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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. Central Limit Theorem
Random walk (usually acceptable) - Constant volatility (unlikely)
Statement of the error or precision of an estimate
For n>30 - sample mean is approximately normal
Returns over time for a combination of assets (combination of time series and cross - sectional data)

2. Two requirements of OVB
If variance of the conditional distribution of u(i) is not constant
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

3. Variance of aX
Price/return tends to run towards a long - run level
(a^2)(variance(x)
i = ln(Si/Si - 1)
We accept a hypothesis that should have been rejected

4. GPD
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance = (1/m) summation(u<n - i>^2)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Least absolute deviations estimator - used when extreme outliers are not uncommon

5. Variance of aX + bY
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
Variance(X) + Variance(Y) - 2*covariance(XY)
(a^2)(variance(x)) + (b^2)(variance(y))
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

6. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Sampling distribution of sample means tend to be normal
Distribution with only two possible outcomes
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

7. Key properties of linear regression
(a^2)(variance(x)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Easy to manipulate
Regression can be non - linear in variables but must be linear in parameters

8. SER
Confidence level
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!)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

9. Skewness
(a^2)(variance(x)) + (b^2)(variance(y))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Has 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

10. Maximum likelihood method
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Easy to manipulate
Choose parameters that maximize the likelihood of what observations occurring
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

11. Inverse transform method
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Least absolute deviations estimator - used when extreme outliers are not uncommon
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

12. Square root rule
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

13. Perfect multicollinearity
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
We reject a hypothesis that is actually true
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
When one regressor is a perfect linear function of the other regressors

14. Poisson distribution equations for mean variance and std deviation
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Low Frequency - High Severity events
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

15. Multivariate probability
Average return across assets on a given day
More than one random variable
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Confidence level

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

17. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Distribution with only two possible outcomes
Low Frequency - High Severity events
Statement of the error or precision of an estimate

18. Exponential distribution
Population denominator = n - Sample denominator = n - 1
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Model dependent - Options with the same underlying assets may trade at different volatilities

19. GEV
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Price/return tends to run towards a long - run level
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

20. WLS
Confidence set for two coefficients - two dimensional analog for the confidence interval
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
P(X=x - Y=y) = P(X=x) * P(Y=y)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

21. Importance sampling technique
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Model dependent - Options with the same underlying assets may trade at different volatilities
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Attempts to sample along more important paths

22. Beta distribution
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Least absolute deviations estimator - used when extreme outliers are not uncommon
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

23. Covariance
E(XY) - E(X)E(Y)
Population denominator = n - Sample denominator = n - 1
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
Concerned with a single random variable (ex. Roll of a die)

24. Regime - switching volatility model
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Choose parameters that maximize the likelihood of what observations occurring
Returns over time for an individual asset
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

25. F distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
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

26. Law of Large Numbers
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
Sample mean will near the population mean as the sample size increases
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

27. Standard normal distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Transformed to a unit variable - Mean = 0 Variance = 1
Nonlinearity
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

28. Simulation models
Sampling distribution of sample means tend to be normal
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"
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

29. Test for unbiasedness
We reject a hypothesis that is actually true
Choose parameters that maximize the likelihood of what observations occurring
For n>30 - sample mean is approximately normal
E(mean) = mean

30. Variance of X+Y assuming dependence
P(Z>t)
Variance(x) + Variance(Y) + 2*covariance(XY)
i = ln(Si/Si - 1)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

31. Single variable (univariate) probability
Confidence level
Transformed to a unit variable - Mean = 0 Variance = 1
Concerned with a single random variable (ex. Roll of a die)
Yi = B0 + B1Xi + ui

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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Only requires two parameters = mean and variance
Probability that the random variables take on certain values simultaneously

33. LFHS
Low Frequency - High Severity events
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Has heavy tails
Variance(x)

34. Variance of X - Y assuming dependence
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
If variance of the conditional distribution of u(i) is not constant
Statement of the error or precision of an estimate
Variance(X) + Variance(Y) - 2*covariance(XY)

35. Sample variance
Attempts to sample along more important paths
Variance(x) + Variance(Y) + 2*covariance(XY)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Concerned with a single random variable (ex. Roll of a die)

36. Unstable return distribution
Attempts to sample along more important paths
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

37. Monte Carlo Simulations
Based on a dataset
Probability that the random variables take on certain values simultaneously
Has heavy tails
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

38. Implied standard deviation for options
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)
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.
Transformed to a unit variable - Mean = 0 Variance = 1

39. Logistic distribution
Has heavy tails
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Confidence level

40. Mean(expected value)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

41. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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!)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Sample mean will near the population mean as the sample size increases

42. Result of combination of two normal with same means
Mean = np - Variance = npq - Std dev = sqrt(npq)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Combine to form distribution with leptokurtosis (heavy tails)

43. Confidence interval for sample mean
When one regressor is a perfect linear function of the other regressors
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

44. Test for statistical independence
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

45. Hazard rate of exponentially distributed random variable
SSR
Easy to manipulate
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Expected value of the sample mean is the population mean

46. Potential reasons for fat tails in return distributions
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Normal - Student's T - Chi - square - F distribution
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Yi = B0 + B1Xi + ui

47. GARCH
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)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
i = ln(Si/Si - 1)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

48. Least squares estimator(m)
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
Easy to manipulate
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Special type of pooled data in which the cross sectional unit is surveyed over time
P(X=x - Y=y) = P(X=x) * P(Y=y)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

50. Biggest (and only real) drawback of GARCH mode
Concerned with a single random variable (ex. Roll of a die)
If variance of the conditional distribution of u(i) is not constant
Nonlinearity
Z = (Y - meany)/(stddev(y)/sqrt(n))