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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. Joint probability functions
Probability that the random variables take on certain values simultaneously
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
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
Expected value of the sample mean is the population mean

2. Variance of X+Y assuming dependence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Confidence set for two coefficients - two dimensional analog for the confidence interval
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance(x) + Variance(Y) + 2*covariance(XY)

3. Gamma 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
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
Based on a dataset

4. Beta distribution
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
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance reverts to a long run level

5. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
More than one random variable
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

6. Covariance calculations using weight sums (lambda)
Distribution with only two possible outcomes
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance reverts to a long run level
Mean = np - Variance = npq - Std dev = sqrt(npq)

7. Pooled data
Variance(y)/n = variance of sample Y
Model dependent - Options with the same underlying assets may trade at different volatilities
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

8. Binomial distribution
SSR
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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!)

9. Standard variable for non - normal distributions
Variance(x) + Variance(Y) + 2*covariance(XY)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Contains variables not explicit in model - Accounts for randomness
Z = (Y - meany)/(stddev(y)/sqrt(n))

10. Implications of homoscedasticity
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
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
Sample mean will near the population mean as the sample size increases

11. Sample correlation
E(XY) - E(X)E(Y)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Rxy = Sxy/(Sx*Sy)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

12. Efficiency
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Among all unbiased estimators - estimator with the smallest variance is efficient

13. Variance of X+Y
E(mean) = mean
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!)
Var(X) + Var(Y)
Z = (Y - meany)/(stddev(y)/sqrt(n))

14. Cholesky factorization (decomposition)
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

15. Regime - switching volatility model
Use historical simulation approach but use the EWMA weighting system
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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

16. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
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)
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

17. Monte Carlo Simulations
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance(y)/n = variance of sample Y
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

18. Overall F - statistic
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

19. Expected future variance rate (t periods forward)
Variance reverts to a long run level
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sampling distribution of sample means tend to be normal
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

20. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Expected value of the sample mean is the population mean
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Average return across assets on a given day

21. ESS
We accept a hypothesis that should have been rejected
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

22. Priori (classical) probability
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
When one regressor is a perfect linear function of the other regressors
Based on an equation - P(A) = # of A/total outcomes
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

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
Only requires two parameters = mean and variance
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
E(mean) = mean

24. Cross - sectional
Average return across assets on a given day
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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 = (1/m) summation(u<n - i>^2)

25. Two ways to calculate historical volatility
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
Variance reverts to a long run level
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

26. SER
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance(x)
Confidence level

27. Key properties of linear regression
P(Z>t)
Regression can be non - linear in variables but must be linear in parameters
When the sample size is large - the uncertainty about the value of the sample is very small
Var(X) + Var(Y)

28. Lognormal
P(Z>t)
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

29. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

30. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Independently and Identically Distributed
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
We accept a hypothesis that should have been rejected

31. i.i.d.
Independently and Identically Distributed
Yi = B0 + B1Xi + ui
P(X=x - Y=y) = P(X=x) * P(Y=y)
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)

32. Continuously compounded return equation
Random walk (usually acceptable) - Constant volatility (unlikely)
i = ln(Si/Si - 1)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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

33. Hybrid method for conditional volatility
Mean of sampling distribution is the population mean
Rxy = Sxy/(Sx*Sy)
Use historical simulation approach but use the EWMA weighting system
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

34. What does the OLS minimize?
Low Frequency - High Severity events
Among all unbiased estimators - estimator with the smallest variance is efficient
SSR
Normal - Student's T - Chi - square - F distribution

35. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
When the sample size is large - the uncertainty about the value of the sample is very small
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

36. Variance of X - Y assuming dependence
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

37. Test for statistical independence
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Confidence level
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

38. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Statement of the error or precision of an estimate
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
Variance = (1/m) summation(u<n - i>^2)

39. Antithetic variable technique
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

40. Tractable
(a^2)(variance(x)
Attempts to sample along more important paths
Easy to manipulate
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

41. Deterministic Simulation
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
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

42. Economical(elegant)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Only requires two parameters = mean and variance
Variance(y)/n = variance of sample Y
Use historical simulation approach but use the EWMA weighting system

43. Consistent
Model dependent - Options with the same underlying assets may trade at different volatilities
When the sample size is large - the uncertainty about the value of the sample is very small
Confidence level
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

44. Chi - squared distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Among all unbiased estimators - estimator with the smallest variance is efficient
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

45. Sample mean
P(X=x - Y=y) = P(X=x) * P(Y=y)
Expected value of the sample mean is the population mean
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)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

46. Adjusted R^2
E(mean) = mean
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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

47. Unbiased
Application of mathematical statistics to economic data to lend empirical support to models
Normal - Student's T - Chi - square - F distribution
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
Mean of sampling distribution is the population mean

48. Test for unbiasedness
Normal - Student's T - Chi - square - F distribution
Attempts to sample along more important paths
E(mean) = mean
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

49. Block maxima
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(X) + Variance(Y) - 2*covariance(XY)
Independently and Identically Distributed
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

50. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Variance(x) + Variance(Y) + 2*covariance(XY)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails