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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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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. Variance of sample mean
Variance(y)/n = variance of sample Y
Z = (Y - meany)/(stddev(y)/sqrt(n))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance reverts to a long run level

2. Control variates technique
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
More than one random variable
Variance(X) + Variance(Y) - 2*covariance(XY)

3. Persistence
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
i = ln(Si/Si - 1)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Statement of the error or precision of an estimate

4. Standard error
More than one random variable
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance(y)/n = variance of sample Y
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

5. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Random walk (usually acceptable) - Constant volatility (unlikely)
Special type of pooled data in which the cross sectional unit is surveyed over time
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

6. Variance of aX + bY
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
(a^2)(variance(x)) + (b^2)(variance(y))

7. Hazard rate of exponentially distributed random variable
Rxy = Sxy/(Sx*Sy)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sampling distribution of sample means tend to be normal
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

8. Variance of sampling distribution of means when n<N
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!)
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

9. Reliability
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Statement of the error or precision of an estimate
Average return across assets on a given day

10. EWMA
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E(XY) - E(X)E(Y)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
More than one random variable

11. Historical std dev
Low Frequency - High Severity events
Has heavy tails
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

12. Sample correlation
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Rxy = Sxy/(Sx*Sy)
Average return across assets on a given day
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

13. What does the OLS minimize?
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E(XY) - E(X)E(Y)
SSR
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

14. Tractable
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(mean) = mean
Confidence level
Easy to manipulate

15. 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
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
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

16. Mean(expected value)
Average return across assets on a given day
Has heavy tails
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

17. Simulation models
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Among all unbiased estimators - estimator with the smallest variance is efficient
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

18. Variance - covariance approach for VaR of a portfolio
Mean = np - Variance = npq - Std dev = sqrt(npq)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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

19. Antithetic variable technique
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
Average return across assets on a given day
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

20. Empirical frequency
Choose parameters that maximize the likelihood of what observations occurring
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Based on a dataset

21. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
We accept a hypothesis that should have been rejected
Average return across assets on a given day
Use historical simulation approach but use the EWMA weighting system

22. ESS
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

23. Variance(discrete)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Confidence level
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

24. Variance of X - Y assuming dependence
Random walk (usually acceptable) - Constant volatility (unlikely)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(X) + Variance(Y) - 2*covariance(XY)
For n>30 - sample mean is approximately normal

25. Continuous representation of the GBM
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
Contains variables not explicit in model - Accounts for randomness
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)
Application of mathematical statistics to economic data to lend empirical support to models

26. GPD
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Rxy = Sxy/(Sx*Sy)
Only requires two parameters = mean and variance

27. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
P - value
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)

28. Variance of X+Y
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
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
Var(X) + Var(Y)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

29. Heteroskedastic
Based on an equation - P(A) = # of A/total outcomes
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
When the sample size is large - the uncertainty about the value of the sample is very small
If variance of the conditional distribution of u(i) is not constant

30. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
When the sample size is large - the uncertainty about the value of the sample is very small
Has heavy tails
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

31. Time series data
Does not depend on a prior event or information
Expected value of the sample mean is the population mean
Based on a dataset
Returns over time for an individual asset

32. SER
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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!)

33. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Model dependent - Options with the same underlying assets may trade at different volatilities
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

34. Cross - sectional
E(mean) = mean
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Average return across assets on a given day
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)

35. P - value
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
P(Z>t)
Average return across assets on a given day
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

36. Biggest (and only real) drawback of GARCH mode
(a^2)(variance(x)) + (b^2)(variance(y))
Contains variables not explicit in model - Accounts for randomness
Average return across assets on a given day
Nonlinearity

37. Variance of weighted scheme
Based on an equation - P(A) = # of A/total outcomes
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

38. Sample mean
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
When one regressor is a perfect linear function of the other regressors
Expected value of the sample mean is the population mean
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

39. Overall F - statistic
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
P(X=x - Y=y) = P(X=x) * P(Y=y)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

40. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Expected value of the sample mean is the population mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

41. Block maxima
Nonlinearity
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 = (1/m) summation(u<n - i>^2)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

42. i.i.d.
Independently and Identically Distributed
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Statement of the error or precision of an estimate
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

43. Confidence interval for sample mean
Expected value of the sample mean is the population mean
Random walk (usually acceptable) - Constant volatility (unlikely)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

44. Mean reversion in asset dynamics
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
Price/return tends to run towards a long - run level
Variance reverts to a long run level
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

45. Standard normal distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(x) + Variance(Y) + 2*covariance(XY)

46. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
For n>30 - sample mean is approximately normal
Distribution with only two possible outcomes
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

47. Confidence interval (from t)
Application of mathematical statistics to economic data to lend empirical support to models
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
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
Sample mean +/ - t*(stddev(s)/sqrt(n))

48. Statistical (or empirical) model
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Yi = B0 + B1Xi + ui
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

49. F distribution
Attempts to sample along more important paths
Easy to manipulate
Variance(X) + Variance(Y) - 2*covariance(XY)
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

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