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Test your basic knowledge 
FRM Foundations Of Risk Management Quantitative Methods
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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.
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 reenforces your understanding as you take the test each time.
1. Implications of homoscedasticity
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Normal  Student's T  Chi  square  F distribution
Change in S = S<t  1>(meanchange in time + stddev E * sqrt(change in time))
2. Block maxima
P  value
P(X=x  Y=y) = P(X=x) * P(Y=y)
Dataset is parsed into blocks with greater length than the periodicity  Observations must be i.i.d.
Sample mean +/  t*(stddev(s)/sqrt(n))
3. Standard error for Monte Carlo replications
Two parameters: alpha(center) and beta(shape)   Popular for modeling recovery rates
SE(predicted std dev) = std dev * sqrt(1/2T)  Ten times more precision needs 100 times more replications
E[(Y  meany)^2] = E(Y^2)  [E(Y)]^2
(a^2)(variance(x)) + (b^2)(variance(y))
4. POT
Compute series of periodic returns  Choose a weighting scheme to translate a series into a single metric
When a distribution switches from high to low volatility  but never in between  Will exhibit fat tails of unaccounted for
Normal  Student's T  Chi  square  F distribution
Peaks over threshold  Collects dataset in excess of some threshold
5. Control variates technique
Conditional mean is time  varying  Conditional volatility is time  varying (more likely)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance(x)
Choose parameters that maximize the likelihood of what observations occurring
6. Variance of X+Y
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
Var(X) + Var(Y)
(a^2)(variance(x)
7. GARCH
Reverse engineer the implied std dev from the market price  Cmarket = f(implied standard deviation)
When one regressor is a perfect linear function of the other regressors
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)
(a^2)(variance(x)
8. SER
Standard error of error term  SER = sqrt(SSR/(n  k  1))  K is the # of slope coefficients
Returns over time for an individual asset
Distribution with only two possible outcomes
Two parameters: alpha(center) and beta(shape)   Popular for modeling recovery rates
9. T distribution
Non  parametric directly uses a historical dataset  Parametric imposes a specific distribution assumption
T = (x  meanx)/(stddev(x)/sqrt(n))  Symmetrical  mean = 0  Variance = k/k  2  Slightly heavy tail (kurtosis>3)
(a^2)(variance(x)
More than one random variable
10. Two requirements of OVB
Compute series of periodic returns  Choose a weighting scheme to translate a series into a single metric
Yi = B0 + B1Xi + ui
Omitted variable is correlated with regressor  Omitted variable is a determinant of the dependent variable
Variance(y)/n = variance of sample Y
11. Empirical frequency
(a^2)(variance(x)
Based on a dataset
Variance(x) + Variance(Y) + 2*covariance(XY)
Omitted variable is correlated with regressor  Omitted variable is a determinant of the dependent variable
12. Covariance
E(XY)  E(X)E(Y)
Independently and Identically Distributed
Summation(Yi  m)^2 = 1  Minimizes the sum of squares gaps
Contains variables not explicit in model  Accounts for randomness
13. Extending the HS approach for computing value of a portfolio
14. LFHS
Generalized exponential distribution  Exponential is a Weibull distribution with alpha = 1.0  F(x) = 1  e^  (x/beta)^alpha
Application of mathematical statistics to economic data to lend empirical support to models
Least absolute deviations estimator  used when extreme outliers are not uncommon
Low Frequency  High Severity events
15. Standard normal distribution
Transformed to a unit variable  Mean = 0 Variance = 1
Mean = np  Variance = npq  Std dev = sqrt(npq)
i = ln(Si/Si  1)
Variance(X) + Variance(Y)  2*covariance(XY)
16. Historical std dev
Simplest and most common way to estimate future volatility  Variance(t) = (1/N) Summation(r^2)
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)
Probability that the random variables take on certain values simultaneously
Coefficent of determination  fraction of variance explained by independent variables  R^2 = ESS/TSS = 1  (SSR/TSS)
17. Square root rule
Easy to manipulate
Simplest approach to extending horizon  J  period VaR = sqrt(J) * 1  period VaR  Only applies under i.i.d
Time to wait until an event takes place  F(x) = lambda e^(  lambdax)  Lambda = 1/beta
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n)  VL)
18. Unbiased
Only requires two parameters = mean and variance
Conditional mean is time  varying  Conditional volatility is time  varying (more likely)
Non  parametric directly uses a historical dataset  Parametric imposes a specific distribution assumption
Mean of sampling distribution is the population mean
19. Economical(elegant)
More than one random variable
Weighted least squares estimator  Weights the squares to account for heteroskedasticity and is BLUE
1/lambda is hazard rate of default intensity  Lambda = 1/beta  f(x) = lambda e^(  lambdax) F(x) = 1  e^(  lambda*x)
Only requires two parameters = mean and variance
20. Continuously compounded return equation
Least absolute deviations estimator  used when extreme outliers are not uncommon
i = ln(Si/Si  1)
Only requires two parameters = mean and variance
Exponentially Weighted Moving Average  Weights decline in constant proportion given by lambda
21. Single variable (univariate) probability
Simplest and most common way to estimate future volatility  Variance(t) = (1/N) Summation(r^2)
Make parametric assumptions about covariances of each position and extend them to entire portfolio  Problem: correlations change during stressful market events
Concerned with a single random variable (ex. Roll of a die)
Weights are not a function of time  but based on the nature of the historic period (more similar to historic stake  greater the weight)
22. Standard error
Sampling distribution of sample means tend to be normal
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Translates a random number into a cumulative standard normal distribution  EXCEL: NORMSINV(RAND())
23. Binomial distribution equations for mean variance and std dev
Low Frequency  High Severity events
Mean = np  Variance = npq  Std dev = sqrt(npq)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Reverse engineer the implied std dev from the market price  Cmarket = f(implied standard deviation)
24. Hazard rate of exponentially distributed random variable
Mean = lambda  Variance = lambda  Std dev = sqrt(lambda)
Based on an equation  P(A) = # of A/total outcomes
1/lambda is hazard rate of default intensity  Lambda = 1/beta  f(x) = lambda e^(  lambdax) F(x) = 1  e^(  lambda*x)
Combine to form distribution with leptokurtosis (heavy tails)
25. Confidence interval for sample mean
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n)  VL)
X  t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n))  Random interval since it will vary by the sample
Returns over time for an individual asset
Sample mean +/  t*(stddev(s)/sqrt(n))
26. Unconditional vs conditional distributions
Only requires two parameters = mean and variance
Unconditional is the same regardless of market or economic conditions (unrealistic)  Conditional depends on the economy  market  or other state
SSR
When the sample size is large  the uncertainty about the value of the sample is very small
27. Variance of X+b
SE(predicted std dev) = std dev * sqrt(1/2T)  Ten times more precision needs 100 times more replications
Low Frequency  High Severity events
Generalized exponential distribution  Exponential is a Weibull distribution with alpha = 1.0  F(x) = 1  e^  (x/beta)^alpha
Variance(x)
28. Inverse transform method
When a distribution switches from high to low volatility  but never in between  Will exhibit fat tails of unaccounted for
E(XY)  E(X)E(Y)
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
Translates a random number into a cumulative standard normal distribution  EXCEL: NORMSINV(RAND())
29. Variance of sample mean
Non  parametric directly uses a historical dataset  Parametric imposes a specific distribution assumption
Concerned with a single random variable (ex. Roll of a die)
Variance = summation(alpha weight)(u<n  i>^2)  alpha weights must sum to one
Variance(y)/n = variance of sample Y
30. Homoskedastic
Generalized Pareto Distribution  Models distribution of POT  Empirical distributions are rarely sufficient for this model
Can Use alpha and beta weights to solve for the long  run average variance  VL = w/(1  alpha  beta)
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
Variance(sample y) = (variance(y)/n)*(N  n/N  1)
31. i.i.d.
F = [(SSR<restricted>  SSR<unrestricted>)/q]/(SSR<unrestricted>/(n  k<unrestricted>  1)
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)
Independently and Identically Distributed
Sampling distribution of sample means tend to be normal
32. Monte Carlo Simulations
Low Frequency  High Severity events
Simplest and most common way to estimate future volatility  Variance(t) = (1/N) Summation(r^2)
Generation of a distribution of returns by use of random numbers  Return path decided by algorithm  Correlation must be modeled
Variance = (1/m) summation(u<n  i>^2)
33. Cholesky factorization (decomposition)
T = (x  meanx)/(stddev(x)/sqrt(n))  Symmetrical  mean = 0  Variance = k/k  2  Slightly heavy tail (kurtosis>3)
More than one random variable
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
Concerned with a single random variable (ex. Roll of a die)
34. What does the OLS minimize?
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
P(X=x  Y=y) = P(X=x) * P(Y=y)
Variance(X) + Variance(Y)  2*covariance(XY)
SSR
35. Poisson distribution equations for mean variance and std deviation
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
Concerned with a single random variable (ex. Roll of a die)
Variance = summation(alpha weight)(u<n  i>^2)  alpha weights must sum to one
Mean = lambda  Variance = lambda  Std dev = sqrt(lambda)
36. Reliability
Statement of the error or precision of an estimate
Assumes a value among a finite set including x1  x2  etc  P(X=xk) = f(xk)
[1/(n  1)]*summation((Xi  X)(Yi  Y))
Variance(X) + Variance(Y)  2*covariance(XY)
37. Type I error
Special type of pooled data in which the cross sectional unit is surveyed over time
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
We reject a hypothesis that is actually true
Has heavy tails
38. Antithetic variable technique
Weighted least squares estimator  Weights the squares to account for heteroskedasticity and is BLUE
Changes the sign of the random samples  appropriate when distribution is symmetric  creates twice as many replications
Simplest approach to extending horizon  J  period VaR = sqrt(J) * 1  period VaR  Only applies under i.i.d
Expected value of the sample mean is the population mean
39. Non  parametric vs parametric calculation of VaR
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
Variance(x)
Population denominator = n  Sample denominator = n  1
40. Poisson Distribution
F = ½ ((t1^2)+(t2^2)  (correlation t1 t2))/(1  2correlation)
Standard error of error term  SER = sqrt(SSR/(n  k  1))  K is the # of slope coefficients
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
41. Efficiency
Returns over time for a combination of assets (combination of time series and cross  sectional data)
Among all unbiased estimators  estimator with the smallest variance is efficient
Variance(x)
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
42. Extreme Value Theory
When the sample size is large  the uncertainty about the value of the sample is very small
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
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
EVT  Fits a separate distribution to the extreme loss tail  Only uses tail
43. Marginal unconditional probability function
1/lambda is hazard rate of default intensity  Lambda = 1/beta  f(x) = lambda e^(  lambdax) F(x) = 1  e^(  lambda*x)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Does not depend on a prior event or information
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
44. Cross  sectional
Generalized exponential distribution  Exponential is a Weibull distribution with alpha = 1.0  F(x) = 1  e^  (x/beta)^alpha
Average return across assets on a given day
Sample mean +/  t*(stddev(s)/sqrt(n))
Confidence set for two coefficients  two dimensional analog for the confidence interval
45. Heteroskedastic
Covariance = (lambda)(cov(n  1)) + (1  lambda)(xn  1)(yn  1)
Average return across assets on a given day
If variance of the conditional distribution of u(i) is not constant
Rxy = Sxy/(Sx*Sy)
46. Sample mean
Expected value of the sample mean is the population mean
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
Infinite number of values within an interval  P(a<x<b) = interval from a to b of f(x)dx
Summation((xi  mean)^k)/n
47. R^2
Coefficent of determination  fraction of variance explained by independent variables  R^2 = ESS/TSS = 1  (SSR/TSS)
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
Rxy = Sxy/(Sx*Sy)
Summation((xi  mean)^k)/n
48. WLS
Transformed to a unit variable  Mean = 0 Variance = 1
Weighted least squares estimator  Weights the squares to account for heteroskedasticity and is BLUE
Average return across assets on a given day
Make parametric assumptions about covariances of each position and extend them to entire portfolio  Problem: correlations change during stressful market events
49. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Confidence level
Sampling distribution of sample means tend to be normal
Independently and Identically Distributed
50. Maximum likelihood method
Distribution with only two possible outcomes
Choose parameters that maximize the likelihood of what observations occurring
Special type of pooled data in which the cross sectional unit is surveyed over time
Coefficent of determination  fraction of variance explained by independent variables  R^2 = ESS/TSS = 1  (SSR/TSS)