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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Empirical frequency
Based on a dataset
i = ln(Si/Si - 1)
Attempts to sample along more important paths
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

2. Key properties of linear regression
Least absolute deviations estimator - used when extreme outliers are not uncommon
Regression can be non - linear in variables but must be linear in parameters
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

3. Continuously compounded return equation
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
i = ln(Si/Si - 1)
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
Population denominator = n - Sample denominator = n - 1

4. Variance of X+Y assuming dependence
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(x) + Variance(Y) + 2*covariance(XY)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

5. Overall F - statistic
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Statement of the error or precision of an estimate
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

6. Central Limit Theorem(CLT)
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
Sampling distribution of sample means tend to be normal
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

7. Covariance
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
E(XY) - E(X)E(Y)

8. Test for unbiasedness
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
E(mean) = mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

9. Gamma distribution
Based on an equation - P(A) = # of A/total outcomes
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
Var(X) + Var(Y)
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)

10. Poisson distribution equations for mean variance and std deviation
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

11. Pooled data
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Price/return tends to run towards a long - run level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Returns over time for a combination of assets (combination of time series and cross - sectional data)

12. Mean reversion in asset dynamics
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
More than one random variable
Price/return tends to run towards a long - run level
Regression can be non - linear in variables but must be linear in parameters

13. i.i.d.
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Independently and Identically Distributed
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

14. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

15. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Least absolute deviations estimator - used when extreme outliers are not uncommon

16. Logistic distribution
Has heavy tails
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

17. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
(a^2)(variance(x)

18. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Peaks over threshold - Collects dataset in excess of some threshold
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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

19. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
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(y)/n = variance of sample Y
Confidence level

20. Joint probability functions
Probability that the random variables take on certain values simultaneously
Model dependent - Options with the same underlying assets may trade at different volatilities
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

21. P - value
Probability that the random variables take on certain values simultaneously
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
P(Z>t)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

22. Type II Error
Confidence level
We accept a hypothesis that should have been rejected
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Among all unbiased estimators - estimator with the smallest variance is efficient

23. EWMA
Variance reverts to a long run level
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Variance(x) + Variance(Y) + 2*covariance(XY)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

24. Non - parametric vs parametric calculation of VaR
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Mean = np - Variance = npq - Std dev = sqrt(npq)

25. Homoskedastic
Returns over time for an individual asset
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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

26. Variance of aX
(a^2)(variance(x)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Regression can be non - linear in variables but must be linear in parameters

27. Implied standard deviation for options
Returns over time for an individual asset
When one regressor is a perfect linear function of the other regressors
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

28. Multivariate probability
More than one random variable
Yi = B0 + B1Xi + ui
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Easy to manipulate

29. Sample mean
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
We accept a hypothesis that should have been rejected
Expected value of the sample mean is the population mean

30. Hazard rate of exponentially distributed random variable
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Choose parameters that maximize the likelihood of what observations occurring
Low Frequency - High Severity events
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

31. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
95% = 1.65 99% = 2.33 For one - tailed tests
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
We reject a hypothesis that is actually true

32. Homoskedastic only F - stat
E(mean) = mean
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

33. What does the OLS minimize?
SSR
Does not depend on a prior event or information
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

34. Multivariate Density Estimation (MDE)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
i = ln(Si/Si - 1)
Variance(x)

35. Econometrics
Low Frequency - High Severity events
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)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Application of mathematical statistics to economic data to lend empirical support to models

36. R^2
Variance(X) + Variance(Y) - 2*covariance(XY)
Yi = B0 + B1Xi + ui
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

37. Extreme Value Theory
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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

38. Time series data
Returns over time for an individual asset
Model dependent - Options with the same underlying assets may trade at different volatilities
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

39. Confidence interval (from t)
Special type of pooled data in which the cross sectional unit is surveyed over time
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance = (1/m) summation(u<n - i>^2)
Sample mean +/ - t*(stddev(s)/sqrt(n))

40. Significance =1
Rxy = Sxy/(Sx*Sy)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Confidence level

41. Reliability
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Statement of the error or precision of an estimate
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

42. Two requirements of OVB
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
P(Z>t)

43. Control variates technique
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Expected value of the sample mean is the population mean
Among all unbiased estimators - estimator with the smallest variance is efficient

44. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
SSR
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
When the sample size is large - the uncertainty about the value of the sample is very small

45. Law of Large Numbers
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
Sample mean will near the population mean as the sample size increases

46. Covariance calculations using weight sums (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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

47. Direction of OVB
Variance(x) + Variance(Y) + 2*covariance(XY)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

48. Exact significance level
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 = (1/m) summation(u<n - i>^2)
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!)
P - value

49. Variance of X+Y
Var(X) + Var(Y)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

50. Confidence ellipse
Mean = np - Variance = npq - Std dev = sqrt(npq)
95% = 1.65 99% = 2.33 For one - tailed tests
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