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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. 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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Normal - Student's T - Chi - square - F distribution
Nonlinearity

2. Statistical (or empirical) model
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Yi = B0 + B1Xi + ui
Variance reverts to a long run level

3. Type I error
Sampling distribution of sample means tend to be normal
We reject a hypothesis that is actually true
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

4. Variance(discrete)
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
Probability that the random variables take on certain values simultaneously
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

5. Central Limit Theorem
Mean = np - Variance = npq - Std dev = sqrt(npq)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
For n>30 - sample mean is approximately normal
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

6. WLS
Does not depend on a prior event or information
Among all unbiased estimators - estimator with the smallest variance is efficient
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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

7. Sample variance
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)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance(y)/n = variance of sample Y

8. Continuously compounded return equation
i = ln(Si/Si - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
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

9. Beta distribution
Attempts to sample along more important paths
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Least absolute deviations estimator - used when extreme outliers are not uncommon

10. Mean(expected value)
Only requires two parameters = mean and variance
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Application of mathematical statistics to economic data to lend empirical support to models

11. Result of combination of two normal with same means
Average return across assets on a given day
Variance reverts to a long run level
Combine to form distribution with leptokurtosis (heavy tails)
Confidence set for two coefficients - two dimensional analog for the confidence interval

12. Simulating for VaR
For n>30 - sample mean is approximately normal
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)
When the sample size is large - the uncertainty about the value of the sample is very small
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

13. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Distribution with only two possible outcomes
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Confidence set for two coefficients - two dimensional analog for the confidence interval

14. Importance sampling 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
Regression can be non - linear in variables but must be linear in parameters
Attempts to sample along more important paths
Population denominator = n - Sample denominator = n - 1

15. Heteroskedastic
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
If variance of the conditional distribution of u(i) is not constant
Sampling distribution of sample means tend to be normal
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

16. Binomial distribution
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!)
Rxy = Sxy/(Sx*Sy)
Model dependent - Options with the same underlying assets may trade at different volatilities
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

17. Non - parametric vs parametric calculation of VaR
Does not depend on a prior event or information
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Concerned with a single random variable (ex. Roll of a die)

18. Variance of sampling distribution of means when n<N
Variance(y)/n = variance of sample Y
Distribution with only two possible outcomes
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

19. Law of Large Numbers
Combine to form distribution with leptokurtosis (heavy tails)
Has heavy tails
Sample mean will near the population mean as the sample size increases
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

20. Skewness
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Application of mathematical statistics to economic data to lend empirical support to models
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
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. Variance of X+Y assuming dependence
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(x) + Variance(Y) + 2*covariance(XY)
Var(X) + Var(Y)

22. Conditional probability functions
Average return across assets on a given day
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Independently and Identically Distributed
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

23. P - value
P(Z>t)
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
For n>30 - sample mean is approximately normal

24. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Sample mean +/ - t*(stddev(s)/sqrt(n))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Mean of sampling distribution is the population mean

25. Kurtosis
Sample mean will near the population mean as the sample size increases
Low Frequency - High Severity events
Sampling distribution of sample means tend to be normal
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

26. Variance of aX + bY
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
(a^2)(variance(x)) + (b^2)(variance(y))

27. Empirical frequency
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Based on a dataset
Choose parameters that maximize the likelihood of what observations occurring
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

28. Simplified standard (un - weighted) variance
Variance reverts to a long run level
We accept a hypothesis that should have been rejected
Statement of the error or precision of an estimate
Variance = (1/m) summation(u<n - i>^2)

29. Simulation models
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
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)
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

30. Variance of sample mean
Variance(y)/n = variance of sample Y
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
We accept a hypothesis that should have been rejected

31. SER
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

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

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33. ESS
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Concerned with a single random variable (ex. Roll of a die)

34. Extending the HS approach for computing value of a portfolio

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35. EWMA
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Contains variables not explicit in model - Accounts for randomness
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

36. Implied standard deviation for options
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Application of mathematical statistics to economic data to lend empirical support to models
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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)

37. Sample correlation
Sample mean +/ - t*(stddev(s)/sqrt(n))
When one regressor is a perfect linear function of the other regressors
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Rxy = Sxy/(Sx*Sy)

38. Stochastic error term
Mean = np - Variance = npq - Std dev = sqrt(npq)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Contains variables not explicit in model - Accounts for randomness
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

39. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Sample mean will near the population mean as the sample size increases
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
Rxy = Sxy/(Sx*Sy)

40. Historical std dev
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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!)

41. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Peaks over threshold - Collects dataset in excess of some threshold
Normal - Student's T - Chi - square - F distribution

42. Unconditional vs conditional distributions
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)
Has heavy tails
Confidence set for two coefficients - two dimensional analog for the confidence interval

43. Potential reasons for fat tails in return distributions
Confidence level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
We accept a hypothesis that should have been rejected

44. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Does not depend on a prior event or information
P - value

45. i.i.d.
Variance reverts to a long run level
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
E(XY) - E(X)E(Y)
Independently and Identically Distributed

46. Test for statistical independence
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
SSR
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

47. Persistence
We reject a hypothesis that is actually true
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Average return across assets on a given day

48. Shortcomings of implied volatility
95% = 1.65 99% = 2.33 For one - tailed tests
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Nonlinearity
Model dependent - Options with the same underlying assets may trade at different volatilities

49. Cross - sectional
Population denominator = n - Sample denominator = n - 1
Application of mathematical statistics to economic data to lend empirical support to models
Average return across assets on a given day
Sampling distribution of sample means tend to be normal

50. Square root rule
Population denominator = n - Sample denominator = n - 1
Based on an equation - P(A) = # of A/total outcomes
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under 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