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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. Homoskedastic
Random walk (usually acceptable) - Constant volatility (unlikely)
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(x)
Model dependent - Options with the same underlying assets may trade at different volatilities

2. SER
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 error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

3. Sample covariance
Confidence set for two coefficients - two dimensional analog for the confidence interval
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Easy to manipulate

4. Inverse transform method
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
E(XY) - E(X)E(Y)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Low Frequency - High Severity events

5. Two assumptions of square root rule
Application of mathematical statistics to economic data to lend empirical support to models
Random walk (usually acceptable) - Constant volatility (unlikely)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Sampling distribution of sample means tend to be normal

6. Biggest (and only real) drawback of GARCH mode
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Among all unbiased estimators - estimator with the smallest variance is efficient
Nonlinearity

7. Regime - switching volatility model
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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

8. Variance of aX + bY
Variance(y)/n = variance of sample Y
(a^2)(variance(x)) + (b^2)(variance(y))
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
E(XY) - E(X)E(Y)

9. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Concerned with a single random variable (ex. Roll of a die)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
(a^2)(variance(x)

10. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Only requires two parameters = mean and variance
Variance(x)

11. WLS
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Var(X) + Var(Y)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

12. Binomial distribution
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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!)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

13. What does the OLS minimize?
SSR
Confidence level
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

14. Two drawbacks of moving average series
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

15. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
We accept a hypothesis that should have been rejected
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Attempts to sample along more important paths

16. Multivariate probability
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
More than one random variable
Least absolute deviations estimator - used when extreme outliers are not uncommon
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

17. Sample mean
Expected value of the sample mean is the population mean
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Confidence set for two coefficients - two dimensional analog for the confidence interval

18. Cross - sectional
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Average return across assets on a given day
Use historical simulation approach but use the EWMA weighting system

19. Lognormal
Least absolute deviations estimator - used when extreme outliers are not uncommon
95% = 1.65 99% = 2.33 For one - tailed tests
Choose parameters that maximize the likelihood of what observations occurring
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

20. Test for unbiasedness
Statement of the error or precision of an estimate
Use historical simulation approach but use the EWMA weighting system
Has heavy tails
E(mean) = mean

21. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Independently and Identically Distributed
Easy to manipulate
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)

22. Homoskedastic only F - stat
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

23. Importance sampling technique
Returns over time for an individual asset
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Attempts to sample along more important paths
Special type of pooled data in which the cross sectional unit is surveyed over time

24. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Z = (Y - meany)/(stddev(y)/sqrt(n))
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

25. Beta distribution
Concerned with a single random variable (ex. Roll of a die)
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

26. Covariance
E(XY) - E(X)E(Y)
Low Frequency - High Severity events
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Model dependent - Options with the same underlying assets may trade at different volatilities

27. Gamma distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
Population denominator = n - Sample denominator = n - 1
Expected value of the sample mean is the population mean

28. Key properties of linear regression
Random walk (usually acceptable) - Constant volatility (unlikely)
Regression can be non - linear in variables but must be linear in parameters
Z = (Y - meany)/(stddev(y)/sqrt(n))
Summation((xi - mean)^k)/n

29. Expected future variance rate (t periods forward)
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Yi = B0 + B1Xi + ui

30. Square root rule
Average return across assets on a given day
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

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

32. Type I error
We reject a hypothesis that is actually true
Variance(X) + Variance(Y) - 2*covariance(XY)
Var(X) + Var(Y)
Special type of pooled data in which the cross sectional unit is surveyed over time

33. P - value
Returns over time for an individual asset
P(Z>t)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(x)

34. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(x)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

35. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Returns over time for an individual asset

36. Bernouli Distribution
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Distribution with only two possible outcomes
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance = (1/m) summation(u<n - i>^2)

37. Potential reasons for fat tails in return distributions
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance(y)/n = variance of sample Y
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

38. Type II Error
We accept a hypothesis that should have been rejected
Peaks over threshold - Collects dataset in excess of some threshold
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

39. Unconditional vs conditional distributions
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Combine to form distribution with leptokurtosis (heavy tails)

40. Unstable return distribution
95% = 1.65 99% = 2.33 For one - tailed tests
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Least absolute deviations estimator - used when extreme outliers are not uncommon
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

41. Sample correlation
Combine to form distribution with leptokurtosis (heavy tails)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance reverts to a long run level
Rxy = Sxy/(Sx*Sy)

42. Confidence ellipse
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Confidence set for two coefficients - two dimensional analog for the confidence interval

43. Variance of weighted scheme
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

44. Maximum likelihood method
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Choose parameters that maximize the likelihood of what observations occurring

45. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
For n>30 - sample mean is approximately normal
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

46. Continuous random variable
Use historical simulation approach but use the EWMA weighting system
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

47. Variance of X+Y
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Var(X) + Var(Y)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

48. Reliability
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Statement of the error or precision of an estimate
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

49. Adjusted R^2
P(X=x - Y=y) = P(X=x) * P(Y=y)
P - value
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

50. Economical(elegant)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Only requires two parameters = mean and variance
Distribution with only two possible outcomes
Among all unbiased estimators - estimator with the smallest variance is efficient