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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. Discrete representation of the GBM
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Choose parameters that maximize the likelihood of what observations occurring
Attempts to sample along more important paths

2. Implied standard deviation for options
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(y)/n = variance of sample Y

3. Conditional probability functions
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Confidence level
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

4. Control variates technique
Average return across assets on a given day
Variance = (1/m) summation(u<n - i>^2)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

5. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

6. Simulating for VaR
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Distribution with only two possible outcomes
Average return across assets on a given day

7. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Z = (Y - meany)/(stddev(y)/sqrt(n))
Based on a dataset
Rxy = Sxy/(Sx*Sy)

8. Antithetic variable technique
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Distribution with only two possible outcomes
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

9. Sample variance
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
(a^2)(variance(x)
We reject a hypothesis that is actually true

10. What does the OLS minimize?
SSR
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)
Confidence level

11. Adjusted R^2
Only requires two parameters = mean and variance
Model dependent - Options with the same underlying assets may trade at different volatilities
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

12. Exact significance level
Sample mean will near the population mean as the sample size increases
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
P - value
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

13. Confidence interval for sample mean
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

14. Heteroskedastic
Use historical simulation approach but use the EWMA weighting system
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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!)
If variance of the conditional distribution of u(i) is not constant

15. Multivariate probability
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Choose parameters that maximize the likelihood of what observations occurring
More than one random variable
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

16. Logistic distribution
Choose parameters that maximize the likelihood of what observations occurring
E(mean) = mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Has heavy tails

17. Type II Error
SSR
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
P - value
We accept a hypothesis that should have been rejected

18. T distribution
Application of mathematical statistics to economic data to lend empirical support to models
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 = (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)

19. Key properties of linear regression
Only requires two parameters = mean and variance
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Regression can be non - linear in variables but must be linear in parameters
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

20. Cross - sectional
Average return across assets on a given day
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sampling distribution of sample means tend to be normal
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

21. Two ways to calculate historical volatility
Normal - Student's T - Chi - square - F distribution
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Use historical simulation approach but use the EWMA weighting system

22. Shortcomings of implied volatility
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Model dependent - Options with the same underlying assets may trade at different volatilities
Peaks over threshold - Collects dataset in excess of some threshold
Statement of the error or precision of an estimate

23. Historical std dev
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Nonlinearity
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

24. Two assumptions of square root rule
Low Frequency - High Severity events
Random walk (usually acceptable) - Constant volatility (unlikely)
Contains variables not explicit in model - Accounts for randomness
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

25. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Distribution with only two possible outcomes
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Easy to manipulate

26. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Var(X) + Var(Y)
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

27. Economical(elegant)
Mean of sampling distribution is the population mean
Peaks over threshold - Collects dataset in excess of some threshold
When the sample size is large - the uncertainty about the value of the sample is very small
Only requires two parameters = mean and variance

28. Potential reasons for fat tails in return distributions
Use historical simulation approach but use the EWMA weighting system
Transformed to a unit variable - Mean = 0 Variance = 1
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

29. WLS
Transformed to a unit variable - Mean = 0 Variance = 1
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Regression can be non - linear in variables but must be linear in parameters

30. Binomial distribution equations for mean variance and std dev
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Only requires two parameters = mean and variance
Mean = np - Variance = npq - Std dev = sqrt(npq)
Peaks over threshold - Collects dataset in excess of some threshold

31. Multivariate Density Estimation (MDE)
Var(X) + Var(Y)
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Variance = (1/m) summation(u<n - i>^2)

32. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Transformed to a unit variable - Mean = 0 Variance = 1
For n>30 - sample mean is approximately normal

33. Standard error for Monte Carlo replications
(a^2)(variance(x)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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!)
Peaks over threshold - Collects dataset in excess of some threshold

34. 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)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Normal - Student's T - Chi - square - F distribution
Peaks over threshold - Collects dataset in excess of some threshold

35. Square root rule
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
P(Z>t)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

36. Regime - switching volatility model
We reject a hypothesis that is actually true
P(Z>t)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
95% = 1.65 99% = 2.33 For one - tailed tests

37. LFHS
Low Frequency - High Severity events
Returns over time for an individual asset
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Peaks over threshold - Collects dataset in excess of some threshold

38. Mean(expected value)
E(XY) - E(X)E(Y)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(x) + Variance(Y) + 2*covariance(XY)
When the sample size is large - the uncertainty about the value of the sample is very small

39. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance reverts to a long run level
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

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

41. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Regression can be non - linear in variables but must be linear in parameters
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
Price/return tends to run towards a long - run level

42. Variance(discrete)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

43. Two requirements of OVB
Variance(y)/n = variance of sample Y
(a^2)(variance(x)) + (b^2)(variance(y))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

44. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance reverts to a long run level
P(Z>t)
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. Test for unbiasedness
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
E(mean) = mean

46. Bernouli Distribution
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Regression can be non - linear in variables but must be linear in parameters
Contains variables not explicit in model - Accounts for randomness
Distribution with only two possible outcomes

47. POT
Sample mean will near the population mean as the sample size increases
Peaks over threshold - Collects dataset in excess of some threshold
(a^2)(variance(x)) + (b^2)(variance(y))
Var(X) + Var(Y)

48. SER
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Independently and Identically Distributed
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. Discrete random variable
If variance of the conditional distribution of u(i) is not constant
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

50. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)