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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Two drawbacks of moving average series
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Application of mathematical statistics to economic data to lend empirical support to models
Average return across assets on a given day

2. What does the OLS minimize?
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
SSR
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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!)

3. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Independently and Identically Distributed
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

4. Maximum likelihood method
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
Combine to form distribution with leptokurtosis (heavy tails)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

5. Variance of X+b
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance(x)

6. Control variates technique
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.
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Rxy = Sxy/(Sx*Sy)

7. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Regression can be non - linear in variables but must be linear in parameters
E(XY) - E(X)E(Y)
Sample mean +/ - t*(stddev(s)/sqrt(n))

8. GEV
Returns over time for an individual asset
Variance(x)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

9. Type II Error
Sampling distribution of sample means tend to be normal
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)
We accept a hypothesis that should have been rejected
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

10. Confidence ellipse
E(mean) = mean
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Confidence set for two coefficients - two dimensional analog for the confidence interval

11. Regime - switching volatility model
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
Variance(y)/n = variance of sample Y
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

12. BLUE
i = ln(Si/Si - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
(a^2)(variance(x)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

13. Chi - squared distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Returns over time for an individual asset
Mean = np - Variance = npq - Std dev = sqrt(npq)

14. Beta distribution
Based on a dataset
E(mean) = mean
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
i = ln(Si/Si - 1)

15. Efficiency
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Among all unbiased estimators - estimator with the smallest variance is efficient
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

16. Reliability
Statement of the error or precision of an estimate
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

17. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Attempts to sample along more important paths

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

19. Central Limit Theorem(CLT)
Distribution with only two possible outcomes
Peaks over threshold - Collects dataset in excess of some threshold
Sampling distribution of sample means tend to be normal
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

20. LAD
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Least absolute deviations estimator - used when extreme outliers are not uncommon

21. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Probability that the random variables take on certain values simultaneously
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

22. Cross - sectional
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Average return across assets on a given day

23. Pooled data
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

24. Bootstrap method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

25. Bernouli Distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Distribution with only two possible outcomes
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

26. Homoskedastic
Does not depend on a prior event or information
Application of mathematical statistics to economic data to lend empirical support to models
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

27. Block maxima
Nonlinearity
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

28. Variance of X+Y assuming dependence
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance(x) + Variance(Y) + 2*covariance(XY)
E(XY) - E(X)E(Y)
Variance reverts to a long run level

29. Variance of sampling distribution of means when n<N
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Use historical simulation approach but use the EWMA weighting system
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance(X) + Variance(Y) - 2*covariance(XY)

30. F distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
Concerned with a single random variable (ex. Roll of a die)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

31. WLS
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
(a^2)(variance(x)
Variance(x) + Variance(Y) + 2*covariance(XY)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

32. Critical z values
Yi = B0 + B1Xi + ui
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Variance(x) + Variance(Y) + 2*covariance(XY)
95% = 1.65 99% = 2.33 For one - tailed tests

33. Time series data
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Sampling distribution of sample means tend to be normal
When the sample size is large - the uncertainty about the value of the sample is very small
Returns over time for an individual asset

34. Potential reasons for fat tails in return distributions
Transformed to a unit variable - Mean = 0 Variance = 1
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

35. Antithetic variable technique
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

36. Difference between population and sample variance
Regression can be non - linear in variables but must be linear in parameters
Population denominator = n - Sample denominator = n - 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
E(mean) = mean

37. GARCH
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)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
(a^2)(variance(x)
Confidence level

38. Empirical frequency
Use historical simulation approach but use the EWMA weighting system
We accept a hypothesis that should have been rejected
E(mean) = mean
Based on a dataset

39. Standard error for Monte Carlo replications
Variance = (1/m) summation(u<n - i>^2)
Combine to form distribution with leptokurtosis (heavy tails)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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. Test for statistical independence
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance(x) + Variance(Y) + 2*covariance(XY)

41. Heteroskedastic
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 a combination of assets (combination of time series and cross - sectional data)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
If variance of the conditional distribution of u(i) is not constant

42. Unstable return distribution
For n>30 - sample mean is approximately normal
Variance(X) + Variance(Y) - 2*covariance(XY)
Model dependent - Options with the same underlying assets may trade at different volatilities
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

43. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
For n>30 - sample mean is approximately normal

44. Skewness
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

45. Square root rule
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

46. POT
Combine to form distribution with leptokurtosis (heavy tails)
Peaks over threshold - Collects dataset in excess of some threshold
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
P(X=x - Y=y) = P(X=x) * P(Y=y)

47. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Normal - Student's T - Chi - square - F distribution
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

48. Key properties of linear regression
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
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
Regression can be non - linear in variables but must be linear in parameters

49. Gamma distribution
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Rxy = Sxy/(Sx*Sy)
Peaks over threshold - Collects dataset in excess of some threshold
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

50. Hybrid method for conditional volatility
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
Use historical simulation approach but use the EWMA weighting system
Returns over time for an individual asset
Model dependent - Options with the same underlying assets may trade at different volatilities