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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(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
For n>30 - sample mean is approximately normal
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

2. Chi - squared distribution
Has heavy tails
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Returns over time for an individual asset
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

3. Shortcomings of implied volatility
Variance(x)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Model dependent - Options with the same underlying assets may trade at different volatilities
Use historical simulation approach but use the EWMA weighting system

4. Maximum likelihood method
Variance(y)/n = variance of sample Y
Choose parameters that maximize the likelihood of what observations occurring
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

5. Variance of sample mean
Average return across assets on a given day
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Variance(y)/n = variance of sample Y

6. Adjusted R^2
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Normal - Student's T - Chi - square - F distribution
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
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. Confidence ellipse
We reject a hypothesis that is actually true
Probability that the random variables take on certain values simultaneously
Confidence set for two coefficients - two dimensional analog for the confidence interval
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

8. Standard normal distribution
Normal - Student's T - Chi - square - F distribution
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
Price/return tends to run towards a long - run level
Transformed to a unit variable - Mean = 0 Variance = 1

9. Control variates technique
E(XY) - E(X)E(Y)
Regression can be non - linear in variables but must be linear in parameters
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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

10. Sample correlation
Rxy = Sxy/(Sx*Sy)
Normal - Student's T - Chi - square - F distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

11. POT
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Peaks over threshold - Collects dataset in excess of some threshold
Expected value of the sample mean is the population mean
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

12. Variance of X+b
E(XY) - E(X)E(Y)
i = ln(Si/Si - 1)
Variance(x)
Statement of the error or precision of an estimate

13. Variance of sampling distribution of means when n<N
Special type of pooled data in which the cross sectional unit is surveyed over time
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))

14. 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)
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
Variance = (1/m) summation(u<n - i>^2)
Statement of the error or precision of an estimate

15. Mean reversion
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Does not depend on a prior event or information
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

16. Two ways to calculate historical volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
We reject a hypothesis that is actually true
Yi = B0 + B1Xi + ui

17. Hybrid method for conditional volatility
E(XY) - E(X)E(Y)
Variance(y)/n = variance of sample Y
Summation((xi - mean)^k)/n
Use historical simulation approach but use the EWMA weighting system

18. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
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)
Returns over time for an individual asset
Expected value of the sample mean is the population mean

19. Non - parametric vs parametric calculation of VaR
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
We reject a hypothesis that is actually true
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Special type of pooled data in which the cross sectional unit is surveyed over time

20. Homoskedastic only F - stat
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

21. Type II Error
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
P(Z>t)
We accept a hypothesis that should have been rejected

22. Statistical (or empirical) model
Statement of the error or precision of an estimate
Yi = B0 + B1Xi + ui
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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

23. Two drawbacks of moving average series
Regression can be non - linear in variables but must be linear in parameters
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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

24. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

25. Key properties of linear regression
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
We accept a hypothesis that should have been rejected
Regression can be non - linear in variables but must be linear in parameters
Does not depend on a prior event or information

26. Central Limit Theorem
For n>30 - sample mean is approximately normal
Independently and Identically Distributed
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

27. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Contains variables not explicit in model - Accounts for randomness
Concerned with a single random variable (ex. Roll of a die)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

28. Variance of aX
(a^2)(variance(x)
Based on an equation - P(A) = # of A/total outcomes
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
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

29. Result of combination of two normal with same means
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Combine to form distribution with leptokurtosis (heavy tails)
Z = (Y - meany)/(stddev(y)/sqrt(n))

30. Importance sampling technique
Variance(x) + Variance(Y) + 2*covariance(XY)
(a^2)(variance(x)
Yi = B0 + B1Xi + ui
Attempts to sample along more important paths

31. Gamma distribution
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
Population denominator = n - Sample denominator = n - 1
Variance(X) + Variance(Y) - 2*covariance(XY)
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

32. Bernouli Distribution
Model dependent - Options with the same underlying assets may trade at different volatilities
Distribution with only two possible outcomes
If variance of the conditional distribution of u(i) is not constant
More than one random variable

33. Square root rule
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Does not depend on a prior event or information

34. 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)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Transformed to a unit variable - Mean = 0 Variance = 1
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

35. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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

36. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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

37. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Does not depend on a prior event or information

38. What does the OLS minimize?
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Sample mean will near the population mean as the sample size increases
SSR

39. Cholesky factorization (decomposition)
(a^2)(variance(x)) + (b^2)(variance(y))
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
Sampling distribution of sample means tend to be normal

40. Perfect multicollinearity
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
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
When one regressor is a perfect linear function of the other regressors

41. Variance of X+Y assuming dependence
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(x) + Variance(Y) + 2*covariance(XY)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

42. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
(a^2)(variance(x)
Random walk (usually acceptable) - Constant volatility (unlikely)

43. Significance =1
Confidence level
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 reverts to a long run level
Yi = B0 + B1Xi + ui

44. LAD
Returns over time for an individual asset
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

45. Implied standard deviation for options
E(XY) - E(X)E(Y)
Choose parameters that maximize the likelihood of what observations occurring
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

46. 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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

47. Expected future variance rate (t periods forward)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
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

48. Exact significance level
P - value
(a^2)(variance(x)) + (b^2)(variance(y))
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

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

50. Marginal unconditional probability function
Nonlinearity
Does not depend on a prior event or information
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Probability that the random variables take on certain values simultaneously