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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Implied standard deviation for options
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Distribution with only two possible outcomes
Z = (Y - meany)/(stddev(y)/sqrt(n))

2. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Sampling distribution of sample means tend to be normal
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance(X) + Variance(Y) - 2*covariance(XY)

3. Expected future variance rate (t periods forward)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Transformed to a unit variable - Mean = 0 Variance = 1

4. Historical std dev
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

5. Variance of weighted scheme
Least absolute deviations estimator - used when extreme outliers are not uncommon
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

6. Kurtosis
Random walk (usually acceptable) - Constant volatility (unlikely)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

7. Covariance
Contains variables not explicit in model - Accounts for randomness
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Concerned with a single random variable (ex. Roll of a die)
E(XY) - E(X)E(Y)

8. Test for unbiasedness
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
E(mean) = mean
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

9. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Yi = B0 + B1Xi + ui
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

10. P - value
P(Z>t)
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
Price/return tends to run towards a long - run level
P - value

11. Deterministic Simulation
Probability that the random variables take on certain values simultaneously
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Choose parameters that maximize the likelihood of what observations occurring

12. Variance of aX
(a^2)(variance(x)
Model dependent - Options with the same underlying assets may trade at different volatilities
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

13. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Contains variables not explicit in model - Accounts for randomness
Among all unbiased estimators - estimator with the smallest variance is efficient
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

14. GEV
Attempts to sample along more important paths
P(Z>t)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E(mean) = mean

15. Poisson distribution equations for mean variance and std deviation
Use historical simulation approach but use the EWMA weighting system
Statement of the error or precision of an estimate
For n>30 - sample mean is approximately normal
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

16. Bernouli Distribution
Distribution with only two possible outcomes
Combine to form distribution with leptokurtosis (heavy tails)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

17. Importance sampling technique
Attempts to sample along more important paths
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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)

18. What does the OLS minimize?
SSR
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Peaks over threshold - Collects dataset in excess of some threshold
Z = (Y - meany)/(stddev(y)/sqrt(n))

19. Economical(elegant)
Only requires two parameters = mean and variance
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)
Independently and Identically Distributed

20. Priori (classical) probability
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Based on an equation - P(A) = # of A/total outcomes
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

21. Sample correlation
Rxy = Sxy/(Sx*Sy)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Mean of sampling distribution is the population mean

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

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23. Multivariate Density Estimation (MDE)
Nonlinearity
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Only requires two parameters = mean and variance

24. Two assumptions of square root rule
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Random walk (usually acceptable) - Constant volatility (unlikely)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

25. POT
Peaks over threshold - Collects dataset in excess of some threshold
We reject a hypothesis that is actually true
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance(X) + Variance(Y) - 2*covariance(XY)

26. Square root rule
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Attempts to sample along more important paths
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

27. Multivariate probability
Distribution with only two possible 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
More than one random variable
Does not depend on a prior event or information

28. Continuous random variable
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
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
Confidence level

29. 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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Probability that the random variables take on certain values simultaneously
Concerned with a single random variable (ex. Roll of a die)

30. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Var(X) + Var(Y)
Independently and Identically Distributed
Normal - Student's T - Chi - square - F distribution

31. Homoskedastic
Distribution with only two possible outcomes
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
Easy to manipulate
P(Z>t)

32. Variance of X - Y assuming dependence
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Model dependent - Options with the same underlying assets may trade at different volatilities
SSR
Variance(X) + Variance(Y) - 2*covariance(XY)

33. Variance of aX + bY
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Summation((xi - mean)^k)/n
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
(a^2)(variance(x)) + (b^2)(variance(y))

34. Variance of X+Y assuming dependence
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(x) + Variance(Y) + 2*covariance(XY)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
If variance of the conditional distribution of u(i) is not constant

35. Binomial distribution
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!)
Based on an equation - P(A) = # of A/total outcomes
Rxy = Sxy/(Sx*Sy)

36. Variance of X+Y
Mean = np - Variance = npq - Std dev = sqrt(npq)
Var(X) + Var(Y)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Rxy = Sxy/(Sx*Sy)

37. 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)
E(mean) = mean
95% = 1.65 99% = 2.33 For one - tailed tests
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

38. Significance =1
Confidence level
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Among all unbiased estimators - estimator with the smallest variance is efficient
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

39. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Choose parameters that maximize the likelihood of what observations occurring
Least absolute deviations estimator - used when extreme outliers are not uncommon
Probability that the random variables take on certain values simultaneously

40. Potential reasons for fat tails in return distributions
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Average return across assets on a given day

41. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

42. Econometrics
Yi = B0 + B1Xi + ui
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Application of mathematical statistics to economic data to lend empirical support to models

43. Critical z values
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
i = ln(Si/Si - 1)
95% = 1.65 99% = 2.33 For one - tailed tests
Rxy = Sxy/(Sx*Sy)

44. Shortcomings of implied volatility
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Model dependent - Options with the same underlying assets may trade at different volatilities

45. Cross - sectional
Sampling distribution of sample means tend to be normal
Average return across assets on a given day
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)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

46. Empirical frequency
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Based on a dataset
(a^2)(variance(x)) + (b^2)(variance(y))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

47. Homoskedastic only F - stat
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!)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(x) + Variance(Y) + 2*covariance(XY)
Concerned with a single random variable (ex. Roll of a die)

48. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Has heavy tails
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Probability that the random variables take on certain values simultaneously

49. Persistence
Regression can be non - linear in variables but must be linear in parameters
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Based on an equation - P(A) = # of A/total outcomes

50. Tractable
Easy to manipulate
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Peaks over threshold - Collects dataset in excess of some threshold
(a^2)(variance(x)