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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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Answer 50 questions in 15 minutes.
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1. Discrete representation of the GBM
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!)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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(x)

2. Heteroskedastic
Confidence set for two coefficients - two dimensional analog for the confidence interval
If variance of the conditional distribution of u(i) is not constant
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Application of mathematical statistics to economic data to lend empirical support to models

3. SER
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(x)
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
Low Frequency - High Severity events

4. LFHS
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Low Frequency - High Severity events
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

5. WLS
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

6. Simulating for VaR
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
When the sample size is large - the uncertainty about the value of the sample is very small
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

7. Kurtosis
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
Statement of the error or precision of an estimate
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

8. Exponential distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Confidence level
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

9. Logistic distribution
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Z = (Y - meany)/(stddev(y)/sqrt(n))
Has heavy tails

10. BLUE
Variance(x)
Application of mathematical statistics to economic data to lend empirical support to models
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Does not depend on a prior event or information

11. Maximum likelihood method
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
More than one random variable
Choose parameters that maximize the likelihood of what observations occurring
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

12. Mean(expected value)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Based on a dataset
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

13. Variance of sampling distribution of means when n<N
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(sample y) = (variance(y)/n)*(N - n/N - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

14. Central Limit Theorem
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
For n>30 - sample mean is approximately normal
E(XY) - E(X)E(Y)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

15. Unbiased
Combine to form distribution with leptokurtosis (heavy tails)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Mean of sampling distribution is the population mean
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

16. SER
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance reverts to a long run level
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
If variance of the conditional distribution of u(i) is not constant

17. Empirical frequency
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Based on a dataset

18. Pooled data
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Sample mean will near the population mean as the sample size increases
Variance reverts to a long run level
Returns over time for a combination of assets (combination of time series and cross - sectional data)

19. Variance of X+Y
(a^2)(variance(x)) + (b^2)(variance(y))
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Var(X) + Var(Y)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

20. Bernouli Distribution
Distribution with only two possible outcomes
Regression can be non - linear in variables but must be linear in parameters
Statement of the error or precision of an estimate
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

21. Variance(discrete)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
E(XY) - E(X)E(Y)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

22. Inverse transform method
When one regressor is a perfect linear function of the other regressors
Price/return tends to run towards a long - run level
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

23. Persistence
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
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
Among all unbiased estimators - estimator with the smallest variance is efficient

24. Variance of sample mean
SSR
Variance(y)/n = variance of sample Y
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

25. Simplified standard (un - weighted) variance
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
SSR
Expected value of the sample mean is the population mean
Variance = (1/m) summation(u<n - i>^2)

26. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Confidence set for two coefficients - two dimensional analog for the confidence interval
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

27. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance = (1/m) summation(u<n - i>^2)
Z = (Y - meany)/(stddev(y)/sqrt(n))

28. Weibul distribution
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Returns over time for a combination of assets (combination of time series and cross - sectional data)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

29. Beta distribution
Yi = B0 + B1Xi + ui
P(X=x - Y=y) = P(X=x) * P(Y=y)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

30. Continuous random variable
E(XY) - E(X)E(Y)
Use historical simulation approach but use the EWMA weighting system
Random walk (usually acceptable) - Constant volatility (unlikely)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

31. Implied standard deviation for options
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Average return across assets on a given day
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

32. POT
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Peaks over threshold - Collects dataset in excess of some threshold
Transformed to a unit variable - Mean = 0 Variance = 1
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

33. Standard error for Monte Carlo replications
Yi = B0 + B1Xi + ui
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

34. Unconditional vs conditional distributions
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

35. Single variable (univariate) probability
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
Concerned with a single random variable (ex. Roll of a die)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
When one regressor is a perfect linear function of the other regressors

36. GARCH
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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)
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)

37. Test for unbiasedness
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
E(mean) = mean
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
Sample mean will near the population mean as the sample size increases

38. Regime - switching volatility model
If variance of the conditional distribution of u(i) is not constant
Variance(X) + Variance(Y) - 2*covariance(XY)
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

39. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
P(Z>t)
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
(a^2)(variance(x)) + (b^2)(variance(y))

40. Gamma 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
Independently and Identically Distributed
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

41. 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
95% = 1.65 99% = 2.33 For one - tailed tests
E(XY) - E(X)E(Y)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

42. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Sampling distribution of sample means tend to be normal
Low Frequency - High Severity events
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

43. Extending the HS approach for computing value of a portfolio

44. Priori (classical) probability
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
Based on an equation - P(A) = # of A/total outcomes
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Transformed to a unit variable - Mean = 0 Variance = 1

45. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Average return across assets on a given day
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

46. Law of Large Numbers
Based on a dataset
Z = (Y - meany)/(stddev(y)/sqrt(n))
Sample mean will near the population mean as the sample size increases
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

47. Simulation models
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
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
P(Z>t)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

48. Continuously compounded return equation
Contains variables not explicit in model - Accounts for randomness
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
i = ln(Si/Si - 1)

49. Potential reasons for fat tails in return distributions
E(XY) - E(X)E(Y)
Has heavy tails
Average return across assets on a given day
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

50. Marginal unconditional probability function
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Does not depend on a prior event or information
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)