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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. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Among all unbiased estimators - estimator with the smallest variance is efficient
Normal - Student's T - Chi - square - F distribution
Population denominator = n - Sample denominator = n - 1

2. Logistic distribution
Rxy = Sxy/(Sx*Sy)
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
Has heavy tails
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

3. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Low Frequency - High Severity events
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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

4. Mean(expected value)
When the sample size is large - the uncertainty about the value of the sample is very small
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!)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

5. Stochastic error term
Sample mean will near the population mean as the sample size increases
Contains variables not explicit in model - Accounts for randomness
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
Var(X) + Var(Y)

6. Pooled data
Independently and Identically Distributed
Variance(x)
Combine to form distribution with leptokurtosis (heavy tails)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

7. Variance of X+b
Variance(x)
Statement of the error or precision of an estimate
Returns over time for an individual asset
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

8. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Price/return tends to run towards a long - run level
We reject a hypothesis that is actually true
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

9. Non - parametric vs parametric calculation of VaR
SSR
Contains variables not explicit in model - Accounts for randomness
E(mean) = mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

10. Critical z values
Model dependent - Options with the same underlying assets may trade at different volatilities
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
95% = 1.65 99% = 2.33 For one - tailed tests
Mean = np - Variance = npq - Std dev = sqrt(npq)

11. Standard normal distribution
Variance(x)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Transformed to a unit variable - Mean = 0 Variance = 1
Confidence set for two coefficients - two dimensional analog for the confidence interval

12. 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
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)
When the sample size is large - the uncertainty about the value of the sample is very small
Returns over time for a combination of assets (combination of time series and cross - sectional data)

13. Implications of homoscedasticity
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

14. Cross - sectional
(a^2)(variance(x)) + (b^2)(variance(y))
P - value
Average return across assets on a given day
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

15. Continuously compounded return equation
Variance = (1/m) summation(u<n - i>^2)
i = ln(Si/Si - 1)
Transformed to a unit variable - Mean = 0 Variance = 1
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

16. Tractable
Summation((xi - mean)^k)/n
Easy to manipulate
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

17. POT
Peaks over threshold - Collects dataset in excess of some threshold
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
P(Z>t)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

18. Empirical frequency
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Among all unbiased estimators - estimator with the smallest variance is efficient
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Based on a dataset

19. Homoskedastic only F - stat
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Only requires two parameters = mean and variance
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

20. Variance of aX
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
(a^2)(variance(x)
More than one random variable

21. Single variable (univariate) probability
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
We reject a hypothesis that is actually true
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

22. 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
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(x)

23. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

24. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Model dependent - Options with the same underlying assets may trade at different volatilities
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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. Least squares estimator(m)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

26. Sample correlation
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Rxy = Sxy/(Sx*Sy)
Based on a dataset
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

27. Unstable return distribution
Returns over time for an individual asset
Sample mean +/ - t*(stddev(s)/sqrt(n))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

28. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Based on a dataset
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

29. Variance of aX + bY
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
(a^2)(variance(x)) + (b^2)(variance(y))

30. SER
Population denominator = n - Sample denominator = n - 1
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Attempts to sample along more important paths

31. Econometrics
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
P(X=x - Y=y) = P(X=x) * P(Y=y)
Application of mathematical statistics to economic data to lend empirical support to models
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

32. Biggest (and only real) drawback of GARCH mode
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Nonlinearity
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Variance(X) + Variance(Y) - 2*covariance(XY)

33. Direction of OVB
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
Attempts to sample along more important paths
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

34. Sample covariance
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

35. Reliability
Statement of the error or precision of an estimate
Contains variables not explicit in model - Accounts for randomness
Model dependent - Options with the same underlying assets may trade at different volatilities
Price/return tends to run towards a long - run level

36. Time series data
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Returns over time for an individual asset
(a^2)(variance(x)) + (b^2)(variance(y))
SSR

37. Two assumptions of square root rule
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Independently and Identically Distributed
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Random walk (usually acceptable) - Constant volatility (unlikely)

38. GEV
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
If variance of the conditional distribution of u(i) is not constant
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

39. Discrete random variable
Summation((xi - mean)^k)/n
When the sample size is large - the uncertainty about the value of the sample is very small
We reject a hypothesis that is actually true
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

40. Implied standard deviation for options
We reject a hypothesis that is actually true
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

41. Monte Carlo Simulations
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

42. Normal distribution
Var(X) + Var(Y)
For n>30 - sample mean is approximately normal
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
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

43. Binomial distribution
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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!)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

44. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Based on an equation - P(A) = # of A/total outcomes
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

45. Block maxima
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Based on an equation - P(A) = # of A/total outcomes
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

46. Discrete representation of the GBM
Based on an equation - P(A) = # of A/total outcomes
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

47. Mean reversion in variance
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Variance reverts to a long run level

48. Continuous representation of the GBM
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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)
Sampling distribution of sample means tend to be normal
Confidence set for two coefficients - two dimensional analog for the confidence interval

49. Importance sampling technique
Price/return tends to run towards a long - run level
If variance of the conditional distribution of u(i) is not constant
Attempts to sample along more important paths
Easy to manipulate

50. Four sampling distributions