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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. Consistent
Mean = np - Variance = npq - Std dev = sqrt(npq)
When the sample size is large - the uncertainty about the value of the sample is very small
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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)

2. Variance of X+b
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(x)

3. Simulating for VaR
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

4. Two requirements of OVB
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Use historical simulation approach but use the EWMA weighting system
Does not depend on a prior event or information
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

5. 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)
Only requires two parameters = mean and variance
Independently and Identically Distributed
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

6. Confidence interval for sample mean
(a^2)(variance(x)
Special type of pooled data in which the cross sectional unit is surveyed over time
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

7. Significance =1
Sample mean +/ - t*(stddev(s)/sqrt(n))
95% = 1.65 99% = 2.33 For one - tailed tests
Confidence level
Expected value of the sample mean is the population mean

8. Central Limit Theorem
Population denominator = n - Sample denominator = n - 1
Peaks over threshold - Collects dataset in excess of some threshold
For n>30 - sample mean is approximately normal
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

9. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Statement of the error or precision of an estimate
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

10. Covariance calculations using weight sums (lambda)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Statement of the error or precision of an estimate
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

11. Variance of aX
More than one random variable
(a^2)(variance(x)
Returns over time for an individual asset
Least absolute deviations estimator - used when extreme outliers are not uncommon

12. Homoskedastic
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
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

13. SER
P(X=x - Y=y) = P(X=x) * P(Y=y)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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)

14. 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)
Variance(X) + Variance(Y) - 2*covariance(XY)
Has heavy tails
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

15. WLS
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
Mean of sampling distribution is the population mean
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Independently and Identically Distributed

16. Potential reasons for fat tails in return distributions
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

17. Importance sampling technique
Attempts to sample along more important paths
Low Frequency - High Severity events
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!)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

18. Implications of homoscedasticity
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Based on a dataset

19. Sample mean
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
Expected value of the sample mean is the population mean
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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

20. Least squares estimator(m)
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Use historical simulation approach but use the EWMA weighting system
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

21. Variance - covariance approach for VaR of a portfolio
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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

22. Perfect multicollinearity
Model dependent - Options with the same underlying assets may trade at different volatilities
When one regressor is a perfect linear function of the other regressors
Population denominator = n - Sample denominator = n - 1
Regression can be non - linear in variables but must be linear in parameters

23. Marginal unconditional probability function
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Does not depend on a prior event or information

24. Difference between population and sample variance
Variance(x)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Choose parameters that maximize the likelihood of what observations occurring
Population denominator = n - Sample denominator = n - 1

25. LAD
Normal - Student's T - Chi - square - F distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon
Peaks over threshold - Collects dataset in excess of some threshold
SSR

26. Historical std dev
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)
Concerned with a single random variable (ex. Roll of a die)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

27. T 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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

28. Non - parametric vs parametric calculation of VaR
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
When one regressor is a perfect linear function of the other regressors
Probability that the random variables take on certain values simultaneously

29. Expected future variance rate (t periods forward)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Confidence set for two coefficients - two dimensional analog for the confidence interval

30. Stochastic error term
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Contains variables not explicit in model - Accounts for randomness

31. BLUE
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Confidence level

32. Economical(elegant)
Only requires two parameters = mean and variance
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Summation((xi - mean)^k)/n
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

33. Bernouli Distribution
Distribution with only two possible outcomes
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Combine to form distribution with leptokurtosis (heavy tails)
Mean = np - Variance = npq - Std dev = sqrt(npq)

34. Unstable return distribution
Population denominator = n - Sample denominator = n - 1
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

35. Multivariate probability
More than one random variable
Peaks over threshold - Collects dataset in excess of some threshold
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

36. GPD
Among all unbiased estimators - estimator with the smallest variance is efficient
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
For n>30 - sample mean is approximately normal
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

37. Cholesky factorization (decomposition)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Nonlinearity

38. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Model dependent - Options with the same underlying assets may trade at different volatilities

39. Time series data
When one regressor is a perfect linear function of the other regressors
Returns over time for an individual asset
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
We reject a hypothesis that is actually true

40. Sample correlation
Sampling distribution of sample means tend to be normal
Rxy = Sxy/(Sx*Sy)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

41. Implied standard deviation for options
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance(X) + Variance(Y) - 2*covariance(XY)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

42. Heteroskedastic
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
If variance of the conditional distribution of u(i) is not constant

43. Two assumptions of square root rule
Confidence set for two coefficients - two dimensional analog for the confidence interval
Random walk (usually acceptable) - Constant volatility (unlikely)
Transformed to a unit variable - Mean = 0 Variance = 1
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

44. Extreme Value Theory
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Concerned with a single random variable (ex. Roll of a die)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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)

45. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Only requires two parameters = mean and variance

46. Covariance
E(XY) - E(X)E(Y)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Transformed to a unit variable - Mean = 0 Variance = 1
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

47. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
95% = 1.65 99% = 2.33 For one - tailed tests

48. Overall F - statistic
Peaks over threshold - Collects dataset in excess of some threshold
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
95% = 1.65 99% = 2.33 For one - tailed tests

49. Poisson 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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Combine to form distribution with leptokurtosis (heavy tails)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

50. Standard variable for non - normal distributions
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Average return across assets on a given day
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Z = (Y - meany)/(stddev(y)/sqrt(n))