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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. 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(Y|X) = P(X -Y)/P(X) - P(B|A) = 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(Y|X) = P(X -Y)/P(X) - P(B|A) = 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