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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Binomial distribution equations for mean variance and std dev
Regression can be non - linear in variables but must be linear in parameters
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

2. Cholesky factorization (decomposition)
Peaks over threshold - Collects dataset in excess of some threshold
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

3. Key properties of linear regression
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Regression can be non - linear in variables but must be linear in parameters
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
Application of mathematical statistics to economic data to lend empirical support to models

4. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Does not depend on a prior event or information
95% = 1.65 99% = 2.33 For one - tailed tests

5. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Contains variables not explicit in model - Accounts for randomness
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

6. i.i.d.
Independently and Identically Distributed
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance reverts to a long run level
Model dependent - Options with the same underlying assets may trade at different volatilities

7. What does the OLS minimize?
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
We accept a hypothesis that should have been rejected
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
SSR

8. F distribution
Nonlinearity
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
When the sample size is large - the uncertainty about the value of the sample is very small

9. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(X) + Variance(Y) - 2*covariance(XY)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

10. Shortcomings of implied volatility
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Model dependent - Options with the same underlying assets may trade at different volatilities
Based on a dataset
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

11. Lognormal
Use historical simulation approach but use the EWMA weighting system
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

12. Discrete random variable
Random walk (usually acceptable) - Constant volatility (unlikely)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
For n>30 - sample mean is approximately normal
Use historical simulation approach but use the EWMA weighting system

13. Confidence interval for sample mean
Mean of sampling distribution is the population mean
P(Z>t)
Based on an equation - P(A) = # of A/total outcomes
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

14. Homoskedastic
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Does not depend on a prior event or information
Average return across assets on a given day

15. Simulation models
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

16. Result of combination of two normal with same means
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Confidence level
Special type of pooled data in which the cross sectional unit is surveyed over time
Combine to form distribution with leptokurtosis (heavy tails)

17. Antithetic variable technique
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Use historical simulation approach but use the EWMA weighting system
When one regressor is a perfect linear function of the other regressors
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

18. Persistence
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Use historical simulation approach but use the EWMA weighting system
P(Z>t)
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

19. Multivariate Density Estimation (MDE)
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
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

20. Gamma distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
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
When one regressor is a perfect linear function of the other regressors

21. Econometrics
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Application of mathematical statistics to economic data to lend empirical support to models
Average return across assets on a given day
Z = (Y - meany)/(stddev(y)/sqrt(n))

22. Economical(elegant)
Only requires two parameters = mean and variance
Transformed to a unit variable - Mean = 0 Variance = 1
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
i = ln(Si/Si - 1)

23. Two drawbacks of moving average series
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
Price/return tends to run towards a long - run level

24. Potential reasons for fat tails in return distributions
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
When the sample size is large - the uncertainty about the value of the sample is very small
Rxy = Sxy/(Sx*Sy)

25. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Only requires two parameters = mean and variance
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

26. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Attempts to sample along more important paths
Variance(x) + Variance(Y) + 2*covariance(XY)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

27. Direction of OVB
Combine to form distribution with leptokurtosis (heavy tails)
E(XY) - E(X)E(Y)
We reject a hypothesis that is actually true
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

28. Unbiased
E(mean) = mean
Variance reverts to a long run level
Mean of sampling distribution is the population mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

29. Exact significance level
Choose parameters that maximize the likelihood of what observations occurring
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
P - value
i = ln(Si/Si - 1)

30. Type I error
We reject a hypothesis that is actually true
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
P - value

31. POT
Peaks over threshold - Collects dataset in excess of some threshold
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

32. Control variates technique
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Rxy = Sxy/(Sx*Sy)
Variance(x) + Variance(Y) + 2*covariance(XY)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

33. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Has heavy tails
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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)

34. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Attempts to sample along more important paths
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Only requires two parameters = mean and variance

35. Historical std dev
Distribution with only two possible outcomes
Normal - Student's T - Chi - square - F distribution
Variance = (1/m) summation(u<n - i>^2)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

36. Consistent
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)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
When the sample size is large - the uncertainty about the value of the sample is very small
Based on a dataset

37. Variance of sampling distribution of means when n<N
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Has heavy tails
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

38. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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

39. Variance of X+b
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!)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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(x)

40. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Normal - Student's T - Chi - square - F distribution
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

41. Empirical frequency
Based on a dataset
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance(x)

42. Variance - covariance approach for VaR of a portfolio
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
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

43. BLUE
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Low Frequency - High Severity events
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Easy to manipulate

44. Importance sampling technique
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Attempts to sample along more important paths
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)

45. Multivariate probability
P(Z>t)
More than one random variable
When one regressor is a perfect linear function of the other regressors
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!)

46. Unconditional vs conditional distributions
Rxy = Sxy/(Sx*Sy)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
If variance of the conditional distribution of u(i) is not constant
Application of mathematical statistics to economic data to lend empirical support to models

47. Efficiency
Combine to form distribution with leptokurtosis (heavy tails)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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)
Among all unbiased estimators - estimator with the smallest variance is efficient

48. Mean reversion in variance
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance reverts to a long run level
Regression can be non - linear in variables but must be linear in parameters
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

49. Four sampling distributions

50. LFHS
Regression can be non - linear in variables but must be linear in parameters
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Low Frequency - High Severity events
Contains variables not explicit in model - Accounts for randomness