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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. Variance of X+b
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance(x)
Combine to form distribution with leptokurtosis (heavy tails)
Variance reverts to a long run level

2. Conditional probability functions
Price/return tends to run towards a long - run level
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

3. Heteroskedastic
Rxy = Sxy/(Sx*Sy)
Statement of the error or precision of an estimate
More than one random variable
If variance of the conditional distribution of u(i) is not constant

4. Variance of weighted scheme
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!)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Based on an equation - P(A) = # of A/total outcomes
(a^2)(variance(x)

5. Block maxima
Rxy = Sxy/(Sx*Sy)
Variance(X) + Variance(Y) - 2*covariance(XY)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Easy to manipulate

6. Efficiency
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Among all unbiased estimators - estimator with the smallest variance is efficient
Sampling distribution of sample means tend to be normal

7. Variance of aX
(a^2)(variance(x)
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
Random walk (usually acceptable) - Constant volatility (unlikely)
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

8. Shortcomings of implied volatility
Variance(X) + Variance(Y) - 2*covariance(XY)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Does not depend on a prior event or information
Model dependent - Options with the same underlying assets may trade at different volatilities

9. Sample variance
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Only requires two parameters = mean and variance
Independently and Identically Distributed
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

10. BLUE
E(XY) - E(X)E(Y)
Only requires two parameters = mean and variance
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

11. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
More than one random variable
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
When one regressor is a perfect linear function of the other regressors

12. WLS
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Based on a dataset
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

13. ESS
Variance(x) + Variance(Y) + 2*covariance(XY)
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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

14. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Probability that the random variables take on certain values simultaneously
Does not depend on a prior event or information

15. Chi - squared distribution
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
Transformed to a unit variable - Mean = 0 Variance = 1
Expected value of the sample mean is the population mean
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

16. Covariance calculations using weight sums (lambda)
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
i = ln(Si/Si - 1)

17. Simulating for VaR
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
When the sample size is large - the uncertainty about the value of the sample is very small
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

18. Mean reversion in asset dynamics
i = ln(Si/Si - 1)
Based on a dataset
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

19. Cross - sectional
Average return across assets on a given day
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Contains variables not explicit in model - Accounts for randomness

20. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

21. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Summation((xi - mean)^k)/n
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
If variance of the conditional distribution of u(i) is not constant

22. Continuous representation of the GBM
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Does not depend on a prior event or information
Summation((xi - mean)^k)/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)

23. Potential reasons for fat tails in return distributions
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Confidence level
Does not depend on a prior event or information
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

24. Adjusted R^2
Special type of pooled data in which the cross sectional unit is surveyed over time
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
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

25. Logistic distribution
Has heavy tails
If variance of the conditional distribution of u(i) is not constant
When one regressor is a perfect linear function of the other regressors
Based on an equation - P(A) = # of A/total outcomes

26. Maximum likelihood method
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Confidence set for two coefficients - two dimensional analog for the confidence interval
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Choose parameters that maximize the likelihood of what observations occurring

27. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Concerned with a single random variable (ex. Roll of a die)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

28. Central Limit Theorem
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
For n>30 - sample mean is approximately normal
Choose parameters that maximize the likelihood of what observations occurring
Price/return tends to run towards a long - run level

29. Test for statistical independence
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
P(X=x - Y=y) = P(X=x) * P(Y=y)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

30. LFHS
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Low Frequency - High Severity events
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!)
Transformed to a unit variable - Mean = 0 Variance = 1

31. Test for unbiasedness
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
When the sample size is large - the uncertainty about the value of the sample is very small
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
E(mean) = mean

32. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Yi = B0 + B1Xi + ui
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!)

33. Mean reversion in variance
Use historical simulation approach but use the EWMA weighting system
More than one random variable
Variance reverts to a long run level
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

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

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35. K - th moment
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Summation((xi - mean)^k)/n
Returns over time for a combination of assets (combination of time series and cross - sectional data)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

36. i.i.d.
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
Price/return tends to run towards a long - run level
Independently and Identically Distributed

37. Marginal unconditional probability function
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Does not depend on a prior event or information
Based on an equation - P(A) = # of A/total outcomes

38. Lognormal
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
E(mean) = mean
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

39. Type I error
SSR
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Model dependent - Options with the same underlying assets may trade at different volatilities
We reject a hypothesis that is actually true

40. Extreme Value Theory
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

41. Economical(elegant)
(a^2)(variance(x)) + (b^2)(variance(y))
Sample mean +/ - t*(stddev(s)/sqrt(n))
Only requires two parameters = mean and variance
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

42. GARCH
We reject a hypothesis that is actually true
Summation((xi - mean)^k)/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)
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)

43. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Easy to manipulate
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

44. SER
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Probability that the random variables take on certain values simultaneously

45. Consistent
Probability that the random variables take on certain values simultaneously
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
When the sample size is large - the uncertainty about the value of the sample is very small
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

46. Exact significance level
E(mean) = mean
P - value
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

47. Confidence interval (from t)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Sample mean +/ - t*(stddev(s)/sqrt(n))

48. LAD
Population denominator = n - Sample denominator = n - 1
Variance(X) + Variance(Y) - 2*covariance(XY)
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

49. Empirical frequency
Variance(y)/n = variance of sample Y
Confidence set for two coefficients - two dimensional analog for the confidence interval
Based on a dataset
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

50. Two assumptions of square root rule
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
E(XY) - E(X)E(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
Random walk (usually acceptable) - Constant volatility (unlikely)