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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. Perfect multicollinearity
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
When one regressor is a perfect linear function of the other regressors
Returns over time for an individual asset
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

2. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Application of mathematical statistics to economic data to lend empirical support to models
Z = (Y - meany)/(stddev(y)/sqrt(n))
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

3. i.i.d.
Independently and Identically Distributed
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

4. Mean reversion in variance
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
Variance reverts to a long run level
(a^2)(variance(x)) + (b^2)(variance(y))
Nonlinearity

5. Block maxima
P - value
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

6. Multivariate probability
i = ln(Si/Si - 1)
When one regressor is a perfect linear function of the other regressors
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
More than one random variable

7. Central Limit Theorem
For n>30 - sample mean is approximately normal
Random walk (usually acceptable) - Constant volatility (unlikely)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Population denominator = n - Sample denominator = n - 1

8. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Variance(sample y) = (variance(y)/n)*(N - n/N - 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)
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

9. Economical(elegant)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Regression can be non - linear in variables but must be linear in parameters
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
Only requires two parameters = mean and variance

10. Multivariate Density Estimation (MDE)
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

11. Variance of aX + bY
Mean of sampling distribution is the population mean
Based on a dataset
(a^2)(variance(x)) + (b^2)(variance(y))
Var(X) + Var(Y)

12. Chi - squared distribution
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Application of mathematical statistics to economic data to lend empirical support to models
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

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

14. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
P(X=x - Y=y) = P(X=x) * P(Y=y)

15. Variance - covariance approach for VaR of a portfolio
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

16. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

17. Overall F - statistic
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
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

18. Mean reversion in asset dynamics
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Price/return tends to run towards a long - run level
When one regressor is a perfect linear function of the other regressors
We accept a hypothesis that should have been rejected

19. Potential reasons for fat tails in return distributions
Easy to manipulate
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

20. Expected future variance rate (t periods forward)
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Distribution with only two possible outcomes
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

21. SER
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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

22. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Random walk (usually acceptable) - Constant volatility (unlikely)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 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

23. Confidence ellipse
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

24. Monte Carlo Simulations
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Model dependent - Options with the same underlying assets may trade at different volatilities
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

25. Kurtosis
Variance(y)/n = variance of sample Y
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

26. Cross - sectional
Average return across assets on a given day
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

27. Test for unbiasedness
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample mean will near the population mean as the sample size increases
E(mean) = mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

28. Sample variance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

29. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Average return across assets on a given day
Special type of pooled data in which the cross sectional unit is surveyed over time
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

30. Continuous random variable
Transformed to a unit variable - Mean = 0 Variance = 1
95% = 1.65 99% = 2.33 For one - tailed tests
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Price/return tends to run towards a long - run level

31. Standard variable for non - normal distributions
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(x) + Variance(Y) + 2*covariance(XY)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

32. Tractable
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Easy to manipulate
Price/return tends to run towards a long - run level

33. Efficiency
i = ln(Si/Si - 1)
Variance(x) + Variance(Y) + 2*covariance(XY)
Among all unbiased estimators - estimator with the smallest variance is efficient
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

34. Bernouli Distribution
Application of mathematical statistics to economic data to lend empirical support to models
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Distribution with only two possible outcomes
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

35. Consistent
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
When the sample size is large - the uncertainty about the value of the sample is very small
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

36. Mean reversion
Confidence set for two coefficients - two dimensional analog for the confidence interval
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
(a^2)(variance(x)) + (b^2)(variance(y))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

37. Hybrid method for conditional volatility
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
Use historical simulation approach but use the EWMA weighting system
P(X=x - Y=y) = P(X=x) * P(Y=y)
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

38. Unconditional vs conditional distributions
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance(X) + Variance(Y) - 2*covariance(XY)
Peaks over threshold - Collects dataset in excess of some threshold

39. Sample mean
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
P(Z>t)
Expected value of the sample mean is the population mean

40. Homoskedastic
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Combine to form distribution with leptokurtosis (heavy tails)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

41. Sample covariance
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
When one regressor is a perfect linear function of the other regressors
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Nonlinearity

42. K - th moment
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Price/return tends to run towards a long - run level
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Summation((xi - mean)^k)/n

43. Normal distribution
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/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
For n>30 - sample mean is approximately normal
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

44. Four sampling distributions

45. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Does not depend on a prior event or information
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!)

46. Significance =1
Confidence level
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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

47. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Contains variables not explicit in model - Accounts for randomness
Var(X) + Var(Y)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

48. Type I error
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
We reject a hypothesis that is actually true
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

49. Confidence interval for sample mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
E(XY) - E(X)E(Y)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

50. Bootstrap method
Variance(x) + Variance(Y) + 2*covariance(XY)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance(x)
Sample mean +/ - t*(stddev(s)/sqrt(n))