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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. Homoskedastic
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
We reject a hypothesis that is actually true
Special type of pooled data in which the cross sectional unit is surveyed over time
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

2. Expected future variance rate (t periods forward)
Application of mathematical statistics to economic data to lend empirical support to models
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
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

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

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4. Maximum likelihood method
Application of mathematical statistics to economic data to lend empirical support to models
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Choose parameters that maximize the likelihood of what observations occurring

5. What does the OLS minimize?
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
SSR
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

6. SER
Least absolute deviations estimator - used when extreme outliers are not uncommon
If variance of the conditional distribution of u(i) is not constant
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Contains variables not explicit in model - Accounts for randomness

7. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

8. Type I error
E(XY) - E(X)E(Y)
Peaks over threshold - Collects dataset in excess of some threshold
We reject a hypothesis that is actually true
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

9. Deterministic Simulation
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
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
(a^2)(variance(x)) + (b^2)(variance(y))

10. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Choose parameters that maximize the likelihood of what observations occurring
Variance(y)/n = variance of sample Y

11. Continuous representation of the GBM
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)
Sample mean will near the population mean as the sample size increases
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

12. Confidence interval for sample mean
Independently and Identically Distributed
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
More than one random variable
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

13. WLS
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Probability that the random variables take on certain values simultaneously
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Variance(x)

14. Mean(expected value)
(a^2)(variance(x)
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
Low Frequency - High Severity events
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

15. Priori (classical) probability
Special type of pooled data in which the cross sectional unit is surveyed over time
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
P(Z>t)
Based on an equation - P(A) = # of A/total outcomes

16. Tractable
Concerned with a single random variable (ex. Roll of a die)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Easy to manipulate
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

17. Potential reasons for fat tails in return distributions
Distribution with only two possible outcomes
We reject a hypothesis that is actually true
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

18. Gamma distribution
Statement of the error or precision of an estimate
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 the sample size is large - the uncertainty about the value of the sample is very small
More than one random variable

19. Mean reversion
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Model dependent - Options with the same underlying assets may trade at different volatilities

20. Hazard rate of exponentially distributed random variable
(a^2)(variance(x)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Choose parameters that maximize the likelihood of what observations occurring

21. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
We reject a hypothesis that is actually true
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

22. SER
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Confidence level
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

23. Bootstrap method
Variance reverts to a long run level
When one regressor is a perfect linear function of the other regressors
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Only requires two parameters = mean and variance

24. Extreme Value Theory
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

25. Critical z values
P(X=x - Y=y) = P(X=x) * P(Y=y)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
95% = 1.65 99% = 2.33 For one - tailed tests
Easy to manipulate

26. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Expected value of the sample mean is the population mean
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Nonlinearity

27. Variance of sample mean
Variance(y)/n = variance of sample Y
Mean of sampling distribution is the population mean
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Based on a dataset

28. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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)

29. Least squares estimator(m)
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
When one regressor is a perfect linear function of the other regressors

30. Result of combination of two normal with same means
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Regression can be non - linear in variables but must be linear in parameters
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Combine to form distribution with leptokurtosis (heavy tails)

31. Difference between population and sample variance
P - value
Easy to manipulate
Var(X) + Var(Y)
Population denominator = n - Sample denominator = n - 1

32. Reliability
Statement of the error or precision of an estimate
Sample mean will near the population mean as the sample size increases
E(mean) = mean
(a^2)(variance(x)

33. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Attempts to sample along more important paths
Special type of pooled data in which the cross sectional unit is surveyed over time
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

34. Two drawbacks of moving average series
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
We reject a hypothesis that is actually true
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

35. R^2
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

36. Joint probability functions
Probability that the random variables take on certain values simultaneously
SSR
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Var(X) + Var(Y)

37. Conditional probability functions
Variance(y)/n = variance of sample Y
Yi = B0 + B1Xi + ui
Only requires two parameters = mean and 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)

38. Binomial distribution
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
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!)
Confidence level
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

39. Two requirements of OVB
Normal - Student's T - Chi - square - F distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

40. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Use historical simulation approach but use the EWMA weighting system

41. Variance - covariance approach for VaR of a portfolio
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)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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)

42. Law of Large Numbers
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)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Sample mean will near the population mean as the sample size increases

43. Statistical (or empirical) model
Sample mean will near the population mean as the sample size increases
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Yi = B0 + B1Xi + ui
(a^2)(variance(x)

44. Economical(elegant)
Normal - Student's T - Chi - square - F distribution
Only requires two parameters = mean and variance
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)

45. Inverse transform method
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
If variance of the conditional distribution of u(i) is not constant

46. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Among all unbiased estimators - estimator with the smallest variance is efficient
Easy to manipulate
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

47. Covariance calculations using weight sums (lambda)
Peaks over threshold - Collects dataset in excess of some threshold
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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

48. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Attempts to sample along more important paths
Sample mean +/ - t*(stddev(s)/sqrt(n))
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

49. Central Limit Theorem(CLT)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Price/return tends to run towards a long - run level
Sampling distribution of sample means tend to be normal
Variance(y)/n = variance of sample Y

50. F distribution
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Confidence level
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