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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. Regime - switching volatility model
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Summation((xi - mean)^k)/n
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)

2. Least squares estimator(m)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

3. Maximum likelihood method
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Random walk (usually acceptable) - Constant volatility (unlikely)
Nonlinearity
Choose parameters that maximize the likelihood of what observations occurring

4. Sample variance
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

5. Standard error for Monte Carlo replications
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

6. Variance of X - Y assuming dependence
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Variance(X) + Variance(Y) - 2*covariance(XY)
When one regressor is a perfect linear function of the other regressors
Special type of pooled data in which the cross sectional unit is surveyed over time

7. Type II Error
Rxy = Sxy/(Sx*Sy)
We accept a hypothesis that should have been rejected
(a^2)(variance(x)
Price/return tends to run towards a long - run level

8. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Attempts to sample along more important paths
Sampling distribution of sample means tend to be normal

9. Standard error
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Special type of pooled data in which the cross sectional unit is surveyed over time
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 deviation of the sampling distribution SE = std dev(y)/sqrt(n)

10. Key properties of linear regression
Low Frequency - High Severity events
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Regression can be non - linear in variables but must be linear in parameters
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

11. Simulating for VaR
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Population denominator = n - Sample denominator = n - 1
Based on an equation - P(A) = # of A/total outcomes

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

13. Statistical (or empirical) model
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Yi = B0 + B1Xi + ui
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)

14. Direction of OVB
Z = (Y - meany)/(stddev(y)/sqrt(n))
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
(a^2)(variance(x)

15. Variance of X+b
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance(x)
Rxy = Sxy/(Sx*Sy)
Least absolute deviations estimator - used when extreme outliers are not uncommon

16. Logistic distribution
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
Has heavy tails
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!)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

17. Poisson Distribution
i = ln(Si/Si - 1)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Sample mean will near the population mean as the sample size increases
Summation((xi - mean)^k)/n

18. Mean reversion in variance
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance reverts to a long run level
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Model dependent - Options with the same underlying assets may trade at different volatilities

19. Tractable
We reject a hypothesis that is actually true
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Easy to manipulate
Variance(x) + Variance(Y) + 2*covariance(XY)

20. GARCH
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
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)

21. Test for unbiasedness
Easy to manipulate
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
E(mean) = mean
Low Frequency - High Severity events

22. Type I error
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
We reject a hypothesis that is actually true

23. Single variable (univariate) probability
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Concerned with a single random variable (ex. Roll of a die)
Rxy = Sxy/(Sx*Sy)

24. Beta distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
P(Z>t)
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

25. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Model dependent - Options with the same underlying assets may trade at different volatilities
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

26. Conditional probability functions
Normal - Student's T - Chi - square - F distribution
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
We reject a hypothesis that is actually true
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

27. Confidence interval for sample mean
Population denominator = n - Sample denominator = n - 1
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
E(mean) = mean
For n>30 - sample mean is approximately normal

28. Two ways to calculate historical volatility
(a^2)(variance(x)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Summation((xi - mean)^k)/n
P(Z>t)

29. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Sampling distribution of sample means tend to be normal
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

30. What does the OLS minimize?
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance(X) + Variance(Y) - 2*covariance(XY)
SSR
Distribution with only two possible outcomes

31. Continuously compounded return equation
i = ln(Si/Si - 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)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

32. Pooled data
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Normal - Student's T - Chi - square - F distribution
Returns over time for a combination of assets (combination of time series and cross - sectional data)
When one regressor is a perfect linear function of the other regressors

33. Sample mean
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Expected value of the sample mean is the population mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Statement of the error or precision of an estimate

34. P - value
P(Z>t)
Does not depend on a prior event or information
Variance(x)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

35. Continuous representation of the GBM
Only requires two parameters = mean and variance
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
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)
Contains variables not explicit in model - Accounts for randomness

36. Cross - sectional
Average return across assets on a given day
Variance(X) + Variance(Y) - 2*covariance(XY)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

37. Chi - squared distribution
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
Application of mathematical statistics to economic data to lend empirical support to models
For n>30 - sample mean is approximately normal
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

38. Confidence ellipse
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(X) + Variance(Y) - 2*covariance(XY)
Confidence set for two coefficients - two dimensional analog for the confidence interval
P(X=x - Y=y) = P(X=x) * P(Y=y)

39. Unbiased
Mean of sampling distribution is the population mean
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Population denominator = n - Sample denominator = n - 1
Mean = np - Variance = npq - Std dev = sqrt(npq)

40. Adjusted R^2
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

41. Mean reversion
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Based on an equation - P(A) = # of A/total outcomes
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Returns over time for an individual asset

42. Mean(expected value)
More than one random variable
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

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

44. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Nonlinearity
More than one random variable
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

45. Variance(discrete)
Low Frequency - High Severity events
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)
Independently and Identically Distributed

46. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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)
Sampling distribution of sample means tend to be normal

47. POT
Peaks over threshold - Collects dataset in excess of some threshold
Sample mean will near the population mean as the sample size increases
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
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

48. Simplified standard (un - weighted) variance
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
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
Among all unbiased estimators - estimator with the smallest variance is efficient

49. 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
Sampling distribution of sample means tend to be 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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

50. Bootstrap method
Concerned with a single random variable (ex. Roll of a die)
Contains variables not explicit in model - Accounts for randomness
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model