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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. Perfect multicollinearity
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
When one regressor is a perfect linear function of the other regressors
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

2. T distribution
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
We accept a hypothesis that should have been rejected
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

3. Unbiased
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Confidence level
Mean of sampling distribution is the population mean

4. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
We accept a hypothesis that should have been rejected
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

5. Marginal unconditional probability function
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)
Does not depend on a prior event or information
P - value

6. K - th moment
Combine to form distribution with leptokurtosis (heavy tails)
For n>30 - sample mean is approximately normal
Summation((xi - mean)^k)/n
Regression can be non - linear in variables but must be linear in parameters

7. Two requirements of OVB
Price/return tends to run towards a long - run level
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
When one regressor is a perfect linear function of the other regressors

8. Sample mean
Expected value of the sample mean is the population mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Normal - Student's T - Chi - square - F distribution

9. Exponential distribution
E(mean) = mean
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
P - value
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

10. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Has heavy tails
Easy to manipulate

11. LAD
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sampling distribution of sample means tend to be 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
Least absolute deviations estimator - used when extreme outliers are not uncommon

12. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Random walk (usually acceptable) - Constant volatility (unlikely)
Normal - Student's T - Chi - square - F distribution
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

13. Biggest (and only real) drawback of GARCH mode
Application of mathematical statistics to economic data to lend empirical support to models
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
Nonlinearity
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

14. Normal distribution
SSR
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

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

16. WLS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

17. Adjusted R^2
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
Based on an equation - P(A) = # of A/total outcomes
Sample mean +/ - t*(stddev(s)/sqrt(n))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

18. Mean reversion in asset dynamics
Low Frequency - High Severity events
Sampling distribution of sample means tend to be normal
Use historical simulation approach but use the EWMA weighting system
Price/return tends to run towards a long - run level

19. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Choose parameters that maximize the likelihood of what observations occurring
i = ln(Si/Si - 1)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

20. GPD
Nonlinearity
Least absolute deviations estimator - used when extreme outliers are not uncommon
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!)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

21. Panel data (longitudinal or micropanel)
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!)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance = (1/m) summation(u<n - i>^2)
Special type of pooled data in which the cross sectional unit is surveyed over time

22. Mean reversion
When one regressor is a perfect linear function of the other regressors
Mean of sampling distribution is the population mean
Variance(X) + Variance(Y) - 2*covariance(XY)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

23. Standard error for Monte Carlo replications
Transformed to a unit variable - Mean = 0 Variance = 1
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

24. POT
Peaks over threshold - Collects dataset in excess of some threshold
Variance reverts to a long run level
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

25. Continuous representation of the GBM
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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)
Application of mathematical statistics to economic data to lend empirical support to models

26. Reliability
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Summation((xi - mean)^k)/n
Statement of the error or precision of an estimate
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

27. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Regression can be non - linear in variables but must be linear in parameters
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())

28. Exact significance level
Least absolute deviations estimator - used when extreme outliers are not uncommon
P - value
Var(X) + Var(Y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

29. Simplified standard (un - weighted) variance
E(mean) = mean
Variance = (1/m) summation(u<n - i>^2)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

30. Direction of OVB
If variance of the conditional distribution of u(i) is not constant
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

31. Four sampling distributions

32. Mean reversion in variance
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Variance reverts to a long run level
Average return across assets on a given day

33. Pooled data
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Sampling distribution of sample means tend to be normal
Low Frequency - High Severity events

34. Logistic distribution
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)
Mean of sampling distribution is the population mean
Has heavy tails
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

35. F 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
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
Based on a dataset
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

36. i.i.d.
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)
Rxy = Sxy/(Sx*Sy)
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

37. Type II Error
Z = (Y - meany)/(stddev(y)/sqrt(n))
We accept a hypothesis that should have been rejected
Low Frequency - High Severity events
Mean of sampling distribution is the population mean

38. Overall F - statistic
Summation((xi - mean)^k)/n
When the sample size is large - the uncertainty about the value of the sample is very small
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Normal - Student's T - Chi - square - F distribution

39. Economical(elegant)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Only requires two parameters = mean and variance
(a^2)(variance(x)

40. Expected future variance rate (t periods forward)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Mean of sampling distribution is the population mean
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

41. Variance of X - Y assuming dependence
Has heavy tails
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
P - value
Variance(X) + Variance(Y) - 2*covariance(XY)

42. Kurtosis
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Expected value of the sample mean is the population mean
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

43. Variance of X+Y assuming dependence
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
Sampling distribution of sample means tend to be normal
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
Variance(x) + Variance(Y) + 2*covariance(XY)

44. Gamma distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
More than one random variable
When the sample size is large - the uncertainty about the value of the sample is very small
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

45. Antithetic variable technique
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Confidence level

46. 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
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)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

47. Confidence interval (from t)
We accept a hypothesis that should have been rejected
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

48. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

49. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Application of mathematical statistics to economic data to lend empirical support to models
Rxy = Sxy/(Sx*Sy)

50. Difference between population and sample variance
Probability that the random variables take on certain values simultaneously
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Population denominator = n - Sample denominator = n - 1
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state