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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. Deterministic Simulation
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
Variance(x) + Variance(Y) + 2*covariance(XY)

2. Confidence interval (from t)
P - value
E(mean) = mean
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

3. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Special type of pooled data in which the cross sectional unit is surveyed over time
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
P - value

4. Difference between population and sample variance
E(XY) - E(X)E(Y)
Population denominator = n - Sample denominator = n - 1
Price/return tends to run towards a long - run level
Variance reverts to a long run level

5. Type II Error
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
We accept a hypothesis that should have been rejected
Summation((xi - mean)^k)/n
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

6. Type I error
Mean = np - Variance = npq - Std dev = sqrt(npq)
We reject a hypothesis that is actually true
Among all unbiased estimators - estimator with the smallest variance is efficient
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

7. Reliability
Statement of the error or precision of an estimate
Variance = (1/m) summation(u<n - i>^2)
Attempts to sample along more important paths
Transformed to a unit variable - Mean = 0 Variance = 1

8. Poisson Distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Based on a dataset
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

9. Conditional probability functions
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
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
(a^2)(variance(x)) + (b^2)(variance(y))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

10. Confidence ellipse
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
P - value
Contains variables not explicit in model - Accounts for randomness
Confidence set for two coefficients - two dimensional analog for the confidence interval

11. What does the OLS minimize?
Distribution with only two possible outcomes
SSR
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

12. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

13. Variance(discrete)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

14. Law of Large Numbers
For n>30 - sample mean is approximately normal
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Sample mean will near the population mean as the sample size increases

15. Tractable
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Easy to manipulate
Variance(x) + Variance(Y) + 2*covariance(XY)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

16. Shortcomings of implied volatility
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)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
If variance of the conditional distribution of u(i) is not constant

17. POT
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Peaks over threshold - Collects dataset in excess of some threshold
Rxy = Sxy/(Sx*Sy)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

18. Result of combination of two normal with same means
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Combine to form distribution with leptokurtosis (heavy tails)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Transformed to a unit variable - Mean = 0 Variance = 1

19. Mean reversion in variance
Variance reverts to a long run level
Variance(x)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Transformed to a unit variable - Mean = 0 Variance = 1

20. Variance of aX + bY
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
(a^2)(variance(x)) + (b^2)(variance(y))
Random walk (usually acceptable) - Constant volatility (unlikely)
Only requires two parameters = mean and variance

21. Lognormal
Regression can be non - linear in variables but must be linear in parameters
E(XY) - E(X)E(Y)
Average return across assets on a given day
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

22. Normal distribution
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)
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

23. Simulation models
Independently and Identically Distributed
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

24. Variance of X+Y
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(y)/n = variance of sample Y
Mean = np - Variance = npq - Std dev = sqrt(npq)
Var(X) + Var(Y)

25. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

26. F distribution
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
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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

27. Overall F - statistic
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E(mean) = mean

28. Block maxima
Low Frequency - High Severity events
When the sample size is large - the uncertainty about the value of the sample is very small
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Summation((xi - mean)^k)/n

29. Extreme Value Theory
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
We accept a hypothesis that should have been rejected
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

30. Variance of weighted scheme
Sample mean +/ - t*(stddev(s)/sqrt(n))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Has heavy tails
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

31. Test for unbiasedness
Contains variables not explicit in model - Accounts for randomness
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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)
E(mean) = mean

32. Test for statistical independence
Concerned with a single random variable (ex. Roll of a die)
Summation((xi - mean)^k)/n
Probability that the random variables take on certain values simultaneously
P(X=x - Y=y) = P(X=x) * P(Y=y)

33. Hazard rate of exponentially distributed random variable
Transformed to a unit variable - Mean = 0 Variance = 1
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Special type of pooled data in which the cross sectional unit is surveyed over time
Does not depend on a prior event or information

34. Poisson distribution equations for mean variance and std deviation
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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

35. Hybrid method for conditional volatility
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Use historical simulation approach but use the EWMA weighting system
When one regressor is a perfect linear function of the other regressors
P(X=x - Y=y) = P(X=x) * P(Y=y)

36. Implied standard deviation for options
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Confidence level
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Choose parameters that maximize the likelihood of what observations occurring

37. BLUE
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

38. Logistic distribution
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
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
We accept a hypothesis that should have been rejected
Has heavy tails

39. Significance =1
Confidence level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Application of mathematical statistics to economic data to lend empirical support to models
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

40. Sample correlation
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Sample mean +/ - t*(stddev(s)/sqrt(n))
Rxy = Sxy/(Sx*Sy)
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

41. Implications of homoscedasticity
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

42. K - th moment
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Expected value of the sample mean is the population mean
Z = (Y - meany)/(stddev(y)/sqrt(n))
Summation((xi - mean)^k)/n

43. Heteroskedastic
P - value
If variance of the conditional distribution of u(i) is not constant
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
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

44. Covariance calculations using weight sums (lambda)
Variance(x)
P(Z>t)
Application of mathematical statistics to economic data to lend empirical support to models
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

45. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Nonlinearity
Does not depend on a prior event or information
Has heavy tails

46. GPD
Variance(x) + Variance(Y) + 2*covariance(XY)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Independently and Identically Distributed
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

47. Variance of sampling distribution of means when n<N
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Least absolute deviations estimator - used when extreme outliers are not uncommon
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

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

49. Beta distribution
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

50. Weibul distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
We accept a hypothesis that should have been rejected
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha