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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. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

2. Logistic distribution
Variance(X) + Variance(Y) - 2*covariance(XY)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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

3. Pooled data
Model dependent - Options with the same underlying assets may trade at different volatilities
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon

4. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
P(Z>t)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

5. Panel data (longitudinal or micropanel)
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Special type of pooled data in which the cross sectional unit is surveyed over time
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

6. Unstable return distribution
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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!)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

7. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
P - value
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

8. Cross - sectional
Sample mean will near the population mean as the sample size increases
P - value
Average return across assets on a given day
If variance of the conditional distribution of u(i) is not constant

9. Standard normal distribution
We reject a hypothesis that is actually true
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Transformed to a unit variable - Mean = 0 Variance = 1

10. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Nonlinearity
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)
P - value

11. Variance - covariance approach for VaR of a portfolio
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

12. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Easy to manipulate
Normal - Student's T - Chi - square - F distribution
Rxy = Sxy/(Sx*Sy)

13. Discrete random variable
Variance(x) + Variance(Y) + 2*covariance(XY)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

14. Test for unbiasedness
Returns over time for a combination of assets (combination of time series and cross - sectional data)
E(mean) = mean
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)
Choose parameters that maximize the likelihood of what observations occurring

15. Economical(elegant)
Normal - Student's T - Chi - square - F distribution
Only requires two parameters = mean and variance
Concerned with a single random variable (ex. Roll of a die)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

16. Mean(expected value)
E(mean) = mean
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
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

17. Expected future variance rate (t periods forward)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
P(Z>t)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Z = (Y - meany)/(stddev(y)/sqrt(n))

18. Sample correlation
Use historical simulation approach but use the EWMA weighting system
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Rxy = Sxy/(Sx*Sy)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

19. Multivariate probability
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
More than one random variable
Average return across assets on a given day

20. Central Limit Theorem(CLT)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Population denominator = n - Sample denominator = n - 1
Sampling distribution of sample means tend to be normal
Special type of pooled data in which the cross sectional unit is surveyed over time

21. SER
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(x) + Variance(Y) + 2*covariance(XY)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

22. Four sampling distributions

23. Variance of weighted scheme
Variance(x)
Expected value of the sample mean is the population mean
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Attempts to sample along more important paths

24. 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
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))

25. Confidence ellipse
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Confidence set for two coefficients - two dimensional analog for the confidence interval
If variance of the conditional distribution of u(i) is not constant

26. Chi - squared distribution
Attempts to sample along more important paths
95% = 1.65 99% = 2.33 For one - tailed tests
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

27. Inverse transform method
Based on an equation - P(A) = # of A/total outcomes
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Price/return tends to run towards a long - run level
P(X=x - Y=y) = P(X=x) * P(Y=y)

28. Time series data
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
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
Returns over time for an individual asset

29. Weibul distribution
Contains variables not explicit in model - Accounts for randomness
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

30. Key properties of linear regression
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Regression can be non - linear in variables but must be linear in parameters

31. Heteroskedastic
Expected value of the sample mean is the population mean
Returns over time for a combination of assets (combination of time series and cross - sectional data)
If variance of the conditional distribution of u(i) is not constant
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

32. Binomial distribution
Use historical simulation approach but use the EWMA weighting system
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
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!)

33. Overall F - statistic
Independently and Identically Distributed
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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)

34. Simplified standard (un - weighted) variance
Choose parameters that maximize the likelihood of what observations occurring
Population denominator = n - Sample denominator = n - 1
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance = (1/m) summation(u<n - i>^2)

35. Block maxima
Attempts to sample along more important paths
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Z = (Y - meany)/(stddev(y)/sqrt(n))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

36. Square root rule
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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!)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

37. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

38. Central Limit Theorem
E(mean) = mean
For n>30 - sample mean is approximately normal
Returns over time for an individual asset
Expected value of the sample mean is the population mean

39. Kurtosis
Rxy = Sxy/(Sx*Sy)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance reverts to a long run level
Based on an equation - P(A) = # of A/total outcomes

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

41. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
P(Z>t)
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

42. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Sample mean will near the population mean as the sample size increases
95% = 1.65 99% = 2.33 For one - tailed tests
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

43. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Variance(y)/n = variance of sample Y
Choose parameters that maximize the likelihood of what observations occurring

44. Beta distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

45. F distribution
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Combine to form distribution with leptokurtosis (heavy tails)
Probability that the random variables take on certain values simultaneously
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

46. Single variable (univariate) probability
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Concerned with a single random variable (ex. Roll of a die)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
For n>30 - sample mean is approximately normal

47. Potential reasons for fat tails in return distributions
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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Random walk (usually acceptable) - Constant volatility (unlikely)

48. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Normal - Student's T - Chi - square - F distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

49. Deterministic Simulation
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
Summation((xi - mean)^k)/n
Only requires two parameters = mean and variance
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

50. Continuous random variable
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Based on an equation - P(A) = # of A/total outcomes