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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Sampling distribution of sample means tend to be normal
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

2. EWMA
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by 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

3. i.i.d.
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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)
Independently and Identically Distributed
P - value

4. Empirical frequency
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Based on a dataset
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

5. Type II Error
Low Frequency - High Severity events
We accept a hypothesis that should have been rejected
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of 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. Sample variance
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
95% = 1.65 99% = 2.33 For one - tailed tests

7. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Probability that the random variables take on certain values simultaneously

8. Variance of X+Y
Var(X) + Var(Y)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Population denominator = n - Sample denominator = n - 1
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

9. Multivariate probability
Variance(X) + Variance(Y) - 2*covariance(XY)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
More than one random variable
Average return across assets on a given day

10. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Attempts to sample along more important paths
Among all unbiased estimators - estimator with the smallest variance is efficient
Expected value of the sample mean is the population mean

11. Least squares estimator(m)
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
Easy to manipulate
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

12. P - value
Low Frequency - High Severity events
P(Z>t)
(a^2)(variance(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)

13. Square root rule
We reject a hypothesis that is actually true
Only requires two parameters = mean and variance
P - value
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

14. Two ways to calculate historical volatility
Distribution with only two possible outcomes
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
If variance of the conditional distribution of u(i) is not constant

15. Type I error
Z = (Y - meany)/(stddev(y)/sqrt(n))
Least absolute deviations estimator - used when extreme outliers are not uncommon
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
We reject a hypothesis that is actually true

16. K - th moment
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
E(XY) - E(X)E(Y)
Summation((xi - mean)^k)/n
Variance = (1/m) summation(u<n - i>^2)

17. Normal distribution
Expected value of the sample mean is the population mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
SSR

18. Deterministic Simulation
If variance of the conditional distribution of u(i) is not constant
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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

19. Confidence ellipse
(a^2)(variance(x)) + (b^2)(variance(y))
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Distribution with only two possible outcomes

20. Pooled data
P(X=x - Y=y) = P(X=x) * P(Y=y)
P(Z>t)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
When one regressor is a perfect linear function of the other regressors

21. Conditional probability functions
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!)
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 reverts to a long run level
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

22. Cholesky factorization (decomposition)
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
Price/return tends to run towards a long - run level

23. Simulating for VaR
Variance(X) + Variance(Y) - 2*covariance(XY)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Variance = (1/m) summation(u<n - i>^2)
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

24. 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!)
Expected value of the sample mean is the population mean
Use historical simulation approach but use the EWMA weighting system
If variance of the conditional distribution of u(i) is not constant

25. Kurtosis
Regression can be non - linear in variables but must be linear in parameters
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

26. Biggest (and only real) drawback of GARCH mode
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Nonlinearity
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Model dependent - Options with the same underlying assets may trade at different volatilities

27. Confidence interval (from t)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance = (1/m) summation(u<n - i>^2)
Sample mean +/ - t*(stddev(s)/sqrt(n))

28. Panel data (longitudinal or micropanel)
Model dependent - Options with the same underlying assets may trade at different volatilities
Special type of pooled data in which the cross sectional unit is surveyed over time
E(mean) = mean
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

29. Heteroskedastic
We reject a hypothesis that is actually true
If variance of the conditional distribution of u(i) is not constant
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)

30. Skewness
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
Concerned with a single random variable (ex. Roll of a die)
We reject a hypothesis that is actually true
Based on an equation - P(A) = # of A/total outcomes

31. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Nonlinearity
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

32. Mean reversion
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Expected value of the sample mean is the population mean

33. Unstable return distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Does not depend on a prior event or information
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

34. Implications of homoscedasticity
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Rxy = Sxy/(Sx*Sy)
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)

35. Gamma distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(x)
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

36. 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)
When one regressor is a perfect linear function of the other regressors
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

37. Significance =1
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Confidence level

38. Continuous representation of the GBM
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)
Statement of the error or precision of an estimate
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Model dependent - Options with the same underlying assets may trade at different volatilities

39. Control variates technique
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
i = ln(Si/Si - 1)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Regression can be non - linear in variables but must be linear in parameters

40. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Population denominator = n - Sample denominator = n - 1
Variance(y)/n = variance of sample Y

41. Extreme Value Theory
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

42. Direction of OVB
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Z = (Y - meany)/(stddev(y)/sqrt(n))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

43. Critical z values
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

44. Result of combination of two normal with same means
Based on a dataset
Combine to form distribution with leptokurtosis (heavy tails)
Peaks over threshold - Collects dataset in excess of some threshold
Variance reverts to a long run level

45. Sample correlation
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Probability that the random variables take on certain values simultaneously
Rxy = Sxy/(Sx*Sy)

46. Covariance
E(XY) - E(X)E(Y)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Random walk (usually acceptable) - Constant volatility (unlikely)

47. Homoskedastic
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Among all unbiased estimators - estimator with the smallest variance is efficient
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

48. Test for statistical independence
P - value
Yi = B0 + B1Xi + ui
Population denominator = n - Sample denominator = n - 1
P(X=x - Y=y) = P(X=x) * P(Y=y)

49. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

50. Continuously compounded return equation
We accept a hypothesis that should have been rejected
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
i = ln(Si/Si - 1)
Returns over time for a combination of assets (combination of time series and cross - sectional data)