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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. Sample covariance
Variance(X) + Variance(Y) - 2*covariance(XY)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Z = (Y - meany)/(stddev(y)/sqrt(n))

2. Type I error
Regression can be non - linear in variables but must be linear in parameters
We reject a hypothesis that is actually true
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

3. Two assumptions of square root rule
Nonlinearity
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance(x) + Variance(Y) + 2*covariance(XY)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

4. Simulation models
P - value
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

5. Empirical frequency
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Based on a dataset
Confidence set for two coefficients - two dimensional analog for the confidence interval
Mean of sampling distribution is the population mean

6. LAD
Based on a dataset
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

7. Unconditional vs conditional distributions
Variance = (1/m) summation(u<n - i>^2)
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
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

8. POT
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
Price/return tends to run towards a long - run level
Peaks over threshold - Collects dataset in excess of some threshold
Probability that the random variables take on certain values simultaneously

9. Time series data
Confidence set for two coefficients - two dimensional analog for the confidence interval
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Attempts to sample along more important paths
Returns over time for an individual asset

10. Bernouli Distribution
We reject a hypothesis that is actually true
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Distribution with only two possible outcomes
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

11. Standard normal distribution
Does not depend on a prior event or information
Regression can be non - linear in variables but must be linear in parameters
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!)
Transformed to a unit variable - Mean = 0 Variance = 1

12. SER
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
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)
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
Only requires two parameters = mean and variance

13. Consistent
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
When the sample size is large - the uncertainty about the value of the sample is very small

14. Reliability
Statement of the error or precision of an estimate
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

15. Significance =1
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Confidence level
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Special type of pooled data in which the cross sectional unit is surveyed over time

16. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Independently and Identically Distributed
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

17. Panel data (longitudinal or micropanel)
95% = 1.65 99% = 2.33 For one - tailed tests
Special type of pooled data in which the cross sectional unit is surveyed over time
Mean = lambda - Variance = lambda - Std dev = sqrt(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)

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

19. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
SSR

20. Extreme Value Theory
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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

21. Difference between population and sample variance
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)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
Population denominator = n - Sample denominator = n - 1

22. Extending the HS approach for computing value of a portfolio

23. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Transformed to a unit variable - Mean = 0 Variance = 1
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

24. Discrete random variable
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance(x)
Model dependent - Options with the same underlying assets may trade at different volatilities

25. GEV
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Confidence level
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

26. Critical z values
Based on an equation - P(A) = # of A/total outcomes
95% = 1.65 99% = 2.33 For one - tailed tests
Variance reverts to a long run level
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

27. Continuous representation of the GBM
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Sampling distribution of sample means tend to be normal
Variance(x) + Variance(Y) + 2*covariance(XY)
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)

28. Overall F - statistic
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
(a^2)(variance(x)
Easy to manipulate
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

29. Marginal unconditional probability function
Population denominator = n - Sample denominator = n - 1
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!)
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
Does not depend on a prior event or information

30. Perfect multicollinearity
E(XY) - E(X)E(Y)
When one regressor is a perfect linear function of the other regressors
Mean of sampling distribution is the population mean
We reject a hypothesis that is actually true

31. Multivariate Density Estimation (MDE)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Expected value of the sample mean is the population mean
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

32. Cross - sectional
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Average return across assets on a given day
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

33. BLUE
Among all unbiased estimators - estimator with the smallest variance is efficient
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(x)

34. Bootstrap method
Nonlinearity
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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!)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

35. Conditional probability functions
Combine to form distribution with leptokurtosis (heavy tails)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Yi = B0 + B1Xi + ui
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

36. Deterministic Simulation
P(Z>t)
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

37. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Has heavy tails
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Low Frequency - High Severity events

38. Continuously compounded return equation
Summation((xi - mean)^k)/n
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
i = ln(Si/Si - 1)
We accept a hypothesis that should have been rejected

39. Four sampling distributions

40. Heteroskedastic
Sampling distribution of sample means tend to be normal
If variance of the conditional distribution of u(i) is not constant
Var(X) + Var(Y)
Transformed to a unit variable - Mean = 0 Variance = 1

41. Poisson Distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Independently and Identically Distributed
95% = 1.65 99% = 2.33 For one - tailed tests

42. Efficiency
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Among all unbiased estimators - estimator with the smallest variance is efficient
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

43. Confidence interval for sample mean
Least absolute deviations estimator - used when extreme outliers are not uncommon
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

44. Standard error for Monte Carlo replications
Based on a dataset
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

45. Economical(elegant)
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
Only requires two parameters = mean and variance
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

46. P - value
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
P(Z>t)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Has heavy tails

47. Chi - squared distribution
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

48. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance = (1/m) summation(u<n - i>^2)
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. Simulating for VaR
P - value
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

50. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha