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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)

8. Two assumptions of square root rule
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Random walk (usually acceptable) - Constant volatility (unlikely)
P - value

9. Unconditional vs conditional distributions
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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))

10. 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
Concerned with a single random variable (ex. Roll of a die)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

11. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

12. Binomial distribution
Combine to form distribution with leptokurtosis (heavy tails)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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!)

13. Variance of sampling distribution of means when n<N
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(sample y) = (variance(y)/n)*(N - n/N - 1)
Special type of pooled data in which the cross sectional unit is surveyed over time
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

14. Standard error
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

15. Expected future variance rate (t periods forward)
Variance(x) + Variance(Y) + 2*covariance(XY)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance(y)/n = variance of sample Y
Regression can be non - linear in variables but must be linear in parameters

16. Test for unbiasedness
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
Average return across assets on a given day
P(X=x - Y=y) = P(X=x) * P(Y=y)
E(mean) = mean

17. Regime - switching volatility model
Special type of pooled data in which the cross sectional unit is surveyed over time
P(X=x - Y=y) = P(X=x) * P(Y=y)
Var(X) + Var(Y)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

18. BLUE
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Returns over time for an individual asset
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

19. F distribution
Regression can be non - linear in variables but must be linear in parameters
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)
Choose parameters that maximize the likelihood of what observations occurring

20. Standard error for Monte Carlo replications
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) + Variance(Y) - 2*covariance(XY)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Returns over time for a combination of assets (combination of time series and cross - sectional data)

21. Simulation models
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
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

22. Covariance calculations using weight sums (lambda)
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
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)
Variance(y)/n = variance of sample Y
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

23. Sample mean
Expected value of the sample mean is the population mean
Probability that the random variables take on certain values simultaneously
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When one regressor is a perfect linear function of the other regressors

24. Confidence interval (from t)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Sample mean +/ - t*(stddev(s)/sqrt(n))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

25. Covariance
E(XY) - E(X)E(Y)
Concerned with a single random variable (ex. Roll of a die)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

26. Poisson Distribution
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Model dependent - Options with the same underlying assets may trade at different volatilities
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

27. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance = (1/m) summation(u<n - i>^2)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
E(XY) - E(X)E(Y)

28. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Concerned with a single random variable (ex. Roll of a die)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

29. Conditional probability functions
(a^2)(variance(x)) + (b^2)(variance(y))
Best Linear Unbiased Estimator - Sample mean for samples that are 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)
E(mean) = mean

30. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Use historical simulation approach but use the EWMA weighting system
Sample mean will near the population mean as the sample size increases
Variance(X) + Variance(Y) - 2*covariance(XY)

31. Four sampling distributions

32. Efficiency
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance reverts to a long run level
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
Among all unbiased estimators - estimator with the smallest variance is efficient

33. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Model dependent - Options with the same underlying assets may trade at different volatilities
Application of mathematical statistics to economic data to lend empirical support to models

34. Poisson distribution equations for mean variance and std deviation
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
Variance reverts to a long run level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

35. Persistence
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
P(X=x - Y=y) = P(X=x) * P(Y=y)
E(mean) = mean

36. Variance of weighted scheme
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

37. Variance(discrete)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Confidence level

38. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Independently and Identically Distributed

39. Test for statistical independence
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
P(X=x - Y=y) = P(X=x) * P(Y=y)

40. Multivariate probability
More than one random variable
95% = 1.65 99% = 2.33 For one - tailed tests
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
If variance of the conditional distribution of u(i) is not constant

41. Variance of aX
(a^2)(variance(x)
Population denominator = n - Sample denominator = n - 1
Returns over time for an individual asset
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

42. GPD
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
(a^2)(variance(x)) + (b^2)(variance(y))
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

43. 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
When one regressor is a perfect linear function of the other regressors
Sampling distribution of sample means tend to be normal
If variance of the conditional distribution of u(i) is not constant

44. Economical(elegant)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Statement of the error or precision of an estimate
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
Only requires two parameters = mean and variance

45. Single variable (univariate) probability
Based on a dataset
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
Concerned with a single random variable (ex. Roll of a die)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

46. Type I error
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
If variance of the conditional distribution of u(i) is not constant
We reject a hypothesis that is actually true
Peaks over threshold - Collects dataset in excess of some threshold

47. Law of Large Numbers
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Sample mean will near the population mean as the sample size increases
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

48. Historical std dev
Random walk (usually acceptable) - Constant volatility (unlikely)
P - value
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

49. GARCH
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)
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)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

50. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"