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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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1. Exact significance level
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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)
Population denominator = n - Sample denominator = n - 1
P - value

2. Sample mean
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Expected value of the sample mean is the population mean
Sampling distribution of sample means tend to be normal

3. Central Limit Theorem(CLT)
Probability that the random variables take on certain values simultaneously
Sampling distribution of sample means tend to be normal
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

4. Deterministic Simulation
Attempts to sample along more important paths
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

5. Type I error
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
We reject a hypothesis that is actually true
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

6. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
When the sample size is large - the uncertainty about the value of the sample is very small

7. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Contains variables not explicit in model - Accounts for randomness
Sampling distribution of sample means tend to be normal
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

8. Continuous representation of the GBM
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Price/return tends to run towards a long - run level

9. Variance of X+Y
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Var(X) + Var(Y)
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
Easy to manipulate

10. Variance of weighted scheme
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
P - value
Distribution with only two possible outcomes

11. BLUE
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Contains variables not explicit in model - Accounts for randomness

12. Simplified standard (un - weighted) variance
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance = (1/m) summation(u<n - i>^2)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Average return across assets on a given day

13. Perfect multicollinearity
Confidence level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
When one regressor is a perfect linear function of the other regressors
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

14. Conditional probability functions
Expected value of the sample mean is the population mean
Has heavy tails
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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

15. Variance of sample mean
P(X=x - Y=y) = P(X=x) * P(Y=y)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance(y)/n = variance of sample Y
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

16. Inverse transform method
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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!)
Price/return tends to run towards a long - run level

17. Limitations of R^2 (what an increase doesn't necessarily imply)

18. SER
Has heavy tails
Yi = B0 + B1Xi + ui
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Nonlinearity

19. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
P(X=x - Y=y) = P(X=x) * P(Y=y)

20. Variance of aX
Confidence set for two coefficients - two dimensional analog for the confidence interval
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
(a^2)(variance(x)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

21. POT
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Confidence level
Peaks over threshold - Collects dataset in excess of some threshold
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

22. Implied standard deviation for options
We reject a hypothesis that is actually true
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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!)

23. Variance(discrete)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Sampling distribution of sample means tend to be normal
(a^2)(variance(x)) + (b^2)(variance(y))

24. GEV
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Has heavy tails

25. Standard normal distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Transformed to a unit variable - Mean = 0 Variance = 1
Combine to form distribution with leptokurtosis (heavy tails)

26. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Choose parameters that maximize the likelihood of what observations occurring
Model dependent - Options with the same underlying assets may trade at different volatilities

27. Priori (classical) probability
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
Based on an equation - P(A) = # of A/total outcomes
Confidence set for two coefficients - two dimensional analog for the confidence interval
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

28. Historical std dev
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Attempts to sample along more important paths
Based on an equation - P(A) = # of A/total outcomes
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

29. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
P(X=x - Y=y) = P(X=x) * P(Y=y)
Independently and Identically Distributed
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

30. Cross - sectional
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Concerned with a single random variable (ex. Roll of a die)
Average return across assets on a given day

31. LAD
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
Choose parameters that maximize the likelihood of what observations occurring
Least absolute deviations estimator - used when extreme outliers are not uncommon
Population denominator = n - Sample denominator = n - 1

32. F distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
Easy to manipulate
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

33. Lognormal
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
(a^2)(variance(x)

34. Adjusted R^2
P - value
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Population denominator = n - Sample denominator = n - 1
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

35. Consistent
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
When the sample size is large - the uncertainty about the value of the sample is very small
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

36. Continuously compounded return equation
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Based on a dataset
i = ln(Si/Si - 1)

37. Variance - covariance approach for VaR of a portfolio
Statement of the error or precision of an estimate
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

38. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
Random walk (usually acceptable) - Constant volatility (unlikely)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Population denominator = n - Sample denominator = n - 1

39. Chi - squared distribution
For n>30 - sample mean is approximately normal
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Distribution with only two possible outcomes

40. Covariance calculations using weight sums (lambda)
Nonlinearity
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

41. Weibul distribution
Population denominator = n - Sample denominator = n - 1
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

42. Poisson Distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

43. Empirical frequency
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Based on a dataset
Concerned with a single random variable (ex. Roll of a die)

44. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
P(X=x - Y=y) = P(X=x) * P(Y=y)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

45. Variance of X - Y assuming dependence
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(X) + Variance(Y) - 2*covariance(XY)

46. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Combine to form distribution with leptokurtosis (heavy tails)

47. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Concerned with a single random variable (ex. Roll of a die)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

48. Two requirements of OVB
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance(x)

49. Reliability
If variance of the conditional distribution of u(i) is not constant
Statement of the error or precision of an estimate
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

50. Result of combination of two normal with same means
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)
Combine to form distribution with leptokurtosis (heavy tails)
Nonlinearity
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