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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Population denominator = n - Sample denominator = n - 1
Price/return tends to run towards a long - run level

2. Square root rule
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Confidence level

3. Confidence interval (from t)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sample mean +/ - t*(stddev(s)/sqrt(n))
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

4. Implications of homoscedasticity
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Transformed to a unit variable - Mean = 0 Variance = 1

5. Mean reversion in asset dynamics
Concerned with a single random variable (ex. Roll of a die)
We accept a hypothesis that should have been rejected
Price/return tends to run towards a long - run level
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

6. Confidence ellipse
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Confidence set for two coefficients - two dimensional analog for the confidence interval
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

7. POT
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Peaks over threshold - Collects dataset in excess of some threshold
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

8. K - th moment
Probability that the random variables take on certain values simultaneously
Summation((xi - mean)^k)/n
When the sample size is large - the uncertainty about the value of the sample is very small
Concerned with a single random variable (ex. Roll of a die)

9. Deterministic Simulation
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
Mean of sampling distribution is the population mean
We accept a hypothesis that should have been rejected
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

10. Least squares estimator(m)
Easy to manipulate
Summation((xi - mean)^k)/n
(a^2)(variance(x)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

11. Panel data (longitudinal or micropanel)
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

12. Variance of X - Y assuming dependence
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Model dependent - Options with the same underlying assets may trade at different volatilities
Use historical simulation approach but use the EWMA weighting system
Variance(X) + Variance(Y) - 2*covariance(XY)

13. Single variable (univariate) probability
Mean of sampling distribution is the population mean
Probability that the random variables take on certain values simultaneously
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Concerned with a single random variable (ex. Roll of a die)

14. BLUE
Based on an equation - P(A) = # of A/total outcomes
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

15. Law of Large Numbers
(a^2)(variance(x)
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Sample mean will near the population mean as the sample size increases

16. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
SSR
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
Returns over time for an individual asset

17. Tractable
Distribution with only two possible outcomes
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Easy to manipulate
Transformed to a unit variable - Mean = 0 Variance = 1

18. Logistic distribution
Independently and Identically Distributed
Distribution with only two possible outcomes
Has heavy tails
E(mean) = mean

19. Binomial distribution
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!)
Transformed to a unit variable - Mean = 0 Variance = 1
(a^2)(variance(x)

20. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Transformed to a unit variable - Mean = 0 Variance = 1
Variance = (1/m) summation(u<n - i>^2)

21. Maximum likelihood method
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E(XY) - E(X)E(Y)
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!)
Choose parameters that maximize the likelihood of what observations occurring

22. Discrete random variable
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

23. Statistical (or empirical) model
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
Rxy = Sxy/(Sx*Sy)
Yi = B0 + B1Xi + ui
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

24. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Statement of the error or precision of an estimate
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

25. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Yi = B0 + B1Xi + ui
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

26. Variance(discrete)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

27. Sample mean
Expected value of the sample mean is the population mean
Attempts to sample along more important paths
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

28. Heteroskedastic
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
If variance of the conditional distribution of u(i) is not constant
Z = (Y - meany)/(stddev(y)/sqrt(n))
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

29. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
If variance of the conditional distribution of u(i) is not constant
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

30. Exponential distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

31. Variance of X+b
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
Variance(x)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Peaks over threshold - Collects dataset in excess of some threshold

32. Four sampling distributions

33. 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
More than one random variable
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Based on a dataset

34. Lognormal
P(X=x - Y=y) = P(X=x) * P(Y=y)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

35. GPD
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Yi = B0 + B1Xi + ui
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Regression can be non - linear in variables but must be linear in parameters

36. P - value
Variance(y)/n = variance of sample Y
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
P(Z>t)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

37. Econometrics
Has heavy tails
Application of mathematical statistics to economic data to lend empirical support to models
Average return across assets on a given day
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

38. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Z = (Y - meany)/(stddev(y)/sqrt(n))
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance reverts to a long run level

39. Central Limit Theorem
For n>30 - sample mean is approximately normal
Returns over time for an individual asset
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

40. WLS
Special type of pooled data in which the cross sectional unit is surveyed over time
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

41. Simulating for VaR
E(mean) = mean
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Based on a dataset
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

42. Sample 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)
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)
We reject a hypothesis that is actually true
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

43. Empirical frequency
95% = 1.65 99% = 2.33 For one - tailed tests
Based on a dataset
E(XY) - E(X)E(Y)
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

44. Shortcomings of implied volatility
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Distribution with only two possible outcomes
Model dependent - Options with the same underlying assets may trade at different volatilities
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

45. Non - parametric vs parametric calculation of VaR
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

46. Extreme Value Theory
(a^2)(variance(x)) + (b^2)(variance(y))
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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)
P - value

47. Sample correlation
Random walk (usually acceptable) - Constant volatility (unlikely)
Rxy = Sxy/(Sx*Sy)
Nonlinearity
Low Frequency - High Severity events

48. Importance sampling technique
Attempts to sample along more important paths
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Peaks over threshold - Collects dataset in excess of some threshold

49. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Does not depend on a prior event or information
Only requires two parameters = mean and variance
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

50. Adjusted R^2
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
(a^2)(variance(x)) + (b^2)(variance(y))
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)