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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Continuous representation of the GBM
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Var(X) + Var(Y)
E(XY) - E(X)E(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)

2. ESS
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Independently and Identically Distributed
Low Frequency - High Severity events

3. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Mean of sampling distribution is the population mean

4. Single variable (univariate) probability
Var(X) + Var(Y)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Does not depend on a prior event or information
Concerned with a single random variable (ex. Roll of a die)

5. Variance - covariance approach for VaR of a portfolio
Normal - Student's T - Chi - square - F distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(XY) - E(X)E(Y)
E(mean) = mean

6. Mean(expected value)
Contains variables not explicit in model - Accounts for randomness
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Yi = B0 + B1Xi + ui

7. Statistical (or empirical) model
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
Based on a dataset
Yi = B0 + B1Xi + ui

8. Block maxima
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
Variance(X) + Variance(Y) - 2*covariance(XY)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

9. Extreme Value Theory
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Based on an equation - P(A) = # of A/total outcomes
Variance(y)/n = variance of sample Y
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

10. Inverse transform method
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
P(Z>t)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

11. WLS
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Confidence set for two coefficients - two dimensional analog for the confidence interval

12. Type II Error
Distribution with only two possible outcomes
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
We accept a hypothesis that should have been rejected
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

13. Variance of aX
(a^2)(variance(x)
If variance of the conditional distribution of u(i) is not constant
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
Random walk (usually acceptable) - Constant volatility (unlikely)

14. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Independently and Identically Distributed
SSR
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

15. Variance of X - Y assuming dependence
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
SSR
Confidence level
Variance(X) + Variance(Y) - 2*covariance(XY)

16. LFHS
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
When the sample size is large - the uncertainty about the value of the sample is very small
Low Frequency - High Severity events
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

17. T distribution
P(Z>t)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

18. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
95% = 1.65 99% = 2.33 For one - tailed tests
Low Frequency - High Severity events
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

19. SER
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
Has heavy tails
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

20. Biggest (and only real) drawback of GARCH mode
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 the sample size is large - the uncertainty about the value of the sample is very small
Nonlinearity
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

21. Multivariate probability
Variance(x) + Variance(Y) + 2*covariance(XY)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Probability that the random variables take on certain values simultaneously
More than one random variable

22. POT
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Peaks over threshold - Collects dataset in excess of some threshold
Mean = np - Variance = npq - Std dev = sqrt(npq)
Statement of the error or precision of an estimate

23. Critical z values
Probability that the random variables take on certain values simultaneously
Only requires two parameters = mean and variance
95% = 1.65 99% = 2.33 For one - tailed tests
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

24. Variance of sample mean
We reject a hypothesis that is actually true
Variance(y)/n = variance of sample Y
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

25. Standard normal distribution
More than one random variable
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
Mean = np - Variance = npq - Std dev = sqrt(npq)

26. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Random walk (usually acceptable) - Constant volatility (unlikely)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

27. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Summation((xi - mean)^k)/n

28. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance(x)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

29. Mean reversion in asset dynamics
Concerned with a single random variable (ex. Roll of a die)
Price/return tends to run towards a long - run level
Independently and Identically Distributed
Transformed to a unit variable - Mean = 0 Variance = 1

30. Least squares estimator(m)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Least absolute deviations estimator - used when extreme outliers are not uncommon

31. i.i.d.
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Independently and Identically Distributed

32. Significance =1
Confidence level
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
Independently and Identically Distributed
Rxy = Sxy/(Sx*Sy)

33. Shortcomings of implied volatility
Sample mean +/ - t*(stddev(s)/sqrt(n))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Distribution with only two possible outcomes
Model dependent - Options with the same underlying assets may trade at different volatilities

34. Time series data
E(mean) = mean
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Returns over time for an individual asset
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!)

35. F distribution
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Does not depend on a prior event or information

36. Simulating for VaR
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

37. Implications of homoscedasticity
More than one random variable
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
Independently and Identically Distributed
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

38. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Attempts to sample along more important paths
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

39. Variance of X+b
Variance(x)
Mean of sampling distribution is the population mean
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
Yi = B0 + B1Xi + ui

40. Monte Carlo Simulations
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Has heavy tails

41. Potential reasons for fat tails in return distributions
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Normal - Student's T - Chi - square - F distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

42. Square root rule
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

43. Variance of X+Y
P(Z>t)
Model dependent - Options with the same underlying assets may trade at different volatilities
Var(X) + Var(Y)
When the sample size is large - the uncertainty about the value of the sample is very small

44. P - value
P(Z>t)
When the sample size is large - the uncertainty about the value of the sample is very small
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

45. LAD
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(y)/n = variance of sample 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
Least absolute deviations estimator - used when extreme outliers are not uncommon

46. Logistic distribution
Has heavy tails
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

47. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
P - value
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

48. Variance of aX + bY
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Probability that the random variables take on certain values simultaneously
(a^2)(variance(x)) + (b^2)(variance(y))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

49. Difference between population and sample variance
Probability that the random variables take on certain values simultaneously
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Population denominator = n - Sample denominator = n - 1
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

50. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Independently and Identically Distributed
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Returns over time for a combination of assets (combination of time series and cross - sectional data)