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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. K - th moment
More than one random variable
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
Summation((xi - mean)^k)/n

2. Variance of sample mean
Variance(y)/n = variance of sample Y
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
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)

3. Type I error
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
We reject a hypothesis that is actually true

4. Variance of sampling distribution of means when n<N
E(mean) = mean
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

5. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Mean of sampling distribution is the population mean
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Z = (Y - meany)/(stddev(y)/sqrt(n))

6. Priori (classical) probability
Contains variables not explicit in model - Accounts for randomness
Based on an equation - P(A) = # of A/total outcomes
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
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

7. Law of Large Numbers
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Sample mean will near the population mean as the sample size increases

8. LFHS
Mean of sampling distribution is the population mean
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Low Frequency - High Severity events
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

9. Control variates technique
Concerned with a single random variable (ex. Roll of a die)
Regression can be non - linear in variables but must be linear in parameters
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
When one regressor is a perfect linear function of the other regressors

10. Poisson Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Average return across assets on a given day
Var(X) + Var(Y)
Variance(X) + Variance(Y) - 2*covariance(XY)

11. Hybrid method for conditional volatility
Confidence set for two coefficients - two dimensional analog for the confidence interval
When the sample size is large - the uncertainty about the value of the sample is very small
E(XY) - E(X)E(Y)
Use historical simulation approach but use the EWMA weighting system

12. Overall F - statistic
Summation((xi - mean)^k)/n
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Combine to form distribution with leptokurtosis (heavy tails)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

13. Exact significance level
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Use historical simulation approach but use the EWMA weighting system
Probability that the random variables take on certain values simultaneously
P - value

14. Unstable return distribution
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
Normal - Student's T - Chi - square - F distribution
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

15. Reliability
Application of mathematical statistics to economic data to lend empirical support to models
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Statement of the error or precision of an estimate

16. Variance of X - Y assuming dependence
Normal - Student's T - Chi - square - F distribution
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance(X) + Variance(Y) - 2*covariance(XY)
Based on an equation - P(A) = # of A/total outcomes

17. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
E(mean) = mean

18. Confidence interval (from t)
We accept a hypothesis that should have been rejected
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!)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Sample mean +/ - t*(stddev(s)/sqrt(n))

19. Adjusted 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
Variance reverts to a long run level
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

20. Sample correlation
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Rxy = Sxy/(Sx*Sy)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

21. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
SSR
Combine to form distribution with leptokurtosis (heavy tails)
Variance(X) + Variance(Y) - 2*covariance(XY)

22. Unbiased
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Mean of sampling distribution is the population mean
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

23. Variance of X+Y
Contains variables not explicit in model - Accounts for randomness
Variance(X) + Variance(Y) - 2*covariance(XY)
Var(X) + Var(Y)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

24. Square root rule
Special type of pooled data in which the cross sectional unit is surveyed over time
Var(X) + Var(Y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

25. Heteroskedastic
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Summation((xi - mean)^k)/n
If variance of the conditional distribution of u(i) is not constant
Special type of pooled data in which the cross sectional unit is surveyed over time

26. Binomial distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Concerned with a single random variable (ex. Roll of a die)
Low Frequency - High Severity events
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!)

27. Variance of aX + bY
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(x)
(a^2)(variance(x)) + (b^2)(variance(y))
Combine to form distribution with leptokurtosis (heavy tails)

28. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
If variance of the conditional distribution of u(i) is not constant
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Peaks over threshold - Collects dataset in excess of some threshold

29. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
If variance of the conditional distribution of u(i) is not constant
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

30. Chi - squared distribution
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
i = ln(Si/Si - 1)
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

31. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
When one regressor is a perfect linear function of the other regressors
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

32. BLUE
Sampling distribution of sample means tend to be normal
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Average return across assets on a given day

33. Importance sampling technique
Attempts to sample along more important paths
P(Z>t)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(y)/n = variance of sample Y

34. Confidence interval for sample mean
Application of mathematical statistics to economic data to lend empirical support to models
i = ln(Si/Si - 1)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

35. Time series data
Returns over time for an individual asset
Expected value of the sample mean is the population mean
Random walk (usually acceptable) - Constant volatility (unlikely)
Only requires two parameters = mean and variance

36. F distribution
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Application of mathematical statistics to economic data to lend empirical support to models
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

37. Critical z values
Sampling distribution of sample means tend to be normal
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
95% = 1.65 99% = 2.33 For one - tailed tests

38. What does the OLS minimize?
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
SSR
Variance(x)
Nonlinearity

39. Non - parametric vs parametric calculation of VaR
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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

40. Marginal unconditional probability function
Does not depend on a prior event or information
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Choose parameters that maximize the likelihood of what observations occurring

41. Econometrics
Variance = (1/m) summation(u<n - i>^2)
We accept a hypothesis that should have been rejected
Variance(x)
Application of mathematical statistics to economic data to lend empirical support to models

42. Variance of aX
Normal - Student's T - Chi - square - F distribution
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
(a^2)(variance(x)

43. Covariance
Least absolute deviations estimator - used when extreme outliers are not uncommon
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Z = (Y - meany)/(stddev(y)/sqrt(n))
E(XY) - E(X)E(Y)

44. Unconditional vs conditional distributions
Model dependent - Options with the same underlying assets may trade at different volatilities
Easy to manipulate
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Sample mean will near the population mean as the sample size increases

45. GARCH
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)
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

46. Direction of OVB
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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. Statistical (or empirical) model
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.
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Yi = B0 + B1Xi + ui

48. 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
Normal - Student's T - Chi - square - F distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

49. Cholesky factorization (decomposition)
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

50. Normal distribution
Nonlinearity
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.