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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. Mean reversion in variance
Variance reverts to a long run level
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!)
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)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

2. Variance of X - Y assuming dependence
Expected value of the sample mean is the population mean
When one regressor is a perfect linear function of the other regressors
P - value
Variance(X) + Variance(Y) - 2*covariance(XY)

3. GPD
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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

4. Maximum likelihood method
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Statement of the error or precision of an estimate
Independently and Identically Distributed
Choose parameters that maximize the likelihood of what observations occurring

5. Sample covariance
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
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Var(X) + Var(Y)

6. Sample variance
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Choose parameters that maximize the likelihood of what observations occurring

7. Variance of weighted scheme
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

8. Hybrid method for conditional volatility
Variance = (1/m) summation(u<n - i>^2)
Rxy = Sxy/(Sx*Sy)
Use historical simulation approach but use the EWMA weighting system
(a^2)(variance(x)

9. LFHS
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Low Frequency - High Severity events
Returns over time for a combination of assets (combination of time series and cross - sectional data)
P(Z>t)

10. Adjusted R^2
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Mean of sampling distribution is the population mean
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

11. Extending the HS approach for computing value of a portfolio
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
If variance of the conditional distribution of u(i) is not constant

12. Covariance
SSR
Returns over time for an individual asset
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
E(XY) - E(X)E(Y)

13. K - th moment
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Summation((xi - mean)^k)/n
P(X=x - Y=y) = P(X=x) * P(Y=y)
SSR

14. Significance =1
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
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
Confidence level
Model dependent - Options with the same underlying assets may trade at different volatilities

15. Cholesky factorization (decomposition)
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

16. Critical z values
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
95% = 1.65 99% = 2.33 For one - tailed tests
Summation((xi - mean)^k)/n

17. Unbiased
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Mean of sampling distribution is the population mean
Variance = (1/m) summation(u<n - i>^2)
Has heavy tails

18. Variance of X+b
Variance = (1/m) summation(u<n - i>^2)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance(x)
When the sample size is large - the uncertainty about the value of the sample is very small

19. Exact significance level
Has heavy tails
Independently and Identically Distributed
P - value
P(Z>t)

20. Hazard rate of exponentially distributed random variable
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
P(X=x - Y=y) = P(X=x) * P(Y=y)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

21. Square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance(x)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

22. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Average return across assets on a given day
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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

23. Overall F - statistic
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

24. Continuously compounded return equation
Concerned with a single random variable (ex. Roll of a die)
i = ln(Si/Si - 1)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Sampling distribution of sample means tend to be normal

25. Standard error for Monte Carlo replications
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance reverts to a long run level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

26. Kurtosis
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance = (1/m) summation(u<n - i>^2)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

27. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Normal - Student's T - Chi - square - F distribution
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

28. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

29. Statistical (or empirical) model
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
95% = 1.65 99% = 2.33 For one - tailed tests
Yi = B0 + B1Xi + ui

30. Key properties of linear regression
Variance(x) + Variance(Y) + 2*covariance(XY)
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
Variance reverts to a long run level

31. BLUE
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance = (1/m) summation(u<n - i>^2)

32. Sample correlation
Mean of sampling distribution is the population mean
Rxy = Sxy/(Sx*Sy)
Concerned with a single random variable (ex. Roll of a die)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

33. Central Limit Theorem(CLT)
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(x)
Peaks over threshold - Collects dataset in excess of some threshold
Sampling distribution of sample means tend to be normal

34. Sample mean
Independently and Identically Distributed
Expected value of the sample mean is the population mean
Sample mean will near the population mean as the sample size increases
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

35. i.i.d.
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(X) + Variance(Y) - 2*covariance(XY)
Independently and Identically Distributed
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

36. Econometrics
Returns over time for an individual asset
Application of mathematical statistics to economic data to lend empirical support to models
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)
Variance(x)

37. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
P(Z>t)

38. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
E(mean) = mean

39. Variance of sampling distribution of means when n<N
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
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Does not depend on a prior event or information

40. WLS
Choose parameters that maximize the likelihood of what observations occurring
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
We accept a hypothesis that should have been rejected
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

41. What does the OLS minimize?
Mean = np - Variance = npq - Std dev = sqrt(npq)
SSR
Returns over time for an individual asset
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

42. Multivariate Density Estimation (MDE)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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)
Confidence set for two coefficients - two dimensional analog for the confidence interval

43. Two drawbacks of moving average series
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
For n>30 - sample mean is approximately normal
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

44. Importance sampling technique
Variance(x) + Variance(Y) + 2*covariance(XY)
Random walk (usually acceptable) - Constant volatility (unlikely)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Attempts to sample along more important paths

45. Conditional probability functions
If variance of the conditional distribution of u(i) is not constant
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Combine to form distribution with leptokurtosis (heavy tails)

46. Non - parametric vs parametric calculation of VaR
Based on an equation - P(A) = # of A/total outcomes
Variance(x) + Variance(Y) + 2*covariance(XY)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

47. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Independently and Identically Distributed
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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

48. Difference between population and sample variance
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Population denominator = n - Sample denominator = n - 1
Based on an equation - P(A) = # of A/total outcomes

49. Variance of aX + bY
Based on a dataset
(a^2)(variance(x)) + (b^2)(variance(y))
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
P(Z>t)

50. Standard error
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Among all unbiased estimators - estimator with the smallest variance is efficient
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)

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FRM Foundations Of Risk Management Quantitative Methods

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