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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. Single variable (univariate) probability
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Concerned with a single random variable (ex. Roll of a die)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Statement of the error or precision of an estimate

2. Unbiased
Mean of sampling distribution is the population mean
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))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

3. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
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)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Random walk (usually acceptable) - Constant volatility (unlikely)

4. Inverse transform method
Least absolute deviations estimator - used when extreme outliers are not uncommon
(a^2)(variance(x)) + (b^2)(variance(y))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

5. Antithetic variable technique
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P(Z>t)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

6. Sample variance
Average return across assets on a given day
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
(a^2)(variance(x)) + (b^2)(variance(y))
For n>30 - sample mean is approximately normal

7. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Among all unbiased estimators - estimator with the smallest variance is efficient
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

8. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Summation((xi - mean)^k)/n
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

9. Variance(discrete)
(a^2)(variance(x)) + (b^2)(variance(y))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Combine to form distribution with leptokurtosis (heavy tails)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

10. Type II Error
We accept a hypothesis that should have been rejected
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
P(Z>t)

11. Continuously compounded return equation
i = ln(Si/Si - 1)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Rxy = Sxy/(Sx*Sy)

12. WLS
Has heavy tails
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

13. Lognormal
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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

14. Control variates technique
Z = (Y - meany)/(stddev(y)/sqrt(n))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Low Frequency - High Severity events

15. Cholesky factorization (decomposition)
Low Frequency - High Severity events
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
Only requires two parameters = mean and variance
Attempts to sample along more important paths

16. P - value
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
P(Z>t)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

17. Simplified standard (un - weighted) 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
Variance = (1/m) summation(u<n - i>^2)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

18. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

19. Key properties of linear regression
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Regression can be non - linear in variables but must be linear in parameters
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

20. Regime - switching volatility model
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Peaks over threshold - Collects dataset in excess of some threshold

21. Skewness
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Easy to manipulate
Concerned with a single random variable (ex. Roll of a die)
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

22. Test for statistical independence
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
P(X=x - Y=y) = P(X=x) * P(Y=y)

23. Simulation models
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
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
If variance of the conditional distribution of u(i) is not constant
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

24. F distribution
Special type of pooled data in which the cross sectional unit is surveyed over time
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^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

25. Tractable
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Easy to manipulate
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

26. Discrete random variable
Combine to form distribution with leptokurtosis (heavy tails)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance(x) + Variance(Y) + 2*covariance(XY)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

27. Covariance
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
More than one random variable
E(XY) - E(X)E(Y)
For n>30 - sample mean is approximately normal

28. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of 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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

29. Result of combination of two normal with same means
Among all unbiased estimators - estimator with the smallest variance is efficient
Combine to form distribution with leptokurtosis (heavy tails)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Use historical simulation approach but use the EWMA weighting system

30. Biggest (and only real) drawback of GARCH mode
When one regressor is a perfect linear function of the other regressors
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Nonlinearity

31. Simulating for VaR
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
Based on a dataset
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

32. Joint probability functions
Probability that the random variables take on certain values simultaneously
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
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

33. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Random walk (usually acceptable) - Constant volatility (unlikely)

34. Two requirements of OVB
Does not depend on a prior event or information
Average return across assets on a given day
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

35. Mean reversion in variance
Statement of the error or precision of an estimate
Variance reverts to a long run level
Concerned with a single random variable (ex. Roll of a die)
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

36. Confidence ellipse
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
When the sample size is large - the uncertainty about the value of the sample is very small
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

37. GEV
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Yi = B0 + B1Xi + ui

38. Central Limit Theorem(CLT)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sampling distribution of sample means tend to be normal
P(X=x - Y=y) = P(X=x) * P(Y=y)
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

39. Mean(expected value)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Var(X) + Var(Y)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

40. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Nonlinearity
We accept a hypothesis that should have been rejected

41. Statistical (or empirical) model
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Yi = B0 + B1Xi + ui
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

42. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Only requires two parameters = mean and variance
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)

43. Homoskedastic only F - stat
Confidence level
Based on a dataset
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

44. Significance =1
Confidence level
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Use historical simulation approach but use the EWMA weighting system

45. Exponential distribution
More than one random variable
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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)

46. Unconditional vs conditional distributions
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Has heavy tails

47. Binomial distribution equations for mean variance and std dev
Based on an equation - P(A) = # of A/total outcomes
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Mean = np - Variance = npq - Std dev = sqrt(npq)
E(mean) = mean

48. Time series data
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
Use historical simulation approach but use the EWMA weighting system
Returns over time for an individual asset
Variance(x) + Variance(Y) + 2*covariance(XY)

49. Normal distribution
Yi = B0 + B1Xi + ui
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
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
Average return across assets on a given day

50. Continuous random variable
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))