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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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1. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Independently and Identically Distributed
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

2. Variance of aX
(a^2)(variance(x)
Use historical simulation approach but use the EWMA weighting system
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)

3. Continuously compounded return equation
Expected value of the sample mean is the population mean
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
i = ln(Si/Si - 1)
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

4. Heteroskedastic
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
If variance of the conditional distribution of u(i) is not constant
Does not depend on a prior event or information
Normal - Student's T - Chi - square - F distribution

5. Antithetic variable technique
Returns over time for an individual asset
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

6. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/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!)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

7. Confidence interval for sample mean
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
SSR

8. Variance - covariance approach for VaR of a portfolio
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)
Variance(x)
Has heavy tails

9. Biggest (and only real) drawback of GARCH mode
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Nonlinearity
Contains variables not explicit in model - Accounts for randomness

10. Econometrics
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Application of mathematical statistics to economic data to lend empirical support to models
Distribution with only two possible outcomes
Does not depend on a prior event or information

11. Potential reasons for fat tails in return distributions
Least absolute deviations estimator - used when extreme outliers are not uncommon
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Probability that the random variables take on certain values simultaneously

12. Unstable return distribution
Based on a dataset
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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

13. Single variable (univariate) probability
Contains variables not explicit in model - Accounts for randomness
Concerned with a single random variable (ex. Roll of a die)
Confidence level
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

14. Unconditional vs conditional distributions
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Among all unbiased estimators - estimator with the smallest variance is efficient
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

15. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth 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
When one regressor is a perfect linear function of the other regressors
Expected value of the sample mean is the population mean

16. Multivariate probability
Distribution with only two possible outcomes
Summation((xi - mean)^k)/n
Z = (Y - meany)/(stddev(y)/sqrt(n))
More than one random variable

17. Binomial distribution equations for mean variance and std dev
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Mean = np - Variance = npq - Std dev = sqrt(npq)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

18. Standard error for Monte Carlo replications
P(Z>t)
Attempts to sample along more important paths
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

19. Direction of OVB
(a^2)(variance(x)) + (b^2)(variance(y))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Returns over time for a combination of assets (combination of time series and cross - sectional data)

20. Implications of homoscedasticity
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
Has heavy tails
Application of mathematical statistics to economic data to lend empirical support to models
Var(X) + Var(Y)

21. Binomial distribution
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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!)
Sampling distribution of sample means tend to be normal
95% = 1.65 99% = 2.33 For one - tailed tests

22. Marginal unconditional probability function
Variance(X) + Variance(Y) - 2*covariance(XY)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Does not depend on a prior event or information
Transformed to a unit variable - Mean = 0 Variance = 1

23. Persistence
Only requires two parameters = mean and variance
Variance reverts to a long run level
Variance(X) + Variance(Y) - 2*covariance(XY)
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

24. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

25. Statistical (or empirical) model
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Use historical simulation approach but use the EWMA weighting system
Yi = B0 + B1Xi + ui
Expected value of the sample mean is the population mean

26. Gamma distribution
SSR
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

27. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Easy to manipulate
Mean of sampling distribution is the population mean

28. Mean reversion in asset dynamics
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Price/return tends to run towards a long - run level
Model dependent - Options with the same underlying assets may trade at different volatilities

29. Cross - sectional
Variance reverts to a long run level
Regression can be non - linear in variables but must be linear in parameters
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Average return across assets on a given day

30. Type II Error
We accept a hypothesis that should have been rejected
Based on a dataset
Sample mean +/ - t*(stddev(s)/sqrt(n))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

31. Normal distribution
E(mean) = mean
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Statement of the error or precision of an estimate
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

32. Bernouli Distribution
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!)
Distribution with only two possible outcomes
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

33. Monte Carlo Simulations
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

34. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Regression can be non - linear in variables but must be linear in parameters
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

35. Importance sampling technique
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Attempts to sample along more important paths
Expected value of the sample mean is the population mean
Variance(X) + Variance(Y) - 2*covariance(XY)

36. Overall F - statistic
P - value
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
Normal - Student's T - Chi - square - F distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

37. Four sampling distributions

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38. Skewness
When one regressor is a perfect linear function of the other regressors
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

39. Hybrid method for conditional volatility
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
Use historical simulation approach but use the EWMA weighting system
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Based on a dataset

40. Unbiased
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
SSR
Mean of sampling distribution is the population mean

41. Variance of X+b
Variance(x)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
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

42. Discrete representation of the GBM
Rxy = Sxy/(Sx*Sy)
Variance(X) + Variance(Y) - 2*covariance(XY)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

43. Efficiency
Only requires two parameters = mean and variance
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Combine to form distribution with leptokurtosis (heavy tails)

44. Type I error
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
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
We reject a hypothesis that is actually true
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

45. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Nonlinearity
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Based on a dataset

46. POT
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
Peaks over threshold - Collects dataset in excess of some threshold
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)
Average return across assets on a given day

47. Variance of weighted scheme
Model dependent - Options with the same underlying assets may trade at different volatilities
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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

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

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49. Poisson distribution equations for mean variance and std deviation
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
(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)

50. Control variates technique
Sample mean +/ - t*(stddev(s)/sqrt(n))
Sampling distribution of sample means tend to be normal
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size