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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Time series data
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Returns over time for an individual asset
(a^2)(variance(x)) + (b^2)(variance(y))

2. LAD
Sampling distribution of sample means tend to be normal
Easy to manipulate
Least absolute deviations estimator - used when extreme outliers are not uncommon
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

3. 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
Has heavy tails
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)
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

4. Limitations of R^2 (what an increase doesn't necessarily imply)

5. Kurtosis
Expected value of the sample mean is the population mean
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Does not depend on a prior event or information
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

6. Variance(discrete)
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance(x) + Variance(Y) + 2*covariance(XY)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

7. 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
Use historical simulation approach but use the EWMA weighting system
SSR
Confidence level

8. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
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

9. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Low Frequency - High Severity events

10. Marginal unconditional probability function
Expected value of the sample mean is the population mean
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
Does not depend on a prior event or information
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

11. POT
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
Peaks over threshold - Collects dataset in excess of some threshold
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance reverts to a long run level

12. Square root rule
Rxy = Sxy/(Sx*Sy)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Transformed to a unit variable - Mean = 0 Variance = 1
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

13. Poisson distribution equations for mean variance and std deviation
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Low Frequency - High Severity events
Variance(y)/n = variance of sample Y
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

14. Homoskedastic
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Z = (Y - meany)/(stddev(y)/sqrt(n))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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

15. GEV
Regression can be non - linear in variables but must be linear in parameters
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
P(Z>t)
(a^2)(variance(x)) + (b^2)(variance(y))

16. ESS
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Price/return tends to run towards a long - run level

17. Variance - covariance approach for VaR of a portfolio
Transformed to a unit variable - Mean = 0 Variance = 1
Attempts to sample along more important paths
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Based on an equation - P(A) = # of A/total outcomes

18. Maximum likelihood method
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Regression can be non - linear in variables but must be linear in parameters
Choose parameters that maximize the likelihood of what observations occurring
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

19. Simulation models
Probability that the random variables take on certain values simultaneously
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Special type of pooled data in which the cross sectional unit is surveyed over time

20. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Concerned with a single random variable (ex. Roll of a die)
More than one random variable
Does not depend on a prior event or information

21. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

22. Beta distribution
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Average return across assets on a given day
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

23. P - value
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
P(Z>t)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

24. SER
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)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(y)/n = variance of sample Y

25. Unconditional vs conditional distributions
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Returns over time for an individual asset
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
When one regressor is a perfect linear function of the other regressors

26. Simulating for VaR
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

27. Critical z values
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
95% = 1.65 99% = 2.33 For one - tailed tests
P - value

28. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Peaks over threshold - Collects dataset in excess of some threshold
E(mean) = mean
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

29. Biggest (and only real) drawback of GARCH mode
Model dependent - Options with the same underlying assets may trade at different volatilities
Nonlinearity
Variance = (1/m) summation(u<n - i>^2)
P(Z>t)

30. Multivariate Density Estimation (MDE)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Based on an equation - P(A) = # of A/total outcomes
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

31. Chi - squared distribution
Regression can be non - linear in variables but must be linear in parameters
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 = (1/m) summation(u<n - i>^2)
Sampling distribution of sample means tend to be normal

32. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Yi = B0 + B1Xi + ui
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

33. Historical std dev
E(XY) - E(X)E(Y)
Rxy = Sxy/(Sx*Sy)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

34. Continuous representation of the GBM
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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)
Variance = (1/m) summation(u<n - i>^2)
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

35. Regime - switching volatility model
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Choose parameters that maximize the likelihood of what observations occurring
Contains variables not explicit in model - Accounts for randomness

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

37. Unstable return distribution
Does not depend on a prior event or information
Nonlinearity
(a^2)(variance(x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

38. Block maxima
Variance = (1/m) summation(u<n - i>^2)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Easy to manipulate

39. T distribution
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
P(Z>t)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

40. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Variance(x) + Variance(Y) + 2*covariance(XY)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

41. Economical(elegant)
Only requires two parameters = mean and variance
Expected value of the sample mean is the population mean
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Contains variables not explicit in model - Accounts for randomness

42. 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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Distribution with only two possible outcomes

43. Monte Carlo Simulations
Application of mathematical statistics to economic data to lend empirical support to models
Sampling distribution of sample means tend to be normal
Variance(y)/n = variance of sample Y
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

44. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Peaks over threshold - Collects dataset in excess of some threshold
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

45. Variance of X - Y assuming dependence
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

46. Binomial distribution equations for mean variance and std dev
Special type of pooled data in which the cross sectional unit is surveyed over time
Mean = np - Variance = npq - Std dev = sqrt(npq)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

47. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
We accept a hypothesis that should have been rejected
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

48. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
95% = 1.65 99% = 2.33 For one - tailed tests
When the sample size is large - the uncertainty about the value of the sample is very small

49. 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)
Nonlinearity
(a^2)(variance(x)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

50. Continuously compounded return equation
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
i = ln(Si/Si - 1)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Random walk (usually acceptable) - Constant volatility (unlikely)