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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. Homoskedastic only F - stat
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Use historical simulation approach but use the EWMA weighting system
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

2. Variance of aX
(a^2)(variance(x)
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

3. Marginal unconditional probability function
Contains variables not explicit in model - Accounts for randomness
Does not depend on a prior event or information
Var(X) + Var(Y)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

4. Potential reasons for fat tails in return distributions
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

5. Test for unbiasedness
E(mean) = mean
(a^2)(variance(x)) + (b^2)(variance(y))
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Returns over time for an individual asset

6. Bootstrap method
Variance(x) + Variance(Y) + 2*covariance(XY)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Concerned with a single random variable (ex. Roll of a die)
Only requires two parameters = mean and variance

7. Mean(expected value)
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

8. Statistical (or empirical) model
(a^2)(variance(x)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Statement of the error or precision of an estimate
Yi = B0 + B1Xi + ui

9. Simplified standard (un - weighted) variance
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Variance = (1/m) summation(u<n - i>^2)
P(X=x - Y=y) = P(X=x) * P(Y=y)

10. Bernouli Distribution
Distribution with only two possible outcomes
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Nonlinearity

11. Confidence interval for sample mean
E(mean) = mean
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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

12. Pooled data
P - value
Variance(y)/n = variance of sample Y
Returns over time for a combination of assets (combination of time series and cross - sectional data)
When one regressor is a perfect linear function of the other regressors

13. Standard error for Monte Carlo replications
Rxy = Sxy/(Sx*Sy)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

14. Priori (classical) probability
For n>30 - sample mean is approximately normal
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Based on an equation - P(A) = # of A/total outcomes
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

15. Binomial distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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!)
When the sample size is large - the uncertainty about the value of the sample is very small

16. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Price/return tends to run towards a long - run level
Nonlinearity
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

17. Sample correlation
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Rxy = Sxy/(Sx*Sy)
Average return across assets on a given day
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!)

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

19. Simulation 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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
Normal - Student's T - Chi - square - F distribution

20. Homoskedastic
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
We accept a hypothesis that should have been rejected
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 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

21. Poisson Distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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!)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

22. Standard error
Variance(y)/n = variance of sample Y
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Rxy = Sxy/(Sx*Sy)

23. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Regression can be non - linear in variables but must be linear in parameters
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

24. Variance of aX + bY
Variance(x) + Variance(Y) + 2*covariance(XY)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
(a^2)(variance(x)) + (b^2)(variance(y))
If variance of the conditional distribution of u(i) is not constant

25. Deterministic Simulation
(a^2)(variance(x)
Distribution with only two possible outcomes
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
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. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Among all unbiased estimators - estimator with the smallest variance is efficient

27. Binomial distribution equations for mean variance and std dev
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean = np - Variance = npq - Std dev = sqrt(npq)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Population denominator = n - Sample denominator = n - 1

28. Hazard rate of exponentially distributed random variable
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

29. Type I error
Mean of sampling distribution is the population mean
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
We reject a hypothesis that is actually true
Easy to manipulate

30. Discrete representation of the GBM
We reject a hypothesis that is actually true
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Confidence level

31. Gamma distribution
We reject a hypothesis that is actually true
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
Choose parameters that maximize the likelihood of what observations occurring
Variance(x)

32. SER
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance reverts to a long run level
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

33. Economical(elegant)
Only requires two parameters = mean and variance
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Combine to form distribution with leptokurtosis (heavy tails)
E(mean) = mean

34. Unstable return distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

35. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
(a^2)(variance(x)
Attempts to sample along more important paths

36. Result of combination of two normal with same means
E(XY) - E(X)E(Y)
Combine to form distribution with leptokurtosis (heavy tails)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Model dependent - Options with the same underlying assets may trade at different volatilities

37. Significance =1
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Confidence level
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
Price/return tends to run towards a long - run level

38. K - th moment
Summation((xi - mean)^k)/n
Concerned with a single random variable (ex. Roll of a die)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

39. Skewness
Independently and Identically Distributed
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
E(mean) = mean

40. SER
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
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Concerned with a single random variable (ex. Roll of a die)

41. Biggest (and only real) drawback of GARCH mode
95% = 1.65 99% = 2.33 For one - tailed tests
Attempts to sample along more important paths
Nonlinearity
Least absolute deviations estimator - used when extreme outliers are not uncommon

42. LAD
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Least absolute deviations estimator - used when extreme outliers are not uncommon

43. Variance of sampling distribution of means when n<N
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
For n>30 - sample mean is approximately normal
P(Z>t)

44. Test for statistical independence
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Peaks over threshold - Collects dataset in excess of some threshold
P(X=x - Y=y) = P(X=x) * P(Y=y)

45. Perfect multicollinearity
Variance = (1/m) summation(u<n - i>^2)
Independently and Identically Distributed
When one regressor is a perfect linear function of the other regressors
SSR

46. Overall F - statistic
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

47. Confidence ellipse
Normal - Student's T - Chi - square - F distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

48. Variance of X+Y
Variance = (1/m) summation(u<n - i>^2)
Var(X) + Var(Y)
If variance of the conditional distribution of u(i) is not constant
Price/return tends to run towards a long - run level

49. Covariance
E(XY) - E(X)E(Y)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Returns over time for an individual asset
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

50. Econometrics
Expected value of the sample mean is the population mean
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
Probability that the random variables take on certain values simultaneously
Application of mathematical statistics to economic data to lend empirical support to models