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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. Binomial distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Among all unbiased estimators - estimator with the smallest variance is efficient
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!)
E(mean) = mean

2. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Random walk (usually acceptable) - Constant volatility (unlikely)
P - value
Does not depend on a prior event or information

3. Unconditional vs conditional distributions
Returns over time for an individual asset
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

4. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Special type of pooled data in which the cross sectional unit is surveyed over time
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

5. Confidence ellipse
Contains variables not explicit in model - Accounts for randomness
Confidence set for two coefficients - two dimensional analog for the confidence interval
Var(X) + Var(Y)
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors

6. 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
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
We accept a hypothesis that should have been rejected

7. Homoskedastic only F - stat
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 = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

8. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Expected value of the sample mean is the population mean
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

9. Variance of X - Y assuming dependence
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
Distribution with only two possible outcomes
Variance(X) + Variance(Y) - 2*covariance(XY)

10. Type I error
We reject a hypothesis that is actually true
i = ln(Si/Si - 1)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Sample mean +/ - t*(stddev(s)/sqrt(n))

11. Mean reversion in variance
SSR
Variance reverts to a long run level
Variance(x) + Variance(Y) + 2*covariance(XY)
Mean of sampling distribution is the population mean

12. Econometrics
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Z = (Y - meany)/(stddev(y)/sqrt(n))
Application of mathematical statistics to economic data to lend empirical support to models

13. Variance of aX
(a^2)(variance(x)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance(y)/n = variance of sample Y

14. Standard error
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Choose parameters that maximize the likelihood of what observations occurring
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

15. Difference between population and sample variance
Var(X) + Var(Y)
Random walk (usually acceptable) - Constant volatility (unlikely)
Yi = B0 + B1Xi + ui
Population denominator = n - Sample denominator = n - 1

16. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Nonlinearity
Among all unbiased estimators - estimator with the smallest variance is efficient

17. Overall F - statistic
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Regression can be non - linear in variables but must be linear in parameters

18. Efficiency
We accept a hypothesis that should have been rejected
Among all unbiased estimators - estimator with the smallest variance is efficient
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

19. T distribution
Returns over time for an individual asset
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 = (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)

20. Critical z values
Mean of sampling distribution is the population mean
95% = 1.65 99% = 2.33 For one - tailed tests
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Regression can be non - linear in variables but must be linear in parameters

21. K - th moment
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
More than one random variable
Summation((xi - mean)^k)/n
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)

22. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Least absolute deviations estimator - used when extreme outliers are not uncommon
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!)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

23. Reliability
Statement of the error or precision of an estimate
Has heavy tails
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

24. Variance of X+Y assuming dependence
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
(a^2)(variance(x)
Variance(x) + Variance(Y) + 2*covariance(XY)
(a^2)(variance(x)) + (b^2)(variance(y))

25. Direction of OVB
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Least absolute deviations estimator - used when extreme outliers are not uncommon

26. Variance(discrete)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

27. Block maxima
More than one random variable
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Summation((xi - mean)^k)/n
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

28. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Based on a dataset
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

29. Maximum likelihood method
Has heavy tails
(a^2)(variance(x)) + (b^2)(variance(y))
Sample mean +/ - t*(stddev(s)/sqrt(n))
Choose parameters that maximize the likelihood of what observations occurring

30. Marginal unconditional probability function
Choose parameters that maximize the likelihood of what observations occurring
Does not depend on a prior event or information
Variance reverts to a long run level
Sample mean +/ - t*(stddev(s)/sqrt(n))

31. Covariance
E(XY) - E(X)E(Y)
Regression can be non - linear in variables but must be linear in parameters
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance reverts to a long run level

32. Mean(expected value)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Distribution with only two possible outcomes
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(X) + Variance(Y) - 2*covariance(XY)

33. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
P(Z>t)
Use historical simulation approach but use the EWMA weighting system
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

34. What does the OLS minimize?
More than one random variable
SSR
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors

35. SER
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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

36. Potential reasons for fat tails in return distributions
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

37. P - value
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
P(Z>t)
Transformed to a unit variable - Mean = 0 Variance = 1

38. Bernouli Distribution
Distribution with only two possible outcomes
Z = (Y - meany)/(stddev(y)/sqrt(n))
If variance of the conditional distribution of u(i) is not constant
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

39. BLUE
Combine to form distribution with leptokurtosis (heavy tails)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Price/return tends to run towards a long - run level
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

40. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Probability that the random variables take on certain values simultaneously

41. Two assumptions of square root rule
Independently and Identically Distributed
95% = 1.65 99% = 2.33 For one - tailed tests
Random walk (usually acceptable) - Constant volatility (unlikely)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

42. Unstable return distribution
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Among all unbiased estimators - estimator with the smallest variance is efficient
For n>30 - sample mean is approximately normal

43. Two ways to calculate historical volatility
Yi = B0 + B1Xi + ui
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Contains variables not explicit in model - Accounts for randomness
95% = 1.65 99% = 2.33 For one - tailed tests

44. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Yi = B0 + B1Xi + ui
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

45. Skewness
Special type of pooled data in which the cross sectional unit is surveyed over time
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors

46. Variance of sample mean
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance(y)/n = variance of sample Y
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

47. Cross - sectional
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Average return across assets on a given day
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

48. Least squares estimator(m)
Among all unbiased estimators - estimator with the smallest variance is efficient
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Special type of pooled data in which the cross sectional unit is surveyed over time

49. GARCH
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(x)

50. Discrete representation of the GBM
(a^2)(variance(x)) + (b^2)(variance(y))
When one regressor is a perfect linear function of the other regressors
We accept a hypothesis that should have been rejected
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))