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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. Priori (classical) probability
More than one random variable
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Based on an equation - P(A) = # of A/total outcomes

2. Variance of sampling distribution of means when n<N
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

3. Variance of X+Y assuming dependence
Independently and Identically Distributed
We accept a hypothesis that should have been rejected
Variance reverts to a long run level
Variance(x) + Variance(Y) + 2*covariance(XY)

4. Conditional probability functions
When the sample size is large - the uncertainty about the value of the sample is very small
Peaks over threshold - Collects dataset in excess of some threshold
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
P - value

5. Variance of sample mean
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(y)/n = variance of sample Y
Probability that the random variables take on certain values simultaneously
Attempts to sample along more important paths

6. Simplified standard (un - weighted) variance
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
Only requires two parameters = mean and variance
Returns over time for an individual asset
Variance = (1/m) summation(u<n - i>^2)

7. Biggest (and only real) drawback of GARCH mode
Variance = (1/m) summation(u<n - i>^2)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Nonlinearity

8. Critical z values
Peaks over threshold - Collects dataset in excess of some threshold
95% = 1.65 99% = 2.33 For one - tailed tests
Var(X) + Var(Y)
Confidence set for two coefficients - two dimensional analog for the confidence interval

9. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
E(XY) - E(X)E(Y)
Among all unbiased estimators - estimator with the smallest variance is efficient
We accept a hypothesis that should have been rejected

10. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance(x)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Average return across assets on a given day

11. Mean reversion in asset dynamics
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Price/return tends to run towards a long - run level
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

12. POT
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)
Peaks over threshold - Collects dataset in excess of some threshold
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Only requires two parameters = mean and variance

13. Mean reversion in variance
More than one random variable
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance reverts to a long run level

14. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Choose parameters that maximize the likelihood of what observations occurring
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

15. Sample variance
Distribution with only two possible outcomes
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Normal - Student's T - Chi - square - F distribution

16. 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
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
Choose parameters that maximize the likelihood of what observations occurring
E(XY) - E(X)E(Y)

17. Square root rule
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Rxy = Sxy/(Sx*Sy)
Variance(x) + Variance(Y) + 2*covariance(XY)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

18. Least squares estimator(m)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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

19. Shortcomings of implied volatility
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
i = ln(Si/Si - 1)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Model dependent - Options with the same underlying assets may trade at different volatilities

20. Discrete random variable
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)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

21. Homoskedastic
Attempts to sample along more important paths
Only requires two parameters = mean and variance
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

22. Two assumptions of square root rule
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Combine to form distribution with leptokurtosis (heavy tails)
Random walk (usually acceptable) - Constant volatility (unlikely)

23. Antithetic variable technique
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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
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

24. Non - parametric vs parametric calculation of VaR
Sampling distribution of sample means tend to be normal
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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

25. Historical std dev
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Based on a dataset
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

26. F distribution
Confidence level
E(mean) = mean
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

27. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
(a^2)(variance(x)) + (b^2)(variance(y))
(a^2)(variance(x)

28. LFHS
Returns over time for an individual asset
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Low Frequency - High Severity events

29. Unstable return distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Among all unbiased estimators - estimator with the smallest variance is efficient
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

30. Reliability
More than one random variable
Statement of the error or precision of an estimate
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance(X) + Variance(Y) - 2*covariance(XY)

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

32. Persistence
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
E(mean) = mean
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

33. What does the OLS minimize?
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance(y)/n = variance of sample Y
SSR
Normal - Student's T - Chi - square - F distribution

34. Lognormal
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
95% = 1.65 99% = 2.33 For one - tailed tests

35. Type II Error
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
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
We accept a hypothesis that should have been rejected
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

36. Multivariate probability
More than one random variable
Application of mathematical statistics to economic data to lend empirical support to models
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Z = (Y - meany)/(stddev(y)/sqrt(n))

37. Poisson distribution equations for mean variance and std deviation
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Returns over time for an individual asset
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

38. Sample mean
Expected value of the sample mean is the population mean
95% = 1.65 99% = 2.33 For one - tailed tests
Independently and Identically Distributed
Only requires two parameters = mean and variance

39. LAD
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(y)/n = variance of sample Y
P(Z>t)
Least absolute deviations estimator - used when extreme outliers are not uncommon

40. Bernouli Distribution
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
Distribution with only two possible outcomes

41. Marginal unconditional probability function
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
Expected value of the sample mean is the population mean
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Does not depend on a prior event or information

42. Confidence ellipse
SSR
Based on a dataset
Confidence set for two coefficients - two dimensional analog for the confidence interval
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

43. Skewness
Among all unbiased estimators - estimator with the smallest variance is efficient
Combine to form distribution with leptokurtosis (heavy tails)
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
Use historical simulation approach but use the EWMA weighting system

44. Key properties of linear regression
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
Sample mean will near the population mean as the sample size increases
Attempts to sample along more important paths
Regression can be non - linear in variables but must be linear in parameters

45. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

46. Unbiased
Sample mean +/ - t*(stddev(s)/sqrt(n))
Mean of sampling distribution is the population mean
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

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

48. GARCH
Concerned with a single random variable (ex. Roll of a die)
(a^2)(variance(x)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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)

49. Adjusted R^2
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Peaks over threshold - Collects dataset in excess of some threshold
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

50. Standard error
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
P(Z>t)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)