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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. i.i.d.
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Choose parameters that maximize the likelihood of what observations occurring
Independently and Identically Distributed

2. Importance sampling technique
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Attempts to sample along more important paths
If variance of the conditional distribution of u(i) is not constant
Returns over time for an individual asset

3. Type II Error
Transformed to a unit variable - Mean = 0 Variance = 1
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)
We accept a hypothesis that should have been rejected
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

4. Normal distribution
Variance = (1/m) summation(u<n - i>^2)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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
Model dependent - Options with the same underlying assets may trade at different volatilities

5. Monte Carlo Simulations
Summation((xi - mean)^k)/n
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

6. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance(x)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

7. 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)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

8. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
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
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)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

9. Variance of sample mean
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(y)/n = variance of sample Y
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

10. Pooled data
If variance of the conditional distribution of u(i) is not constant
Variance(x) + Variance(Y) + 2*covariance(XY)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

11. Simulation models
Expected value of the sample mean is the population mean
Use historical simulation approach but use the EWMA weighting system
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
Random walk (usually acceptable) - Constant volatility (unlikely)

12. 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)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(y)/n = variance of sample Y
Variance = (1/m) summation(u<n - i>^2)

13. Sample covariance
E(XY) - E(X)E(Y)
Application of mathematical statistics to economic data to lend empirical support to models
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

14. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
More than one random variable
Peaks over threshold - Collects dataset in excess of some threshold
Does not depend on a prior event or information

15. Control variates technique
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Sample mean will near the population mean as the sample size increases
E(XY) - E(X)E(Y)

16. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

17. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Does not depend on a prior event or information
Model dependent - Options with the same underlying assets may trade at different volatilities
Independently and Identically Distributed

18. P - value
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
P(Z>t)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

19. Standard variable for non - normal distributions
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Choose parameters that maximize the likelihood of what observations occurring
Z = (Y - meany)/(stddev(y)/sqrt(n))

20. Persistence
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

21. 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
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
Var(X) + Var(Y)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

22. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

23. 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
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
Based on an equation - P(A) = # of A/total outcomes
If variance of the conditional distribution of u(i) is not constant

24. Implied standard deviation for options
Statement of the error or precision of an estimate
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

25. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

26. Standard normal distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Transformed to a unit variable - Mean = 0 Variance = 1
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

27. Variance of aX + bY
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
(a^2)(variance(x)) + (b^2)(variance(y))
Probability that the random variables take on certain values simultaneously
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

28. Simplified standard (un - weighted) variance
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!)
Variance = (1/m) summation(u<n - i>^2)
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

29. Potential reasons for fat tails in return distributions
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance reverts to a long run level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

30. Two requirements of OVB
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
Variance(x)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

31. Implications of homoscedasticity
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
E(XY) - E(X)E(Y)

32. Time series data
Returns over time for an individual asset
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

33. Expected future variance rate (t periods forward)
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!)
Variance(x) + Variance(Y) + 2*covariance(XY)
(a^2)(variance(x)) + (b^2)(variance(y))
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

34. Variance of X+Y
For n>30 - sample mean is approximately normal
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Var(X) + Var(Y)
Choose parameters that maximize the likelihood of what observations occurring

35. Block maxima
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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.
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

36. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Has heavy tails
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Confidence level

37. Kurtosis
Variance(x) + Variance(Y) + 2*covariance(XY)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

38. Continuously compounded return equation
i = ln(Si/Si - 1)
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

39. Variance of aX
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
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
(a^2)(variance(x)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

40. Economical(elegant)
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
Low Frequency - High Severity events
Only requires two parameters = mean and variance

41. Adjusted R^2
Distribution with only two possible outcomes
Rxy = Sxy/(Sx*Sy)
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

42. Joint probability functions
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Probability that the random variables take on certain values simultaneously
More than one random variable
E(XY) - E(X)E(Y)

43. Variance of weighted scheme
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Sampling distribution of sample means tend to be normal

44. Empirical frequency
Based on a dataset
More than one random variable
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

45. Confidence ellipse
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Confidence set for two coefficients - two dimensional analog for the confidence interval
95% = 1.65 99% = 2.33 For one - tailed tests
Confidence level

46. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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!)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

47. Econometrics
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Application of mathematical statistics to economic data to lend empirical support to models
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

48. Statistical (or empirical) model
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Yi = B0 + B1Xi + ui
Based on a dataset
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

49. POT
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Peaks over threshold - Collects dataset in excess of some threshold

50. Maximum likelihood method
Variance = (1/m) summation(u<n - i>^2)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Rxy = Sxy/(Sx*Sy)
Choose parameters that maximize the likelihood of what observations occurring