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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. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Easy to manipulate
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

2. Hybrid method for conditional volatility
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Among all unbiased estimators - estimator with the smallest variance is efficient
Use historical simulation approach but use the EWMA weighting system

3. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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)
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

4. Variance of aX
Variance reverts to a long run level
For n>30 - sample mean is approximately normal
Model dependent - Options with the same underlying assets may trade at different volatilities
(a^2)(variance(x)

5. Sample variance
Sampling distribution of sample means tend to be normal
E(XY) - E(X)E(Y)
(a^2)(variance(x)) + (b^2)(variance(y))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

6. Two requirements of OVB
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Choose parameters that maximize the likelihood of what observations occurring
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

7. Mean reversion
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Yi = B0 + B1Xi + ui
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

8. Confidence interval for sample mean
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Has heavy tails

9. 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
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
Statement of the error or precision of an estimate

10. Perfect multicollinearity
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
When one regressor is a perfect linear function of the other regressors
Summation((xi - mean)^k)/n

11. Adjusted R^2
P(Z>t)
We accept a hypothesis that should have been rejected
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
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

12. Lognormal
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Random walk (usually acceptable) - Constant volatility (unlikely)
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!)

13. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Easy to manipulate

14. Two assumptions of square root rule
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Random walk (usually acceptable) - Constant volatility (unlikely)
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

15. Poisson Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
P(Z>t)
If variance of the conditional distribution of u(i) is not constant
Concerned with a single random variable (ex. Roll of a die)

16. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Mean of sampling distribution is the population mean
Transformed to a unit variable - Mean = 0 Variance = 1
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

17. Historical std dev
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

18. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

19. Continuous representation of the GBM
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

20. Discrete representation of the GBM
95% = 1.65 99% = 2.33 For one - tailed tests
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

21. Sample mean
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)
Expected value of the sample mean is the population mean
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance = (1/m) summation(u<n - i>^2)

22. Extreme Value Theory
Z = (Y - meany)/(stddev(y)/sqrt(n))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
(a^2)(variance(x)) + (b^2)(variance(y))

23. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance ratio distribution F = (variance(x)/variance(y)) - Greater sample variance is numerator - Nonnegative and skewed right - Approaches normal as df increases - Square of t - distribution has a F distribution with 1 -k df - M*F(m -n) = Chi - s
Regression can be non - linear in variables but must be linear in parameters

24. Variance - covariance approach for VaR of a portfolio
If variance of the conditional distribution of u(i) is not constant
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Nonlinearity
Easy to manipulate

25. Persistence
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(mean) = mean
P(Z>t)
Mean of sampling distribution is the population mean

26. POT
Peaks over threshold - Collects dataset in excess of some threshold
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Average return across assets on a given day
Variance = (1/m) summation(u<n - i>^2)

27. GEV
Confidence level
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Expected value of the sample mean is the population mean

28. Unbiased
Mean of sampling distribution is the population mean
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance reverts to a long run level

29. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Based on a dataset
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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

30. Tractable
Distribution with only two possible outcomes
E(XY) - E(X)E(Y)
Easy to manipulate
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

31. R^2
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance(y)/n = variance of sample Y

32. Variance(discrete)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

33. Importance sampling technique
Attempts to sample along more important paths
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

34. Panel data (longitudinal or micropanel)
Price/return tends to run towards a long - run level
Special type of pooled data in which the cross sectional unit is surveyed over time
P(X=x - Y=y) = P(X=x) * P(Y=y)
When the sample size is large - the uncertainty about the value of the sample is very small

35. Square root rule
When one regressor is a perfect linear function of the other regressors
i = ln(Si/Si - 1)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

36. Result of combination of two normal with same means
Easy to manipulate
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Combine to form distribution with leptokurtosis (heavy tails)
Has heavy tails

37. Exponential distribution
Var(X) + Var(Y)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Peaks over threshold - Collects dataset in excess of some threshold

38. Implied standard deviation for options
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
We reject a hypothesis that is actually true
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
We accept a hypothesis that should have been rejected

39. Logistic distribution
Has heavy tails
Sample mean will near the population mean as the sample size increases
Price/return tends to run towards a long - run level
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

40. Variance of X+Y assuming dependence
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(x) + Variance(Y) + 2*covariance(XY)
Average return across assets on a given day

41. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Least absolute deviations estimator - used when extreme outliers are not uncommon
P(X=x - Y=y) = P(X=x) * P(Y=y)
Returns over time for an individual asset

42. Central Limit Theorem
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
For n>30 - sample mean is approximately normal
Choose parameters that maximize the likelihood of what observations occurring

43. T distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
E(XY) - E(X)E(Y)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

44. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
95% = 1.65 99% = 2.33 For one - tailed tests
Price/return tends to run towards a long - run level
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

45. Implications of homoscedasticity
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Confidence level
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

46. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
We reject a hypothesis that is actually true
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Sampling distribution of sample means tend to be normal

47. Discrete random variable
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Use historical simulation approach but use the EWMA weighting system
Peaks over threshold - Collects dataset in excess of some threshold
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

48. K - th moment
Returns over time for an individual asset
We accept a hypothesis that should have been rejected
Summation((xi - mean)^k)/n
Contains variables not explicit in model - Accounts for randomness

49. Potential reasons for fat tails in return distributions
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Price/return tends to run towards a long - run level
Regression can be non - linear in variables but must be linear in parameters

50. Single variable (univariate) probability
P - value
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Concerned with a single random variable (ex. Roll of a die)
Price/return tends to run towards a long - run level