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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. GARCH
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance reverts to a long run level
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)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

2. i.i.d.
Independently and Identically Distributed
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
P(Z>t)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

3. 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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Summation((xi - mean)^k)/n
Nonlinearity

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

5. Variance - covariance approach for VaR of a portfolio
Mean = np - Variance = npq - Std dev = sqrt(npq)
Low Frequency - High Severity events
(a^2)(variance(x)) + (b^2)(variance(y))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

6. Sample covariance
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
P(Z>t)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

7. Economical(elegant)
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Only requires two parameters = mean and variance
95% = 1.65 99% = 2.33 For one - tailed tests

8. Confidence interval (from t)
Z = (Y - meany)/(stddev(y)/sqrt(n))
P - value
Sample mean +/ - t*(stddev(s)/sqrt(n))
Confidence set for two coefficients - two dimensional analog for the confidence interval

9. Sample variance
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

10. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

11. Stochastic error term
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Contains variables not explicit in model - Accounts for randomness
When the sample size is large - the uncertainty about the value of the sample is very small

12. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
We accept a hypothesis that should have been rejected
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

13. EWMA
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Least absolute deviations estimator - used when extreme outliers are not uncommon
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

14. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Application of mathematical statistics to economic data to lend empirical support to models
E(mean) = mean

15. Statistical (or empirical) model
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Yi = B0 + B1Xi + ui
Sample mean +/ - t*(stddev(s)/sqrt(n))
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

16. Control variates technique
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

17. Two requirements of OVB
Only requires two parameters = mean and variance
When the sample size is large - the uncertainty about the value of the sample is very small
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Price/return tends to run towards a long - run level

18. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
We accept a hypothesis that should have been rejected
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E(mean) = mean

19. Joint probability functions
Var(X) + Var(Y)
Based on an equation - P(A) = # of A/total outcomes
Mean of sampling distribution is the population mean
Probability that the random variables take on certain values simultaneously

20. Exact significance level
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Summation((xi - mean)^k)/n
P - value

21. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
P - value
Peaks over threshold - Collects dataset in excess of some threshold

22. Hybrid method for conditional volatility
We accept a hypothesis that should have been rejected
Use historical simulation approach but use the EWMA weighting system
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

23. SER
Choose parameters that maximize the likelihood of what observations occurring
Z = (Y - meany)/(stddev(y)/sqrt(n))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean = np - Variance = npq - Std dev = sqrt(npq)

24. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Peaks over threshold - Collects dataset in excess of some threshold
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

25. Reliability
Statement of the error or precision of an estimate
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Population denominator = n - Sample denominator = n - 1
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

26. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Confidence set for two coefficients - two dimensional analog for the confidence interval
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
E(mean) = mean

27. Logistic distribution
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
i = ln(Si/Si - 1)
Has heavy tails
Mean = np - Variance = npq - Std dev = sqrt(npq)

28. Normal distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Nonlinearity
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
Only requires two parameters = mean and variance

29. Variance of aX + bY
Variance(x)
Variance(x) + Variance(Y) + 2*covariance(XY)
Summation((xi - mean)^k)/n
(a^2)(variance(x)) + (b^2)(variance(y))

30. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Mean of sampling distribution is the population mean
Based on a dataset
Choose parameters that maximize the likelihood of what observations occurring

31. Weibul distribution
Use historical simulation approach but use the EWMA weighting system
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

32. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Nonlinearity
Sampling distribution of sample means tend to be normal
Transformed to a unit variable - Mean = 0 Variance = 1

33. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

34. SER
P(Z>t)
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
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
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

35. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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)
Statement of the error or precision of an estimate

36. Consistent
Independently and Identically Distributed
Attempts to sample along more important paths
When the sample size is large - the uncertainty about the value of the sample is very small
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

37. Implied standard deviation for options
We accept a hypothesis that should have been rejected
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

38. Unconditional vs conditional distributions
Independently and Identically Distributed
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

39. Confidence ellipse
Special type of pooled data in which the cross sectional unit is surveyed over time
Confidence set for two coefficients - two dimensional analog for the confidence interval
For n>30 - sample mean is approximately normal
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

40. GPD
Peaks over threshold - Collects dataset in excess of some threshold
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Distribution with only two possible outcomes
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

41. Variance of X+b
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Based on an equation - P(A) = # of A/total outcomes
Variance(x)
Combine to form distribution with leptokurtosis (heavy tails)

42. Least squares estimator(m)
Model dependent - Options with the same underlying assets may trade at different volatilities
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Application of mathematical statistics to economic data to lend empirical support to models
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

43. Central Limit Theorem
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance = (1/m) summation(u<n - i>^2)
For n>30 - sample mean is approximately normal

44. Shortcomings of implied volatility
Regression can be non - linear in variables but must be linear in parameters
Model dependent - Options with the same underlying assets may trade at different volatilities
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
(a^2)(variance(x)) + (b^2)(variance(y))

45. Type II Error
Average return across assets on a given day
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
We accept a hypothesis that should have been rejected
Based on an equation - P(A) = # of A/total outcomes

46. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(x) + Variance(Y) + 2*covariance(XY)
Concerned with a single random variable (ex. Roll of a die)

47. Two assumptions of square root rule
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Random walk (usually acceptable) - Constant volatility (unlikely)
Application of mathematical statistics to economic data to lend empirical support to models
Probability that the random variables take on certain values simultaneously

48. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Model dependent - Options with the same underlying assets may trade at different volatilities
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Sampling distribution of sample means tend to be normal

49. Poisson Distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
E(XY) - E(X)E(Y)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Nonlinearity

50. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Does not depend on a prior event or information
Special type of pooled data in which the cross sectional unit is surveyed over time
Expected value of the sample mean is the population mean