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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. Deterministic Simulation
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
We reject a hypothesis that is actually true
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

3. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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)

4. Variance of aX
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
(a^2)(variance(x)
Distribution with only two possible outcomes
Variance = (1/m) summation(u<n - i>^2)

5. Test for unbiasedness
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Mean of sampling distribution is the population mean
E(mean) = mean
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

6. Simplified standard (un - weighted) variance
Choose parameters that maximize the likelihood of what observations occurring
Normal - Student's T - Chi - square - F distribution
Price/return tends to run towards a long - run level
Variance = (1/m) summation(u<n - i>^2)

7. Standard normal distribution
Variance(x) + Variance(Y) + 2*covariance(XY)
Transformed to a unit variable - Mean = 0 Variance = 1
E(XY) - E(X)E(Y)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

8. Continuous representation of the GBM
Variance(x)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
P(Z>t)
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)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Mean of sampling distribution is the population mean
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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

10. Mean(expected value)
Variance = (1/m) summation(u<n - i>^2)
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)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

11. Key properties of linear regression
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
95% = 1.65 99% = 2.33 For one - tailed tests
Regression can be non - linear in variables but must be linear in parameters
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

12. Consistent
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Only requires two parameters = mean and variance
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
When the sample size is large - the uncertainty about the value of the sample is very small

13. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
We accept a hypothesis that should have been rejected
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
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

14. Poisson Distribution
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

15. Difference between population and sample variance
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
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
Population denominator = n - Sample denominator = n - 1

16. Variance - covariance approach for VaR of a portfolio
Distribution with only two possible outcomes
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

17. Non - parametric vs parametric calculation of VaR
Transformed to a unit variable - Mean = 0 Variance = 1
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(x) + Variance(Y) + 2*covariance(XY)

18. Simulation models
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Statement of the error or precision of an estimate
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
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. Skewness
Expected value of the sample mean is the population mean
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
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

20. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(x) + Variance(Y) + 2*covariance(XY)
95% = 1.65 99% = 2.33 For one - tailed tests
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

21. Empirical frequency
Special type of pooled data in which the cross sectional unit is surveyed over time
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean = np - Variance = npq - Std dev = sqrt(npq)
Based on a dataset

22. Type II Error
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
We accept a hypothesis that should have been rejected
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

23. Homoskedastic only F - stat
Sample mean will near the population mean as the sample size increases
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
We reject a hypothesis that is actually true

24. T 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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

25. Implications of homoscedasticity
Variance(y)/n = variance of sample Y
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
If variance of the conditional distribution of u(i) is not constant

26. Gamma distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
Application of mathematical statistics to economic data to lend empirical support to models
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

27. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

28. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(x) + Variance(Y) + 2*covariance(XY)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

29. What does the OLS minimize?
Has heavy tails
SSR
Confidence level
Independently and Identically Distributed

30. Stochastic error term
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
Contains variables not explicit in model - Accounts for randomness
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

31. i.i.d.
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Independently and Identically Distributed
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

32. Binomial distribution equations for mean variance and std dev
Concerned with a single random variable (ex. Roll of a die)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Returns over time for an individual asset
Mean = np - Variance = npq - Std dev = sqrt(npq)

33. Historical std dev
Regression can be non - linear in variables but must be linear in parameters
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(x) + Variance(Y) + 2*covariance(XY)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

34. Two assumptions of square root rule
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Random walk (usually acceptable) - Constant volatility (unlikely)

35. Persistence
Nonlinearity
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
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
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

36. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance(y)/n = variance of sample Y

37. P - value
Special type of pooled data in which the cross sectional unit is surveyed over time
More than one random variable
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
P(Z>t)

38. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
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)
Regression can be non - linear in variables but must be linear in parameters
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

39. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

40. Two requirements of OVB
Rxy = Sxy/(Sx*Sy)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

41. Cross - sectional
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Average return across assets on a given day
Transformed to a unit variable - Mean = 0 Variance = 1
Among all unbiased estimators - estimator with the smallest variance is efficient

42. Sample mean
Does not depend on a prior event or information
Yi = B0 + B1Xi + ui
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
Expected value of the sample mean is the population mean

43. Variance of aX + bY
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
(a^2)(variance(x)) + (b^2)(variance(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)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

44. Significance =1
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
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
Confidence level
Random walk (usually acceptable) - Constant volatility (unlikely)

45. Bernouli Distribution
Combine to form distribution with leptokurtosis (heavy tails)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Choose parameters that maximize the likelihood of what observations occurring
Distribution with only two possible outcomes

46. Mean reversion
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
SSR
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

47. Inverse transform method
Variance(y)/n = variance of sample Y
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

48. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Based on an equation - P(A) = # of A/total outcomes
Z = (Y - meany)/(stddev(y)/sqrt(n))
Sampling distribution of sample means tend to be normal

49. F distribution
Low Frequency - High Severity events
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)
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
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

50. Extreme Value Theory
Distribution with only two possible outcomes
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Special type of pooled data in which the cross sectional unit is surveyed over time