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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Multivariate probability
i = ln(Si/Si - 1)
More than one random variable
When the sample size is large - the uncertainty about the value of the sample is very small
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

2. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Summation((xi - mean)^k)/n

3. Inverse transform method
Based on a dataset
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
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

4. Adjusted R^2
Special type of pooled data in which the cross sectional unit is surveyed over time
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

5. Reliability
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(x) + Variance(Y) + 2*covariance(XY)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Statement of the error or precision of an estimate

6. Homoskedastic only F - stat
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
i = ln(Si/Si - 1)

7. Mean reversion in variance
Variance reverts to a long run level
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

8. GARCH
More than one random variable
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)
Does not depend on a prior event or information
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

9. Mean reversion
For n>30 - sample mean is approximately normal
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

10. Overall F - statistic
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

11. Least squares estimator(m)
Choose parameters that maximize the likelihood of what observations occurring
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Returns over time for an individual asset

12. Mean(expected value)
(a^2)(variance(x)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
P(Z>t)
SSR

13. Covariance
E(XY) - E(X)E(Y)
Average return across assets on a given day
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

14. Persistence
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)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

15. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance = (1/m) summation(u<n - i>^2)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

16. EWMA
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

17. Covariance calculations using weight sums (lambda)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Least absolute deviations estimator - used when extreme outliers are not uncommon
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Normal - Student's T - Chi - square - F distribution

18. Simulation models
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
Statement of the error or precision of an estimate
Average return across assets on a given day
Distribution with only two possible outcomes

19. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Population denominator = n - Sample denominator = n - 1
Random walk (usually acceptable) - Constant volatility (unlikely)
P(Z>t)

20. LFHS
Mean of sampling distribution is the population mean
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
For n>30 - sample mean is approximately normal
Low Frequency - High Severity events

21. Panel data (longitudinal or micropanel)
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
Probability that the random variables take on certain values simultaneously
Has heavy tails

22. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

23. WLS
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
Only requires two parameters = mean and variance
Independently and Identically Distributed
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

24. 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)
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
Average return across assets on a given day
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

25. K - th moment
Low Frequency - High Severity events
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Summation((xi - mean)^k)/n
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

26. Control variates technique
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
95% = 1.65 99% = 2.33 For one - tailed tests

27. Statistical (or empirical) model
Probability that the random variables take on certain values simultaneously
95% = 1.65 99% = 2.33 For one - tailed tests
Yi = B0 + B1Xi + ui
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

28. Antithetic variable technique
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Summation((xi - mean)^k)/n
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

29. Confidence interval (from t)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Does not depend on a prior event or information
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))

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

31. Two drawbacks of moving average series
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(x) + Variance(Y) + 2*covariance(XY)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
We reject a hypothesis that is actually true

32. Economical(elegant)
Only requires two parameters = mean and variance
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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 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

33. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Summation((xi - mean)^k)/n
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

34. Test for unbiasedness
E(mean) = mean
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Use historical simulation approach but use the EWMA weighting system
Mean of sampling distribution is the population mean

35. Standard normal distribution
Price/return tends to run towards a long - run level
Transformed to a unit variable - Mean = 0 Variance = 1
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

36. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
(a^2)(variance(x)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Least absolute deviations estimator - used when extreme outliers are not uncommon

37. P - value
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
P(Z>t)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

38. Beta distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance reverts to a long run level
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

39. Regime - switching volatility model
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)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

40. Variance of X - Y assuming dependence
Regression can be non - linear in variables but must be linear in parameters
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
Rxy = Sxy/(Sx*Sy)
Variance(X) + Variance(Y) - 2*covariance(XY)

41. Confidence interval for sample mean
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Nonlinearity
Does not depend on a prior event or information

42. SER
We accept a hypothesis that should have been rejected
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
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

43. Test for statistical independence
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Z = (Y - meany)/(stddev(y)/sqrt(n))
P(X=x - Y=y) = P(X=x) * P(Y=y)

44. Non - parametric vs parametric calculation of VaR
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Yi = B0 + B1Xi + ui
Returns over time for an individual asset
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

45. Confidence ellipse
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(X) + Variance(Y) - 2*covariance(XY)

46. ESS
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
When one regressor is a perfect linear function of the other regressors

47. Two ways to calculate historical volatility
Sampling distribution of sample means tend to be normal
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

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

49. Expected future variance rate (t periods forward)
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)
Only requires two parameters = mean and variance
Easy to manipulate
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

50. Continuous representation of the GBM
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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)
Nonlinearity