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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Multivariate Density Estimation (MDE)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

2. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Sample mean +/ - t*(stddev(s)/sqrt(n))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

3. Bernouli Distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Distribution with only two possible outcomes
(a^2)(variance(x)
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

4. Type I error
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
We reject a hypothesis that is actually true
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

5. Variance of sample mean
If variance of the conditional distribution of u(i) is not constant
P(Z>t)
Variance(y)/n = variance of sample Y
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

6. Type II Error
Confidence set for two coefficients - two dimensional analog for the confidence interval
We accept a hypothesis that should have been rejected
Use historical simulation approach but use the EWMA weighting system
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

7. Standard error
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Z = (Y - meany)/(stddev(y)/sqrt(n))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

8. Bootstrap method
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Special type of pooled data in which the cross sectional unit is surveyed over time
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

9. GPD
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Expected value of the sample mean is the population mean
For n>30 - sample mean is approximately normal
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

10. Empirical frequency
Based on a dataset
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(x)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

11. Cholesky factorization (decomposition)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

12. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Normal - Student's T - Chi - square - F distribution
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

13. Binomial distribution
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!)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
We accept a hypothesis that should have been rejected
P(X=x - Y=y) = P(X=x) * P(Y=y)

14. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Based on a dataset
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
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

15. Control variates technique
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
When the sample size is large - the uncertainty about the value of the sample is very small
SSR

16. Implications of homoscedasticity
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
Var(X) + Var(Y)
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
Does not depend on a prior event or information

17. Poisson Distribution
95% = 1.65 99% = 2.33 For one - tailed tests
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

18. Shortcomings of implied volatility
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
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
Summation((xi - mean)^k)/n

19. Sample correlation
Rxy = Sxy/(Sx*Sy)
Attempts to sample along more important paths
P - value
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

20. Exponential distribution
Var(X) + Var(Y)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Mean = np - Variance = npq - Std dev = sqrt(npq)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

21. Hybrid method for conditional volatility
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Peaks over threshold - Collects dataset in excess of some threshold
Use historical simulation approach but use the EWMA weighting system
SSR

22. Continuous representation of the GBM
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

23. Maximum likelihood method
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Choose parameters that maximize the likelihood of what observations occurring
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

24. Monte Carlo Simulations
Special type of pooled data in which the cross sectional unit is surveyed over time
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

25. Unconditional vs conditional distributions
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

26. Single variable (univariate) probability
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Z = (Y - meany)/(stddev(y)/sqrt(n))
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)

27. Variance of X - Y assuming dependence
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
i = ln(Si/Si - 1)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(X) + Variance(Y) - 2*covariance(XY)

28. Multivariate probability
(a^2)(variance(x)
More than one random variable
Variance reverts to a long run level
Special type of pooled data in which the cross sectional unit is surveyed over time

29. Extending the HS approach for computing value of a portfolio

30. Central Limit Theorem
For n>30 - sample mean is approximately normal
(a^2)(variance(x)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

31. Importance sampling technique
P - value
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Attempts to sample along more important paths
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

32. ESS
Variance(X) + Variance(Y) - 2*covariance(XY)
Normal - Student's T - Chi - square - F distribution
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

33. Continuous random variable
Variance(X) + Variance(Y) - 2*covariance(XY)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

34. Beta distribution
Among all unbiased estimators - estimator with the smallest variance is efficient
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Nonlinearity
Choose parameters that maximize the likelihood of what observations occurring

35. Homoskedastic
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Easy to manipulate
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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

36. Two drawbacks of moving average series
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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

37. Efficiency
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Among all unbiased estimators - estimator with the smallest variance is efficient

38. Discrete random variable
When the sample size is large - the uncertainty about the value of the sample is very small
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance(x) + Variance(Y) + 2*covariance(XY)
Model dependent - Options with the same underlying assets may trade at different volatilities

39. Gamma distribution
Variance(x) + Variance(Y) + 2*covariance(XY)
Easy to manipulate
Confidence 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

40. Confidence ellipse
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Confidence set for two coefficients - two dimensional analog for the confidence interval
Random walk (usually acceptable) - Constant volatility (unlikely)
If variance of the conditional distribution of u(i) is not constant

41. Marginal unconditional probability function
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Var(X) + Var(Y)
E(XY) - E(X)E(Y)
Does not depend on a prior event or information

42. Mean reversion in asset dynamics
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Sample mean will near the population mean as the sample size increases
Price/return tends to run towards a long - run level
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

43. Persistence
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
(a^2)(variance(x)) + (b^2)(variance(y))
E(XY) - E(X)E(Y)
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

44. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
E(XY) - E(X)E(Y)

45. Heteroskedastic
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
If variance of the conditional distribution of u(i) is not constant
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!)

46. What does the OLS minimize?
Confidence level
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
SSR
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

47. P - value
P(Z>t)
Contains variables not explicit in model - Accounts for randomness
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Combine to form distribution with leptokurtosis (heavy tails)

48. T distribution
Independently and Identically Distributed
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Normal - Student's T - Chi - square - F distribution

49. Extreme Value Theory
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

50. LFHS
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
P(X=x - Y=y) = P(X=x) * P(Y=y)
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)
Low Frequency - High Severity events