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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. Sample mean
Concerned with a single random variable (ex. Roll of a die)
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
Expected value of the sample mean is the population mean
Var(X) + Var(Y)

2. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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

3. POT
Peaks over threshold - Collects dataset in excess of some threshold
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Summation((xi - mean)^k)/n

4. Variance of X+Y
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Only requires two parameters = mean and variance
Var(X) + Var(Y)

5. Heteroskedastic
Regression can be non - linear in variables but must be linear in parameters
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
If variance of the conditional distribution of u(i) is not constant
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

6. Mean reversion
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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)
Attempts to sample along more important paths

7. Joint probability functions
If variance of the conditional distribution of u(i) is not constant
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Z = (Y - meany)/(stddev(y)/sqrt(n))
Probability that the random variables take on certain values simultaneously

8. Antithetic variable technique
Based on a dataset
Normal - Student's T - Chi - square - F distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

9. GPD
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance = (1/m) summation(u<n - i>^2)
i = ln(Si/Si - 1)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

10. Kurtosis
Variance reverts to a long run level
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Regression can be non - linear in variables but must be linear in parameters

11. Exact significance 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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
P - value
Independently and Identically Distributed

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

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

14. SER
Use historical simulation approach but use the EWMA weighting system
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

15. Economical(elegant)
Peaks over threshold - Collects dataset in excess of some threshold
If variance of the conditional distribution of u(i) is not constant
Only requires two parameters = mean and variance
P(X=x - Y=y) = P(X=x) * P(Y=y)

16. Pooled data
Summation((xi - mean)^k)/n
Normal - Student's T - Chi - square - F 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

17. Panel data (longitudinal or micropanel)
Sampling distribution of sample means tend to be normal
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

18. Poisson Distribution
Choose parameters that maximize the likelihood of what observations occurring
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
When one regressor is a perfect linear function of the other regressors
Variance reverts to a long run level

19. Marginal unconditional probability function
Does not depend on a prior event or information
Concerned with a single random variable (ex. Roll of a die)
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

20. Importance sampling technique
Application of mathematical statistics to economic data to lend empirical support to models
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
Based on an equation - P(A) = # of A/total outcomes
Attempts to sample along more important paths

21. Two ways to calculate historical volatility
Variance = (1/m) summation(u<n - i>^2)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

22. GARCH
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market 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!)
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)

23. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance reverts to a long run level

24. Variance(discrete)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Least absolute deviations estimator - used when extreme outliers are not uncommon
If variance of the conditional distribution of u(i) is not constant

25. Confidence ellipse
More than one random variable
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Confidence set for two coefficients - two dimensional analog for the confidence interval
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

26. Cross - sectional
Has heavy tails
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Average return across assets on a given day

27. Stochastic error term
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Contains variables not explicit in model - Accounts for randomness
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

28. Regime - switching volatility model
Z = (Y - meany)/(stddev(y)/sqrt(n))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Probability that the random variables take on certain values simultaneously
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

29. Variance of X+b
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance(x)
Independently and Identically Distributed
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)

30. Unstable return distribution
Does not depend on a prior event or information
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Regression can be non - linear in variables but must be linear in parameters

31. Mean reversion in variance
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Variance reverts to a long run level
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

32. Block maxima
Among all unbiased estimators - estimator with the smallest variance is efficient
If variance of the conditional distribution of u(i) is not constant
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

33. Sample correlation
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Rxy = Sxy/(Sx*Sy)
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
We accept a hypothesis that should have been rejected

34. Gamma distribution
Has heavy tails
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

35. Two assumptions of square root rule
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Random walk (usually acceptable) - Constant volatility (unlikely)
Population denominator = n - Sample denominator = n - 1
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

36. ESS
Use historical simulation approach but use the EWMA weighting system
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Application of mathematical statistics to economic data to lend empirical support to models

37. Chi - squared distribution
Mean of sampling distribution is the population mean
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
Random walk (usually acceptable) - Constant volatility (unlikely)
Distribution with only two possible outcomes

38. Type I error
Only requires two parameters = mean and variance
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
We reject a hypothesis that is actually true
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

39. Bootstrap method
E(XY) - E(X)E(Y)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

40. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
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)
Low Frequency - High Severity events
Concerned with a single random variable (ex. Roll of a die)

41. Result of combination of two normal with same means
Sampling distribution of sample means tend to be normal
Combine to form distribution with leptokurtosis (heavy tails)
Population denominator = n - Sample denominator = n - 1
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

42. Statistical (or empirical) model
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of 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)
Yi = B0 + B1Xi + ui
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

43. Variance of aX
(a^2)(variance(x)
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

44. Confidence interval for sample mean
Variance = (1/m) summation(u<n - i>^2)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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(Z>t)

45. Overall F - statistic
Special type of pooled data in which the cross sectional unit is surveyed over time
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Random walk (usually acceptable) - Constant volatility (unlikely)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

46. Skewness
We reject a hypothesis that is actually true
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!)
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
SSR

47. Single variable (univariate) probability
Only requires two parameters = mean and variance
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Concerned with a single random variable (ex. Roll of a die)
95% = 1.65 99% = 2.33 For one - tailed tests

48. Test for unbiasedness
Based on a dataset
E(mean) = mean
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

49. Binomial distribution
i = ln(Si/Si - 1)
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
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!)

50. Unbiased
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)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Special type of pooled data in which the cross sectional unit is surveyed over time
Mean of sampling distribution is the population mean