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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. Inverse transform method
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
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 Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

2. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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!)

3. Monte Carlo Simulations
Model dependent - Options with the same underlying assets may trade at different volatilities
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
E(mean) = mean

4. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Does not depend on a prior event or information
Average return across assets on a given day
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

5. F distribution
Combine to form distribution with leptokurtosis (heavy tails)
Based on a dataset
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
Based on an equation - P(A) = # of A/total outcomes

6. Cross - sectional
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Only requires two parameters = mean and variance
Average return across assets on a given day
Low Frequency - High Severity events

7. Implied standard deviation for options
Average return across assets on a given day
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Contains variables not explicit in model - Accounts for randomness

8. Normal distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
When the sample size is large - the uncertainty about the value of the sample is very small
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

9. Two ways to calculate historical volatility
Concerned with a single random variable (ex. Roll of a die)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

10. Tractable
Transformed to a unit variable - Mean = 0 Variance = 1
Easy to manipulate
Population denominator = n - Sample denominator = n - 1
When one regressor is a perfect linear function of the other regressors

11. Law of Large Numbers
Does not depend on a prior event or information
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Sample mean will near the population mean as the sample size increases
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

12. Maximum likelihood method
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Variance(x)
Choose parameters that maximize the likelihood of what observations occurring

13. Potential reasons for fat tails in return distributions
Variance = (1/m) summation(u<n - i>^2)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

14. Kurtosis
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Application of mathematical statistics to economic data to lend empirical support to models

15. Central Limit Theorem(CLT)
Yi = B0 + B1Xi + ui
Sampling distribution of sample means tend to be normal
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

16. Variance of weighted scheme
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

17. Type I error
We reject a hypothesis that is actually true
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

18. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
When one regressor is a perfect linear function of the other regressors
Special type of pooled data in which the cross sectional unit is surveyed over time

19. Difference between population and sample variance
Variance = (1/m) summation(u<n - i>^2)
Population denominator = n - Sample denominator = n - 1
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Attempts to sample along more important paths

20. Continuous representation of the GBM
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of 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)
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)

21. Result of combination of two normal with same means
Only requires two parameters = mean and variance
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Combine to form distribution with leptokurtosis (heavy tails)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

22. Mean reversion in variance
Variance reverts to a long run level
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
If variance of the conditional distribution of u(i) is not constant
Application of mathematical statistics to economic data to lend empirical support to models

23. Empirical frequency
Concerned with a single random variable (ex. Roll of a die)
Based on a dataset
Yi = B0 + B1Xi + ui
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

24. Cholesky factorization (decomposition)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Easy to manipulate

25. LFHS
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
If variance of the conditional distribution of u(i) is not constant
Low Frequency - High Severity events

26. Unbiased
E(mean) = mean
Mean of sampling distribution is the population mean
When one regressor is a perfect linear function of the other regressors
Based on a dataset

27. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^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
For n>30 - sample mean is approximately normal
P(X=x - Y=y) = P(X=x) * P(Y=y)

28. Central Limit Theorem
Sampling distribution of sample means tend to be normal
For n>30 - sample mean is approximately normal
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Model dependent - Options with the same underlying assets may trade at different volatilities

29. Historical std dev
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Z = (Y - meany)/(stddev(y)/sqrt(n))
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

30. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(x) + Variance(Y) + 2*covariance(XY)
Contains variables not explicit in model - Accounts for randomness
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

31. Biggest (and only real) drawback of GARCH mode
Nonlinearity
E(mean) = mean
Special type of pooled data in which the cross sectional unit is surveyed over time
Mean = np - Variance = npq - Std dev = sqrt(npq)

32. Sample mean
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Has heavy tails
Expected value of the sample mean is the population mean

33. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Least absolute deviations estimator - used when extreme outliers are not uncommon
P(X=x - Y=y) = P(X=x) * P(Y=y)
Transformed to a unit variable - Mean = 0 Variance = 1

34. Chi - squared distribution
95% = 1.65 99% = 2.33 For one - tailed tests
Use historical simulation approach but use the EWMA weighting system
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

35. GPD
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

36. SER
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

37. Persistence
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Population denominator = n - Sample denominator = n - 1

38. Variance - covariance approach for VaR of a portfolio
Transformed to a unit variable - Mean = 0 Variance = 1
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Expected value of the sample mean is the population mean

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

40. Test for unbiasedness
E(mean) = mean
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance reverts to a long run level
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)

41. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

42. Two assumptions of square root rule
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Random walk (usually acceptable) - Constant volatility (unlikely)
Statement of the error or precision of an estimate
Low Frequency - High Severity events

43. Sample correlation
Transformed to a unit variable - Mean = 0 Variance = 1
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)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Rxy = Sxy/(Sx*Sy)

44. Marginal unconditional probability function
Variance = (1/m) summation(u<n - i>^2)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Does not depend on a prior event or information
P(X=x - Y=y) = P(X=x) * P(Y=y)

45. Covariance
Attempts to sample along more important paths
E(XY) - E(X)E(Y)
i = ln(Si/Si - 1)
For n>30 - sample mean is approximately normal

46. Confidence interval for sample mean
Population denominator = n - Sample denominator = n - 1
Based on an equation - P(A) = # of A/total outcomes
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Normal - Student's T - Chi - square - F distribution

47. Single variable (univariate) probability
Mean of sampling distribution is the population mean
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Normal - Student's T - Chi - square - F distribution
Concerned with a single random variable (ex. Roll of a die)

48. Mean(expected value)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
P(X=x - Y=y) = P(X=x) * P(Y=y)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

49. Time series data
For n>30 - sample mean is approximately normal
Returns over time for an individual asset
i = ln(Si/Si - 1)
Has heavy tails

50. Exponential distribution
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Price/return tends to run towards a long - run level
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta