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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. Two assumptions of square root rule
(a^2)(variance(x)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Random walk (usually acceptable) - Constant volatility (unlikely)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

2. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Variance(x) + Variance(Y) + 2*covariance(XY)
For n>30 - sample mean is approximately normal
(a^2)(variance(x)

3. Multivariate Density Estimation (MDE)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Contains variables not explicit in model - Accounts for randomness
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

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

5. 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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Variance(X) + Variance(Y) - 2*covariance(XY)
Statement of the error or precision of an estimate

6. What does the OLS minimize?
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)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
SSR

7. Central Limit Theorem
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
Low Frequency - High Severity events
For n>30 - sample mean is approximately normal
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

8. Econometrics
Rxy = Sxy/(Sx*Sy)
Low Frequency - High Severity events
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
Application of mathematical statistics to economic data to lend empirical support to models

9. Significance =1
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Confidence level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

10. Confidence ellipse
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

11. 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
Peaks over threshold - Collects dataset in excess of some threshold
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(x)

12. Importance sampling technique
Probability that the random variables take on certain values simultaneously
Attempts to sample along more important paths
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

13. Variance of X+Y assuming dependence
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Least absolute deviations estimator - used when extreme outliers are not uncommon
Variance(x) + Variance(Y) + 2*covariance(XY)

14. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Confidence set for two coefficients - two dimensional analog for the confidence interval
We reject a hypothesis that is actually true
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

15. Variance of X+Y
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Var(X) + Var(Y)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

16. Unbiased
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Mean of sampling distribution is the population 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!)

17. Marginal unconditional probability function
Does not depend on a prior event or information
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
Independently and Identically Distributed
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

18. Critical z values
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)
Concerned with a single random variable (ex. Roll of a die)
E(mean) = mean
95% = 1.65 99% = 2.33 For one - tailed tests

19. i.i.d.
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Rxy = Sxy/(Sx*Sy)
Independently and Identically Distributed
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

20. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Transformed to a unit variable - Mean = 0 Variance = 1
Regression can be non - linear in variables but must be linear in parameters
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

21. Unstable return distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance(x) + Variance(Y) + 2*covariance(XY)

22. Simplified standard (un - weighted) variance
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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!)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance = (1/m) summation(u<n - i>^2)

23. Standard variable for non - normal distributions
SSR
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Z = (Y - meany)/(stddev(y)/sqrt(n))
Summation((xi - mean)^k)/n

24. Discrete random variable
Special type of pooled data in which the cross sectional unit is surveyed over time
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sample mean +/ - t*(stddev(s)/sqrt(n))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

25. Sample variance
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
When the sample size is large - the uncertainty about the value of the sample is very small
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

26. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Only requires two parameters = mean and variance
Based on an equation - P(A) = # of A/total outcomes

27. Multivariate probability
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
More than one random variable
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Special type of pooled data in which the cross sectional unit is surveyed over time

28. Maximum likelihood method
Concerned with a single random variable (ex. Roll of a die)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Choose parameters that maximize the likelihood of what observations occurring

29. Type II Error
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
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
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

30. POT
Peaks over threshold - Collects dataset in excess of some threshold
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Population denominator = n - Sample denominator = n - 1

31. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance = (1/m) summation(u<n - i>^2)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

32. Mean reversion
Distribution with only two possible outcomes
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

33. Skewness
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)
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

34. Implications of homoscedasticity
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

35. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Combine to form distribution with leptokurtosis (heavy tails)
Mean = np - Variance = npq - Std dev = sqrt(npq)
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

36. Central Limit Theorem(CLT)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
If variance of the conditional distribution of u(i) is not constant
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
Sampling distribution of sample means tend to be normal

37. Gamma distribution
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
When the sample size is large - the uncertainty about the value of the sample is very small

38. Non - parametric vs parametric calculation of VaR
Contains variables not explicit in model - Accounts for randomness
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

39. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Based on a dataset
Variance(y)/n = variance of sample Y
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

40. Least squares estimator(m)
Variance reverts to a long run level
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Nonlinearity

41. Test for statistical independence
If variance of the conditional distribution of u(i) is not constant
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Var(X) + Var(Y)
P(X=x - Y=y) = P(X=x) * P(Y=y)

42. Lognormal
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Random walk (usually acceptable) - Constant volatility (unlikely)
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

43. Type I error
Average return across assets on a given day
Mean = np - Variance = npq - Std dev = sqrt(npq)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
We reject a hypothesis that is actually true

44. Bernouli Distribution
Distribution with only two possible outcomes
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

45. Priori (classical) probability
Attempts to sample along more important paths
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Based on an equation - P(A) = # of A/total outcomes

46. K - th moment
Summation((xi - mean)^k)/n
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

47. Shortcomings of implied volatility
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
i = ln(Si/Si - 1)
Model dependent - Options with the same underlying assets may trade at different volatilities

48. Variance of aX + bY
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
(a^2)(variance(x)) + (b^2)(variance(y))

49. Monte Carlo Simulations
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(x) + Variance(Y) + 2*covariance(XY)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance(X) + Variance(Y) - 2*covariance(XY)

50. Block maxima
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Returns over time for an individual asset
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2