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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. Kurtosis
Confidence level
Probability that the random variables take on certain values simultaneously
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Sampling distribution of sample means tend to be normal

2. Confidence interval for sample mean
Attempts to sample along more important paths
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Probability that the random variables take on certain values simultaneously

3. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Special type of pooled data in which the cross sectional unit is surveyed over time
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

4. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Confidence level
Statement of the error or precision of an estimate
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

5. Persistence
Variance(x)
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
Mean of sampling distribution is the population mean
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

6. LFHS
Only requires two parameters = mean and variance
Low Frequency - High Severity events
We reject a hypothesis that is actually true
Choose parameters that maximize the likelihood of what observations occurring

7. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
E(mean) = mean
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Variance(x)

8. Adjusted R^2
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Contains variables not explicit in model - Accounts for randomness
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

9. Reliability
If variance of the conditional distribution of u(i) is not constant
Statement of the error or precision of an estimate
Confidence set for two coefficients - two dimensional analog for the confidence interval
SSR

10. Gamma distribution
Expected value of the sample mean is the population mean
Sample mean will near the population mean as the sample size increases
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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

11. GPD
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Probability that the random variables take on certain values simultaneously
Variance(X) + Variance(Y) - 2*covariance(XY)

12. Non - parametric vs parametric calculation of VaR
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Z = (Y - meany)/(stddev(y)/sqrt(n))

13. Continuous random variable
Attempts to sample along more important paths
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
P(X=x - Y=y) = P(X=x) * P(Y=y)

14. Simplified standard (un - weighted) variance
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
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
Variance = (1/m) summation(u<n - i>^2)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

15. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
95% = 1.65 99% = 2.33 For one - tailed tests
E(mean) = mean
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

16. Stochastic error term
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Contains variables not explicit in model - Accounts for randomness
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

17. Continuously compounded return equation
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Contains variables not explicit in model - Accounts for randomness
i = ln(Si/Si - 1)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

18. Two drawbacks of moving average series
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Only requires two parameters = mean and variance
Random walk (usually acceptable) - Constant volatility (unlikely)

19. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Yi = B0 + B1Xi + ui

20. Central Limit Theorem(CLT)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sampling distribution of sample means tend to be normal
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

21. Panel data (longitudinal or micropanel)
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
Combine to form distribution with leptokurtosis (heavy tails)
Special type of pooled data in which the cross sectional unit is surveyed over time
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

22. Importance sampling technique
Var(X) + Var(Y)
Has heavy tails
P(X=x - Y=y) = P(X=x) * P(Y=y)
Attempts to sample along more important paths

23. Key properties of linear regression
Variance(x) + Variance(Y) + 2*covariance(XY)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 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
Regression can be non - linear in variables but must be linear in parameters

24. Exponential distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

25. Deterministic Simulation
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
Sampling distribution of sample means tend to be normal
Special type of pooled data in which the cross sectional unit is surveyed over time
Returns over time for an individual asset

26. Cross - sectional
(a^2)(variance(x)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Rxy = Sxy/(Sx*Sy)
Average return across assets on a given day

27. Implications of homoscedasticity
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sampling distribution of sample means tend to be normal
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

28. Statistical (or empirical) model
Nonlinearity
Special type of pooled data in which the cross sectional unit is surveyed over time
Yi = B0 + B1Xi + ui
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

29. Block maxima
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

30. Sample variance
SSR
Confidence level
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
When one regressor is a perfect linear function of the other regressors

31. Potential reasons for fat tails in return distributions
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Probability that the random variables take on certain values simultaneously

32. Type II Error
Distribution with only two possible outcomes
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
We accept a hypothesis that should have been rejected
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

33. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
95% = 1.65 99% = 2.33 For one - tailed tests
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

34. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Concerned with a single random variable (ex. Roll of a die)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Sample mean will near the population mean as the sample size increases

35. Central Limit Theorem
Based on an equation - P(A) = # of A/total outcomes
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance reverts to a long run level
For n>30 - sample mean is approximately normal

36. P - value
P(Z>t)
Variance(y)/n = variance of sample Y
SSR
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

37. Variance(discrete)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Yi = B0 + B1Xi + ui
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

38. Discrete representation of the GBM
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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))

39. Bernouli Distribution
Distribution with only two possible outcomes
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(x)
P(X=x - Y=y) = P(X=x) * P(Y=y)

40. BLUE
Summation((xi - mean)^k)/n
Price/return tends to run towards a long - run level
Returns over time for an individual asset
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

41. Law of Large Numbers
More than one random variable
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Sample mean will near the population mean as the sample size increases

42. Variance of aX + bY
Easy to manipulate
(a^2)(variance(x)) + (b^2)(variance(y))
For n>30 - sample mean is approximately normal
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

43. POT
Peaks over threshold - Collects dataset in excess of some threshold
Price/return tends to run towards a long - run level
95% = 1.65 99% = 2.33 For one - tailed tests
Var(X) + Var(Y)

44. Sample mean
Random walk (usually acceptable) - Constant volatility (unlikely)
Expected value of the sample mean is the population mean
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

45. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
We reject a hypothesis that is actually true
Sample mean +/ - t*(stddev(s)/sqrt(n))
Mean of sampling distribution is the population mean

46. Pooled data
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Nonlinearity
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Mean = np - Variance = npq - Std dev = sqrt(npq)

47. Poisson Distribution
Var(X) + Var(Y)
Does not depend on a prior event or information
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)

48. i.i.d.
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
If variance of the conditional distribution of u(i) is not constant
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

49. Econometrics
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Application of mathematical statistics to economic data to lend empirical support to models
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

50. Difference between population and sample variance
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
Population denominator = n - Sample denominator = n - 1
Distribution with only two possible outcomes
Parameters (mean - volatility - etc) vary over time due to variability in market conditions