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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. Extreme Value Theory
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!)
Random walk (usually acceptable) - Constant volatility (unlikely)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

2. Continuous representation of the GBM
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)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
When the sample size is large - the uncertainty about the value of the sample is very small
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

3. 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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
We accept a hypothesis that should have been rejected

4. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Low Frequency - High Severity events
Variance reverts to a long run level
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)

5. Least squares estimator(m)
For n>30 - sample mean is approximately normal
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

6. Simulating for VaR
Does not depend on a prior event or information
Sample mean will near the population mean as the sample size increases
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

7. Mean reversion
Sampling distribution of sample means tend to be normal
Sample mean will near the population mean as the sample size increases
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Random walk (usually acceptable) - Constant volatility (unlikely)

8. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Choose parameters that maximize the likelihood of what observations occurring
Variance(y)/n = variance of sample Y
Returns over time for an individual asset

9. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Confidence set for two coefficients - two dimensional analog for the confidence interval
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

10. Central Limit Theorem
Independently and Identically Distributed
Application of mathematical statistics to economic data to lend empirical support to models
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
For n>30 - sample mean is approximately normal

11. Expected future variance rate (t periods forward)
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(X=x - Y=y) = P(X=x) * P(Y=y)
Combine to form distribution with leptokurtosis (heavy tails)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

12. Confidence interval (from t)
Combine to form distribution with leptokurtosis (heavy tails)
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sample mean +/ - t*(stddev(s)/sqrt(n))

13. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Regression can be non - linear in variables but must be linear in parameters
Confidence level
Distribution with only two possible outcomes

14. Time series data
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Returns over time for an individual asset

15. Type II Error
SSR
Easy to manipulate
We accept a hypothesis that should have been rejected
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

16. Historical std dev
Price/return tends to run towards a long - run level
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
i = ln(Si/Si - 1)

17. Biggest (and only real) drawback of GARCH mode
We accept a hypothesis that should have been rejected
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Nonlinearity
Model dependent - Options with the same underlying assets may trade at different volatilities

18. Exponential distribution
Statement of the error or precision of an estimate
Independently and Identically Distributed
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

19. Cholesky factorization (decomposition)
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)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

20. Variance of aX
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
(a^2)(variance(x)

21. Bootstrap method
Population denominator = n - Sample denominator = n - 1
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!)
Returns over time for an individual asset
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

22. Variance of X+Y
i = ln(Si/Si - 1)
Mean of sampling distribution is the population mean
Var(X) + Var(Y)
P - value

23. Key properties of linear regression
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
More than one random variable
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Regression can be non - linear in variables but must be linear in parameters

24. Mean reversion in variance
Variance reverts to a long run level
Choose parameters that maximize the likelihood of what observations occurring
Variance(X) + Variance(Y) - 2*covariance(XY)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

25. Mean reversion in asset dynamics
P(X=x - Y=y) = P(X=x) * P(Y=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)
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
Price/return tends to run towards a long - run level

26. Marginal unconditional probability function
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Does not depend on a prior event or information
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Concerned with a single random variable (ex. Roll of a die)

27. Test for unbiasedness
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
E(mean) = mean

28. Mean(expected value)
We reject a hypothesis that is actually true
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Mean = np - Variance = npq - Std dev = sqrt(npq)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

29. Sample mean
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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

30. R^2
Has heavy tails
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
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

31. Chi - squared distribution
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Expected value of the sample mean is the population mean
[1/(n - 1)]*summation((Xi - X)(Yi - 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

32. Standard error for Monte Carlo replications
Variance(y)/n = variance of sample Y
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Average return across assets on a given day

33. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Confidence set for two coefficients - two dimensional analog for the confidence interval
Statement of the error or precision of an estimate
Variance(x) + Variance(Y) + 2*covariance(XY)

34. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Normal - Student's T - Chi - square - F distribution

35. Gamma distribution
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

36. Direction of OVB
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
Attempts to sample along more important paths
Confidence level

37. Sample correlation
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Rxy = Sxy/(Sx*Sy)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

38. Simplified standard (un - weighted) variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance = (1/m) summation(u<n - i>^2)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

39. Implied standard deviation for options
Sample mean +/ - t*(stddev(s)/sqrt(n))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

40. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of 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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance = (1/m) summation(u<n - i>^2)

41. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
E(XY) - E(X)E(Y)

42. Sample covariance
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
P(Z>t)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

43. Regime - switching volatility model
Use historical simulation approach but use the EWMA weighting system
Least absolute deviations estimator - used when extreme outliers are not uncommon
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

44. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Rxy = Sxy/(Sx*Sy)

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

46. Beta distribution
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Peaks over threshold - Collects dataset in excess of some threshold
Confidence set for two coefficients - two dimensional analog for the confidence interval
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

47. Antithetic variable technique
SSR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Z = (Y - meany)/(stddev(y)/sqrt(n))

48. Efficiency
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Model dependent - Options with the same underlying assets may trade at different volatilities
Concerned with a single random variable (ex. Roll of a die)

49. Shortcomings of implied volatility
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E(XY) - E(X)E(Y)
Model dependent - Options with the same underlying assets may trade at different volatilities

50. Simulation models
Only requires two parameters = mean and variance
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
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE