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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. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Random walk (usually acceptable) - Constant volatility (unlikely)
Easy to manipulate
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

2. Chi - squared distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
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
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
Concerned with a single random variable (ex. Roll of a die)

3. Variance of sample mean
E(mean) = mean
Variance(y)/n = variance of sample Y
Transformed to a unit variable - Mean = 0 Variance = 1
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

4. Difference between population and sample variance
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Independently and Identically Distributed
When the sample size is large - the uncertainty about the value of the sample is very small
Population denominator = n - Sample denominator = n - 1

5. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
We accept a hypothesis that should have been rejected
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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!)

6. Hazard rate of exponentially distributed random variable
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Special type of pooled data in which the cross sectional unit is surveyed over time

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

8. Weibul distribution
Summation((xi - mean)^k)/n
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
E(XY) - E(X)E(Y)

9. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Normal - Student's T - Chi - square - F distribution

10. Sample correlation
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Random walk (usually acceptable) - Constant volatility (unlikely)
Rxy = Sxy/(Sx*Sy)
Choose parameters that maximize the likelihood of what observations occurring

11. Variance of X - Y assuming dependence
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(X) + Variance(Y) - 2*covariance(XY)
Confidence level
Sample mean +/ - t*(stddev(s)/sqrt(n))

12. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Expected value of the sample mean is the population mean
Returns over time for an individual asset

13. Mean reversion in asset dynamics
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Price/return tends to run towards a long - run level
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

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

15. Implications of homoscedasticity
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Yi = B0 + B1Xi + ui

16. F distribution
Random walk (usually acceptable) - Constant volatility (unlikely)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
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)

17. Type II Error
We accept a hypothesis that should have been rejected
Among all unbiased estimators - estimator with the smallest variance is efficient
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

18. Unconditional vs conditional distributions
When one regressor is a perfect linear function of the other regressors
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Sample mean +/ - t*(stddev(s)/sqrt(n))

19. Variance of aX + bY
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Use historical simulation approach but use the EWMA weighting system
(a^2)(variance(x)) + (b^2)(variance(y))

20. Normal distribution
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 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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Does not depend on a prior event or information

21. Inverse transform method
When one regressor is a perfect linear function of the other regressors
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

22. R^2
Confidence set for two coefficients - two dimensional analog for the confidence interval
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Only requires two parameters = mean and variance
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

23. Homoskedastic
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Choose parameters that maximize the likelihood of what observations occurring
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
Average return across assets on a given day

24. Econometrics
SSR
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Attempts to sample along more important paths
Application of mathematical statistics to economic data to lend empirical support to models

25. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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 a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

26. Adjusted R^2
Average return across assets on a given day
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
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
Variance = (1/m) summation(u<n - i>^2)

27. Type I error
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
SSR
We reject a hypothesis that is actually true

28. SER
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

29. Conditional probability functions
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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

30. Block maxima
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
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
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

31. Mean reversion in variance
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance reverts to a long run level
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

32. T distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)
Confidence level
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

33. Test for unbiasedness
Confidence set for two coefficients - two dimensional analog for the confidence interval
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Probability that the random variables take on certain values simultaneously
E(mean) = mean

34. Perfect multicollinearity
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
When one regressor is a perfect linear function of the other regressors
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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

35. Unbiased
Mean of sampling distribution is the population mean
Rxy = Sxy/(Sx*Sy)
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

36. GPD
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 Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

37. Empirical frequency
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Based on a dataset

38. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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!)

39. Regime - switching volatility model
Summation((xi - mean)^k)/n
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
SSR
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

40. Cholesky factorization (decomposition)
Attempts to sample along more important paths
Variance(X) + Variance(Y) - 2*covariance(XY)
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
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

41. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

42. Simulation models
Only requires two parameters = mean and variance
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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

43. Law of Large Numbers
P - value
Variance(x) + Variance(Y) + 2*covariance(XY)
Sample mean will near the population mean as the sample size increases
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

44. LAD
Transformed to a unit variable - Mean = 0 Variance = 1
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Least absolute deviations estimator - used when extreme outliers are not uncommon

45. Standard error for Monte Carlo replications
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
(a^2)(variance(x)) + (b^2)(variance(y))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

46. i.i.d.
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Independently and Identically Distributed
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
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

47. Square root rule
Var(X) + Var(Y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

48. Two ways to calculate historical volatility
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance = (1/m) summation(u<n - i>^2)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

49. Critical z values
i = ln(Si/Si - 1)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
95% = 1.65 99% = 2.33 For one - tailed tests
More than one random variable

50. Continuous random variable
When the sample size is large - the uncertainty about the value of the sample is very small
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)