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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Four sampling distributions

2. Mean reversion
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
SSR
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

3. Mean reversion in asset dynamics
Sample mean +/ - t*(stddev(s)/sqrt(n))
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Price/return tends to run towards a long - run level

4. Joint probability functions
(a^2)(variance(x)) + (b^2)(variance(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
i = ln(Si/Si - 1)
Probability that the random variables take on certain values simultaneously

5. Central Limit Theorem
Expected value of the sample mean is the population mean
Use historical simulation approach but use the EWMA weighting system
Variance(x)
For n>30 - sample mean is approximately normal

6. Block maxima
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!)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Statement of the error or precision of an estimate
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

7. Panel data (longitudinal or micropanel)
Confidence level
Special type of pooled data in which the cross sectional unit is surveyed over time
Does not depend on a prior event or information
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

8. Significance =1
Application of mathematical statistics to economic data to lend empirical support to models
Confidence level
Price/return tends to run towards a long - run level
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)

9. Variance(discrete)
P - value
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

10. Non - parametric vs parametric calculation of VaR
Attempts to sample along more important paths
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(y)/n = variance of sample Y
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

11. Reliability
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Statement of the error or precision of an estimate
95% = 1.65 99% = 2.33 For one - tailed tests

12. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
i = ln(Si/Si - 1)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
P(Z>t)

13. Variance of weighted scheme
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Mean of sampling distribution is the population mean

14. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Peaks over threshold - Collects dataset in excess of some threshold
Contains variables not explicit in model - Accounts for randomness

15. Cross - sectional
Does not depend on a prior event or information
Concerned with a single random variable (ex. Roll of a die)
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)

16. Exponential distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
P - value
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

17. Implied standard deviation for options
P(Z>t)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Confidence level
Rxy = Sxy/(Sx*Sy)

18. Chi - squared distribution
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

19. Type II Error
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
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
We accept a hypothesis that should have been rejected
Nonlinearity

20. Poisson Distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Choose parameters that maximize the likelihood of what observations occurring
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

21. Stochastic error term
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Contains variables not explicit in model - Accounts for randomness
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

22. Variance of aX + bY
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
(a^2)(variance(x)) + (b^2)(variance(y))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

23. Statistical (or empirical) model
If variance of the conditional distribution of u(i) is not constant
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Yi = B0 + B1Xi + ui

24. Deterministic Simulation
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

25. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance(y)/n = variance of sample Y

26. i.i.d.
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Confidence level
Independently and Identically Distributed
Application of mathematical statistics to economic data to lend empirical support to models

27. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Var(X) + Var(Y)
Among all unbiased estimators - estimator with the smallest variance is efficient
Concerned with a single random variable (ex. Roll of a die)

28. Inverse transform method
Expected value of the sample mean is the population mean
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!)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
When one regressor is a perfect linear function of the other regressors

29. Mean reversion in variance
Sample mean will near the population mean as the sample size increases
Does not depend on a prior event or information
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance reverts to a long run level

30. Discrete random variable
Contains variables not explicit in model - Accounts for randomness
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Confidence set for two coefficients - two dimensional analog for the confidence interval

31. 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)
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
Only requires two parameters = mean and variance
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

32. Historical std dev
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)
Variance = (1/m) summation(u<n - i>^2)
Special type of pooled data in which the cross sectional unit is surveyed over time
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

33. Marginal unconditional probability function
Easy to manipulate
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Does not depend on a prior event or information
If variance of the conditional distribution of u(i) is not constant

34. Sample correlation
Var(X) + Var(Y)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Rxy = Sxy/(Sx*Sy)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

35. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

36. Potential reasons for fat tails in return distributions
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
E(mean) = mean
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

37. Result of combination of two normal with same means
Application of mathematical statistics to economic data to lend empirical support to models
Combine to form distribution with leptokurtosis (heavy tails)
SSR
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

38. What does the OLS minimize?
Based on a dataset
Mean = np - Variance = npq - Std dev = sqrt(npq)
SSR
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

39. Test for statistical independence
95% = 1.65 99% = 2.33 For one - tailed tests
Mean = np - Variance = npq - Std dev = sqrt(npq)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

40. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

41. Perfect multicollinearity
Variance reverts to a long run level
When one regressor is a perfect linear function of the other regressors
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Among all unbiased estimators - estimator with the smallest variance is efficient

42. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Random walk (usually acceptable) - Constant volatility (unlikely)
Easy to manipulate
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

43. Sample mean
Expected value of the sample mean is the population mean
Random walk (usually acceptable) - Constant volatility (unlikely)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

44. Persistence
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
Based on a dataset
Independently and Identically Distributed
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

45. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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
Low Frequency - High Severity events
Peaks over threshold - Collects dataset in excess of some threshold

46. Expected future variance rate (t periods forward)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Only requires two parameters = mean and variance

47. Two ways to calculate historical volatility
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

48. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Variance(X) + Variance(Y) - 2*covariance(XY)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

49. Economical(elegant)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Only requires two parameters = mean and variance
We reject a hypothesis that is actually true

50. 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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Returns over time for an individual asset
Sampling distribution of sample means tend to be normal