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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. Simplified standard (un  weighted) variance
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!)
Application of mathematical statistics to economic data to lend empirical support to models
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance = (1/m) summation(u<n  i>^2)
2. Central Limit Theorem(CLT)
Can Use alpha and beta weights to solve for the long  run average variance  VL = w/(1  alpha  beta)
Probability that the random variables take on certain values simultaneously
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
Sampling distribution of sample means tend to be normal
3. Confidence interval for sample mean
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
X  t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n))  Random interval since it will vary by the sample
Can Use alpha and beta weights to solve for the long  run average variance  VL = w/(1  alpha  beta)
Weights are not a function of time  but based on the nature of the historic period (more similar to historic stake  greater the weight)
4. Homoskedastic
Depends upon lambda  which indicates the rate of occurrence of the random events (binomial) over a time interval  (lambda^k)/(k!) * e^(  lambda)
Normal  Student's T  Chi  square  F distribution
i = ln(Si/Si  1)
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
5. Cross  sectional
1/lambda is hazard rate of default intensity  Lambda = 1/beta  f(x) = lambda e^(  lambdax) F(x) = 1  e^(  lambda*x)
Average return across assets on a given day
Based on a dataset
Statement of the error or precision of an estimate
6. Confidence ellipse
Assumes a value among a finite set including x1  x2  etc  P(X=xk) = f(xk)
Attempts to sample along more important paths
Has heavy tails
Confidence set for two coefficients  two dimensional analog for the confidence interval
7. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Variance(y)/n = variance of sample Y
Best Linear Unbiased Estimator  Sample mean for samples that are i.i.d.
Variance(sample y) = (variance(y)/n)*(N  n/N  1)
8. Shortcomings of implied volatility
Model dependent  Options with the same underlying assets may trade at different volatilities
Mean = np  Variance = npq  Std dev = sqrt(npq)
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
Translates a random number into a cumulative standard normal distribution  EXCEL: NORMSINV(RAND())
9. Test for unbiasedness
E(mean) = 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
Independently and Identically Distributed
Mean = np  Variance = npq  Std dev = sqrt(npq)
10. Hybrid method for conditional volatility
Returns over time for an individual asset
Weights are not a function of time  but based on the nature of the historic period (more similar to historic stake  greater the weight)
If variance of the conditional distribution of u(i) is not constant
Use historical simulation approach but use the EWMA weighting system
11. Mean reversion in asset dynamics
[1/(n  1)]*summation((Xi  X)(Yi  Y))
E[(Y  meany)^2] = E(Y^2)  [E(Y)]^2
Price/return tends to run towards a long  run level
Summation(Yi  m)^2 = 1  Minimizes the sum of squares gaps
12. BLUE
Best Linear Unbiased Estimator  Sample mean for samples that are i.i.d.
Returns over time for an individual asset
Variance(X) + Variance(Y)  2*covariance(XY)
Conditional mean is time  varying  Conditional volatility is time  varying (more likely)
13. Variance of X+Y assuming dependence
Distribution with only two possible outcomes
(a^2)(variance(x)
Variance(x) + Variance(Y) + 2*covariance(XY)
Explained sum of squares  Summation[(predicted yi  meany)^2]  Squared distance between the predicted y and the mean of y
14. Time series data
Standard error of error term  SER = sqrt(SSR/(n  k  1))  K is the # of slope coefficients
Returns over time for an individual asset
Population denominator = n  Sample denominator = n  1
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
15. Stochastic error term
Variance(X) + Variance(Y)  2*covariance(XY)
Distribution with only two possible outcomes
Contains variables not explicit in model  Accounts for randomness
X  t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n))  Random interval since it will vary by the sample
16. Two drawbacks of moving average series
When one regressor is a perfect linear function of the other regressors
Ignores order of observations (no weight for most recent observations)  Has a ghosting feature where data points are dropped due to length of window
Exponentially Weighted Moving Average  Weights decline in constant proportion given by lambda
Population denominator = n  Sample denominator = n  1
17. GARCH
Low Frequency  High Severity events
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)
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
Weighted least squares estimator  Weights the squares to account for heteroskedasticity and is BLUE
18. Priori (classical) probability
95% = 1.65 99% = 2.33 For one  tailed tests
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)
Based on an equation  P(A) = # of A/total outcomes
Price/return tends to run towards a long  run level
19. Sample covariance
[1/(n  1)]*summation((Xi  X)(Yi  Y))
P(X=x  Y=y) = P(X=x) * P(Y=y)
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
20. Antithetic variable technique
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
Changes the sign of the random samples  appropriate when distribution is symmetric  creates twice as many replications
Depends on whether X and mean are positively or negatively correlated  Beta1 = beta1 + correlation(x mean)*(stddev(mean)/stddev(x))
More than one random variable
21. Mean(expected value)
Reverse engineer the implied std dev from the market price  Cmarket = f(implied standard deviation)
Discrete: E(Y) = Summation(xi*pi)  Continuous: E(X) = integral(x*f(x)dx)
Generation of a distribution of returns by use of random numbers  Return path decided by algorithm  Correlation must be modeled
T = (x  meanx)/(stddev(x)/sqrt(n))  Symmetrical  mean = 0  Variance = k/k  2  Slightly heavy tail (kurtosis>3)
22. Biggest (and only real) drawback of GARCH mode
Use historical simulation approach but use the EWMA weighting system
Nonlinearity
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Summation((xi  mean)^k)/n
23. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Confidence set for two coefficients  two dimensional analog for the confidence interval
Omitted variable is correlated with regressor  Omitted variable is a determinant of the dependent variable
Model dependent  Options with the same underlying assets may trade at different volatilities
24. Type I error
We reject a hypothesis that is actually true
P(Z>t)
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
25. Binomial distribution equations for mean variance and std dev
Mean = np  Variance = npq  Std dev = sqrt(npq)
Attempts to sample along more important paths
Variance = (1/m) summation(u<n  i>^2)
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
26. Variance of sample mean
Low Frequency  High Severity events
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
1/lambda is hazard rate of default intensity  Lambda = 1/beta  f(x) = lambda e^(  lambdax) F(x) = 1  e^(  lambda*x)
Variance(y)/n = variance of sample Y
27. LAD
Mean = lambda  Variance = lambda  Std dev = sqrt(lambda)
Yi = B0 + B1Xi + ui
Least absolute deviations estimator  used when extreme outliers are not uncommon
Assumes a value among a finite set including x1  x2  etc  P(X=xk) = f(xk)
28. Perfect multicollinearity
Least absolute deviations estimator  used when extreme outliers are not uncommon
When one regressor is a perfect linear function of the other regressors
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
Probability that the random variables take on certain values simultaneously
29. Econometrics
(a^2)(variance(x)
Application of mathematical statistics to economic data to lend empirical support to models
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
Random walk (usually acceptable)  Constant volatility (unlikely)
30. Gamma distribution
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
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
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)
31. Key properties of linear regression
Regression can be non  linear in variables but must be linear in parameters
Sample variance = (1/(k  1))Summation(Yi  mean)^2
Transformed to a unit variable  Mean = 0 Variance = 1
Explained sum of squares  Summation[(predicted yi  meany)^2]  Squared distance between the predicted y and the mean of y
32. Variance of weighted scheme
Variance = summation(alpha weight)(u<n  i>^2)  alpha weights must sum to one
Probability of an outcome given another outcome P(YX) = P(X Y)/P(X)  P(BA) = P(A and B)/P(A)
Changes the sign of the random samples  appropriate when distribution is symmetric  creates twice as many replications
Statement of the error or precision of an estimate
33. Weibul distribution
Unconditional is the same regardless of market or economic conditions (unrealistic)  Conditional depends on the economy  market  or other state
Standard error of error term  SER = sqrt(SSR/(n  k  1))  K is the # of slope coefficients
Generalized exponential distribution  Exponential is a Weibull distribution with alpha = 1.0  F(x) = 1  e^  (x/beta)^alpha
Among all unbiased estimators  estimator with the smallest variance is efficient
34. Efficiency
Sample mean +/  t*(stddev(s)/sqrt(n))
Among all unbiased estimators  estimator with the smallest variance is efficient
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
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
35. Multivariate Density Estimation (MDE)
We accept a hypothesis that should have been rejected
Variance(x)
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))
36. Law of Large Numbers
Exponentially Weighted Moving Average  Weights decline in constant proportion given by lambda
Nonlinearity
Application of mathematical statistics to economic data to lend empirical support to models
Sample mean will near the population mean as the sample size increases
37. Monte Carlo Simulations
Expected value of the sample mean is the population mean
Generation of a distribution of returns by use of random numbers  Return path decided by algorithm  Correlation must be modeled
T = (x  meanx)/(stddev(x)/sqrt(n))  Symmetrical  mean = 0  Variance = k/k  2  Slightly heavy tail (kurtosis>3)
Coefficent of determination  fraction of variance explained by independent variables  R^2 = ESS/TSS = 1  (SSR/TSS)
38. POT
Peaks over threshold  Collects dataset in excess of some threshold
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Sample variance = (1/(k  1))Summation(Yi  mean)^2
Change in S = S<t  1>(meanchange in time + stddev E * sqrt(change in time))
39. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Mean of sampling distribution is the population mean
Probability of an outcome given another outcome P(YX) = P(X Y)/P(X)  P(BA) = P(A and B)/P(A)
Conditional mean is time  varying  Conditional volatility is time  varying (more likely)
40. Implications of homoscedasticity
Mean = np  Variance = npq  Std dev = sqrt(npq)
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
Sampling distribution of sample means tend to be normal
Average return across assets on a given day
41. Empirical frequency
T = (x  meanx)/(stddev(x)/sqrt(n))  Symmetrical  mean = 0  Variance = k/k  2  Slightly heavy tail (kurtosis>3)
Attempts to sample along more important paths
Confidence set for two coefficients  two dimensional analog for the confidence interval
Based on a dataset
42. Normal distribution
F = ½ ((t1^2)+(t2^2)  (correlation t1 t2))/(1  2correlation)
Conditional mean is time  varying  Conditional volatility is time  varying (more likely)
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(X) + Variance(Y)  2*covariance(XY)
43. Variance  covariance approach for VaR of a portfolio
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
Has heavy tails
Variance(X) + Variance(Y)  2*covariance(XY)
Make parametric assumptions about covariances of each position and extend them to entire portfolio  Problem: correlations change during stressful market events
44. Continuous random variable
Infinite number of values within an interval  P(a<x<b) = interval from a to b of f(x)dx
Based on an equation  P(A) = # of A/total outcomes
Returns over time for an individual asset
Depends on whether X and mean are positively or negatively correlated  Beta1 = beta1 + correlation(x mean)*(stddev(mean)/stddev(x))
45. Sample variance
Random walk (usually acceptable)  Constant volatility (unlikely)
Standard error of error term  SER = sqrt(SSR/(n  k  1))  K is the # of slope coefficients
Sample variance = (1/(k  1))Summation(Yi  mean)^2
Reverse engineer the implied std dev from the market price  Cmarket = f(implied standard deviation)
46. ESS
Explained sum of squares  Summation[(predicted yi  meany)^2]  Squared distance between the predicted y and the mean of y
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n)  VL)
Make parametric assumptions about covariances of each position and extend them to entire portfolio  Problem: correlations change during stressful market events
Combine to form distribution with leptokurtosis (heavy tails)
47. Continuously compounded return equation
Sampling distribution of sample means tend to be normal
i = ln(Si/Si  1)
F = ½ ((t1^2)+(t2^2)  (correlation t1 t2))/(1  2correlation)
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
48. Sample correlation
Make parametric assumptions about covariances of each position and extend them to entire portfolio  Problem: correlations change during stressful market events
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)
Rxy = Sxy/(Sx*Sy)
49. T distribution
T = (x  meanx)/(stddev(x)/sqrt(n))  Symmetrical  mean = 0  Variance = k/k  2  Slightly heavy tail (kurtosis>3)
Probability that the random variables take on certain values simultaneously
Generate sequence of variables from which price is computed  Calculate value of asset with these prices  Repeat to form distribution
Independently and Identically Distributed
50. 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
We accept a hypothesis that should have been rejected
Mean = lambda  Variance = lambda  Std dev = sqrt(lambda)
Independently and Identically Distributed