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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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Match each statement with the correct term.
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1. Unstable return distribution
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Population denominator = n - Sample denominator = n - 1
Nonlinearity
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

2. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Yi = B0 + B1Xi + ui
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Returns over time for an individual asset

3. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Does not depend on a prior event or information

4. Simulating for VaR
Variance(y)/n = variance of sample Y
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

5. Lognormal
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Model dependent - Options with the same underlying assets may trade at different volatilities
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

6. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Model dependent - Options with the same underlying assets may trade at different volatilities

7. Limitations of R^2 (what an increase doesn't necessarily imply)

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8. Implied standard deviation for options
Population denominator = n - Sample denominator = n - 1
Sampling distribution of sample means tend to be normal
For n>30 - sample mean is approximately normal
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

9. Mean reversion in asset dynamics
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Random walk (usually acceptable) - Constant volatility (unlikely)
Price/return tends to run towards a long - run level
Choose parameters that maximize the likelihood of what observations occurring

10. Joint probability functions
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
Probability that the random variables take on certain values simultaneously
Based on a dataset
Contains variables not explicit in model - Accounts for randomness

11. Confidence interval (from t)
P(Z>t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

12. Inverse transform method
Contains variables not explicit in model - Accounts for randomness
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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) + Variance(Y) - 2*covariance(XY)

13. BLUE
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Attempts to sample along more important paths
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

14. Logistic distribution
Does not depend on a prior event or information
i = ln(Si/Si - 1)
Has heavy tails
(a^2)(variance(x)) + (b^2)(variance(y))

15. Continuous random variable
E(mean) = mean
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Regression can be non - linear in variables but must be linear in parameters
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

16. Standard error for Monte Carlo replications
Probability that the random variables take on certain values simultaneously
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Mean of sampling distribution is the population mean
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

17. Unbiased
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Mean of sampling distribution is the population mean
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

18. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E(XY) - E(X)E(Y)
Application of mathematical statistics to economic data to lend empirical support to models
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

19. P - value
Application of mathematical statistics to economic data to lend empirical support to 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
P(Z>t)
Based on a dataset

20. Heteroskedastic
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
If variance of the conditional distribution of u(i) is not constant
Based on a dataset
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

21. Discrete random variable
Variance = (1/m) summation(u<n - i>^2)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

22. Simulation models
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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
More than one random variable
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

23. WLS
Attempts to sample along more important paths
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Returns over time for an individual asset
(a^2)(variance(x)) + (b^2)(variance(y))

24. K - th moment
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(x) + Variance(Y) + 2*covariance(XY)
Summation((xi - mean)^k)/n
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

25. Two drawbacks of moving average series
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
(a^2)(variance(x)) + (b^2)(variance(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

26. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Peaks over threshold - Collects dataset in excess of some threshold
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

27. Central Limit Theorem
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
We accept a hypothesis that should have been rejected
Has heavy tails
For n>30 - sample mean is approximately normal

28. LFHS
Low Frequency - High Severity events
Application of mathematical statistics to economic data to lend empirical support to models
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
Price/return tends to run towards a long - run level

29. 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)
95% = 1.65 99% = 2.33 For one - tailed tests
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

30. Type II Error
We accept a hypothesis that should have been rejected
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Sample mean +/ - t*(stddev(s)/sqrt(n))

31. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Least absolute deviations estimator - used when extreme outliers are not uncommon

32. Difference between population and sample variance
(a^2)(variance(x)) + (b^2)(variance(y))
Population denominator = n - Sample denominator = n - 1
Variance(x) + Variance(Y) + 2*covariance(XY)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

33. Continuously compounded return equation
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
We reject a hypothesis that is actually true
Var(X) + Var(Y)
i = ln(Si/Si - 1)

34. R^2
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Statement of the error or precision of an estimate
Concerned with a single random variable (ex. Roll of a die)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

35. Variance of sampling distribution of means when n<N
SSR
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Summation((xi - mean)^k)/n

36. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Has heavy tails
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

37. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Transformed to a unit variable - Mean = 0 Variance = 1
Average return across assets on a given day
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

38. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(x)
Transformed to a unit variable - Mean = 0 Variance = 1
Special type of pooled data in which the cross sectional unit is surveyed over time

39. Control variates technique
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sampling distribution of sample means tend to be normal
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Sample mean will near the population mean as the sample size increases

40. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Rxy = Sxy/(Sx*Sy)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
We reject a hypothesis that is actually true

41. Variance of aX
(a^2)(variance(x)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Choose parameters that maximize the likelihood of what observations occurring
Sample mean will near the population mean as the sample size increases

42. Unconditional vs conditional distributions
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
SSR
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Based on a dataset

43. Binomial distribution equations for mean variance and std dev
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Price/return tends to run towards a long - run level
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean = np - Variance = npq - Std dev = sqrt(npq)

44. Poisson Distribution
Sample mean will near the population mean as the sample size increases
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Contains variables not explicit in model - Accounts for randomness
Expected value of the sample mean is the population mean

45. Sample mean
Based on a dataset
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Expected value of the sample mean is the population mean
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

46. Direction of OVB
Variance(x)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
If variance of the conditional distribution of u(i) is not constant
Normal - Student's T - Chi - square - F distribution

47. Importance sampling technique
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Attempts to sample along more important paths

48. Chi - squared distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Nonlinearity
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
Mean = np - Variance = npq - Std dev = sqrt(npq)

49. LAD
Variance(X) + Variance(Y) - 2*covariance(XY)
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
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

50. T 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)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"