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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. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Nonlinearity
Choose parameters that maximize the likelihood of what observations occurring
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

2. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Var(X) + Var(Y)
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
Concerned with a single random variable (ex. Roll of a die)

3. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
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
Summation((xi - mean)^k)/n

4. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Has heavy tails
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
Statement of the error or precision of an estimate

5. Gamma distribution
We reject a hypothesis that is actually true
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[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

6. Overall F - statistic
Contains variables not explicit in model - Accounts for randomness
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
(a^2)(variance(x)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

7. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

8. Control variates technique
Z = (Y - meany)/(stddev(y)/sqrt(n))
Independently and Identically Distributed
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Contains variables not explicit in model - Accounts for randomness

9. Homoskedastic only F - stat
Special type of pooled data in which the cross sectional unit is surveyed over time
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance reverts to a long run level
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

10. Logistic distribution
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Regression can be non - linear in variables but must be linear in parameters
Has heavy tails

11. Continuously compounded return equation
i = ln(Si/Si - 1)
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Peaks over threshold - Collects dataset in excess of some threshold

12. Perfect multicollinearity
Only requires two parameters = mean and variance
(a^2)(variance(x)) + (b^2)(variance(y))
Confidence level
When one regressor is a perfect linear function of the other regressors

13. Mean(expected value)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Yi = B0 + B1Xi + ui

14. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Expected value of the sample mean is the population mean

15. Variance(discrete)
Population denominator = n - Sample denominator = n - 1
Z = (Y - meany)/(stddev(y)/sqrt(n))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Yi = B0 + B1Xi + ui

16. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Population denominator = n - Sample denominator = n - 1
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
i = ln(Si/Si - 1)

17. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
E(XY) - E(X)E(Y)
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)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

18. Variance of weighted scheme
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E(mean) = mean
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

19. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(x) + Variance(Y) + 2*covariance(XY)

20. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Does not depend on a prior event or information

21. Stochastic error term
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(y)/n = variance of sample Y
Nonlinearity
Contains variables not explicit in model - Accounts for randomness

22. Empirical frequency
Combine to form distribution with leptokurtosis (heavy tails)
Based on a dataset
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)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

23. Sample variance
Var(X) + Var(Y)
Low Frequency - High Severity events
We accept a hypothesis that should have been rejected
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

24. Homoskedastic
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

25. GPD
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Rxy = Sxy/(Sx*Sy)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

26. Joint probability functions
For n>30 - sample mean is approximately normal
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Probability that the random variables take on certain values simultaneously
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

27. Variance of X+Y
Random walk (usually acceptable) - Constant volatility (unlikely)
Var(X) + Var(Y)
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
Yi = B0 + B1Xi + ui

28. GARCH
Choose parameters that maximize the likelihood of what observations occurring
Average return across assets on a given day
We reject a hypothesis that is actually true
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)

29. Unstable return distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

30. POT
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Peaks over threshold - Collects dataset in excess of some threshold
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

31. Sample covariance
Based on a dataset
Use historical simulation approach but use the EWMA weighting system
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

32. Variance of X+Y assuming dependence
Combine to form distribution with leptokurtosis (heavy tails)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Variance(x) + Variance(Y) + 2*covariance(XY)
Returns over time for an individual asset

33. Type II Error
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

34. Continuous representation of the GBM
Normal - Student's T - Chi - square - F distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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)
Low Frequency - High Severity events

35. Adjusted R^2
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
Concerned with a single random variable (ex. Roll of a die)
Variance reverts to a long run level
Yi = B0 + B1Xi + ui

36. Exponential distribution
Statement of the error or precision of an estimate
(a^2)(variance(x)) + (b^2)(variance(y))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

37. Mean reversion in asset dynamics
Population denominator = n - Sample denominator = n - 1
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
Price/return tends to run towards a long - run level
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

38. Implications of homoscedasticity
Use historical simulation approach but use the EWMA weighting system
Combine to form distribution with leptokurtosis (heavy tails)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

39. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

40. Mean reversion in variance
Variance reverts to a long run level
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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

41. Non - parametric vs parametric calculation of VaR
E(XY) - E(X)E(Y)
Expected value of the sample mean is the population mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Independently and Identically Distributed

42. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Sampling distribution of sample means tend to be normal
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Nonlinearity

43. Variance of X+b
Variance(y)/n = variance of sample Y
Variance(x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Application of mathematical statistics to economic data to lend empirical support to models

44. Tractable
When the sample size is large - the uncertainty about the value of the sample is very small
Easy to manipulate
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

45. Reliability
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Statement of the error or precision of an estimate
Only requires two parameters = mean and variance

46. Poisson distribution equations for mean variance and std deviation
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sampling distribution of sample means tend to be normal

47. EWMA
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

48. Central Limit Theorem
For n>30 - sample mean is approximately normal
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

49. Economical(elegant)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Only requires two parameters = mean and variance
(a^2)(variance(x)) + (b^2)(variance(y))
Among all unbiased estimators - estimator with the smallest variance is efficient

50. Antithetic variable technique
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Use historical simulation approach but use the EWMA weighting system
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y