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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. Statistical (or empirical) model
Variance = (1/m) summation(u<n - i>^2)
Yi = B0 + B1Xi + ui
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

2. Importance sampling technique
Attempts to sample along more important paths
Statement of the error or precision of an estimate
Summation((xi - mean)^k)/n
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

3. Binomial distribution
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!)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

4. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
We reject a hypothesis that is actually true
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

5. Skewness
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
Regression can be non - linear in variables but must be linear in parameters
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Has heavy tails

6. Logistic distribution
Has heavy tails
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Based on an equation - P(A) = # of A/total outcomes
Choose parameters that maximize the likelihood of what observations occurring

7. Continuous representation of the GBM
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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)
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

8. Tractable
Easy to manipulate
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)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

9. Mean reversion in asset dynamics
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
(a^2)(variance(x)) + (b^2)(variance(y))
P(X=x - Y=y) = P(X=x) * P(Y=y)
Price/return tends to run towards a long - run level

10. Reliability
Based on an equation - P(A) = # of A/total outcomes
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Statement of the error or precision of an estimate

11. Normal distribution
Based on an equation - P(A) = # of A/total outcomes
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

12. Type I error
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
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
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)

13. Two requirements of OVB
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)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

14. Central Limit Theorem
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Peaks over threshold - Collects dataset in excess of some threshold
For n>30 - sample mean is approximately normal

15. Result of combination of two normal with same means
Expected value of the sample mean is the population mean
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Combine to form distribution with leptokurtosis (heavy tails)

16. Econometrics
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Contains variables not explicit in model - Accounts for randomness
When the sample size is large - the uncertainty about the value of the sample is very small
Application of mathematical statistics to economic data to lend empirical support to models

17. Covariance
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
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
E(XY) - E(X)E(Y)

18. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Confidence level
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

19. EWMA
Z = (Y - meany)/(stddev(y)/sqrt(n))
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

20. LFHS
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
Based on a dataset
Special type of pooled data in which the cross sectional unit is surveyed over time
Low Frequency - High Severity events

21. Shortcomings of implied volatility
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance reverts to a long run level

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

23. Implications of homoscedasticity
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 - value
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

24. Gamma distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance reverts to a long run level
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

25. 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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Distribution with only two possible outcomes
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

26. Test for unbiasedness
Use historical simulation approach but use the EWMA weighting system
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
E(mean) = mean

27. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Choose parameters that maximize the likelihood of what observations occurring

28. Monte Carlo Simulations
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

29. Adjusted R^2
Model dependent - Options with the same underlying assets may trade at different volatilities
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)
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
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

30. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Population denominator = n - Sample denominator = n - 1
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

31. Square root rule
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Choose parameters that maximize the likelihood of what observations occurring

32. Weibul distribution
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Has heavy tails
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

33. R^2
Z = (Y - meany)/(stddev(y)/sqrt(n))
Combine to form distribution with leptokurtosis (heavy tails)
E(XY) - E(X)E(Y)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

34. Cross - sectional
Yi = B0 + B1Xi + ui
Variance(X) + Variance(Y) - 2*covariance(XY)
Combine to form distribution with leptokurtosis (heavy tails)
Average return across assets on a given day

35. Two drawbacks of moving average series
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Independently and Identically Distributed
Mean = np - Variance = npq - Std dev = sqrt(npq)

36. Bernouli Distribution
Distribution with only two possible outcomes
Only requires two parameters = mean and variance
Low Frequency - High Severity events
Based on an equation - P(A) = # of A/total outcomes

37. Variance(discrete)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

38. Priori (classical) probability
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(X) + Variance(Y) - 2*covariance(XY)
Var(X) + Var(Y)
Based on an equation - P(A) = # of A/total outcomes

39. Significance =1
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Contains variables not explicit in model - Accounts for randomness
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
Confidence level

40. Variance - covariance approach for VaR of a portfolio
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

41. Mean(expected value)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Mean = np - Variance = npq - Std dev = sqrt(npq)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Easy to manipulate

42. i.i.d.
Average return across assets on a given day
Statement of the error or precision of an estimate
Independently and Identically Distributed
P(X=x - Y=y) = P(X=x) * P(Y=y)

43. Continuously compounded return equation
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
i = ln(Si/Si - 1)
P - value
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

44. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
P(Z>t)
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
If variance of the conditional distribution of u(i) is not constant

45. Continuous random variable
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Low Frequency - High Severity events

46. Regime - switching volatility model
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
We reject a hypothesis that is actually true
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 a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

47. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
(a^2)(variance(x)
P(Z>t)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

48. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Regression can be non - linear in variables but must be linear in parameters
E(XY) - E(X)E(Y)

49. Sample variance
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Expected value of the sample mean is the population mean
(a^2)(variance(x)

50. Variance of X+b
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(x)
Variance(x) + Variance(Y) + 2*covariance(XY)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda