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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Choose parameters that maximize the likelihood of what observations occurring

2. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
P - value
Contains variables not explicit in model - Accounts for randomness
Concerned with a single random variable (ex. Roll of a die)

3. Marginal unconditional probability function
Special type of pooled data in which the cross sectional unit is surveyed over time
Does not depend on a prior event or information
Choose parameters that maximize the likelihood of what observations occurring
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

4. 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
Only requires two parameters = mean and variance
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

5. GARCH
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

6. Binomial distribution
Attempts to sample along more important paths
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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!)
More than one random variable

7. Standard error
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
Var(X) + Var(Y)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

8. Reliability
Statement of the error or precision of an estimate
E(XY) - E(X)E(Y)
Normal - Student's T - Chi - square - F distribution
Variance(x)

9. Implied standard deviation for options
Confidence set for two coefficients - two dimensional analog for the confidence interval
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Based on an equation - P(A) = # of A/total outcomes
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

10. Kurtosis
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Rxy = Sxy/(Sx*Sy)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

11. Importance sampling technique
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Attempts to sample along more important paths
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

12. Variance of X - Y assuming dependence
E(mean) = mean
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Normal - Student's T - Chi - square - F distribution
Variance(X) + Variance(Y) - 2*covariance(XY)

13. Statistical (or empirical) model
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Yi = B0 + B1Xi + ui
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

14. Consistent
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
We accept a hypothesis that should have been rejected
When the sample size is large - the uncertainty about the value of the sample is very small
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

15. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(x) + Variance(Y) + 2*covariance(XY)
For n>30 - sample mean is approximately normal
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

16. Joint probability functions
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
(a^2)(variance(x)
Probability that the random variables take on certain values simultaneously
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

17. BLUE
Does not depend on a prior event or information
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Statement of the error or precision of an estimate
Variance(y)/n = variance of sample Y

18. Persistence
Choose parameters that maximize the likelihood of what observations occurring
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

19. Unconditional vs conditional distributions
We reject a hypothesis that is actually true
(a^2)(variance(x)) + (b^2)(variance(y))
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

20. Conditional probability functions
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance(X) + Variance(Y) - 2*covariance(XY)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Sample mean will near the population mean as the sample size increases

21. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
When one regressor is a perfect linear function of the other regressors
Based on a dataset
Price/return tends to run towards a long - run level

22. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

23. LFHS
E(mean) = mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Low Frequency - High Severity events

24. EWMA
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

25. GPD
Independently and Identically Distributed
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
We reject a hypothesis that is actually true
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

26. Homoskedastic only F - stat
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
P - value
We accept a hypothesis that should have been rejected
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

27. Standard variable for non - normal distributions
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Among all unbiased estimators - estimator with the smallest variance is efficient
Z = (Y - meany)/(stddev(y)/sqrt(n))

28. Variance of aX
(a^2)(variance(x)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Among all unbiased estimators - estimator with the smallest variance is efficient
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

29. Variance of X+b
Variance(x)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

30. Potential reasons for fat tails in return distributions
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Random walk (usually acceptable) - Constant volatility (unlikely)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

31. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Least absolute deviations estimator - used when extreme outliers are not uncommon
If variance of the conditional distribution of u(i) is not constant
Mean of sampling distribution is the population mean

32. Critical z values
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
95% = 1.65 99% = 2.33 For one - tailed tests
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

33. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

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

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35. LAD
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
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Least absolute deviations estimator - used when extreme outliers are not uncommon

36. Standard error for Monte Carlo replications
Variance(y)/n = variance of sample Y
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Has heavy tails
Combine to form distribution with leptokurtosis (heavy tails)

37. Continuously compounded return equation
P - value
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
i = ln(Si/Si - 1)
95% = 1.65 99% = 2.33 For one - tailed tests

38. Discrete random variable
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
(a^2)(variance(x)) + (b^2)(variance(y))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

39. Simulating for VaR
(a^2)(variance(x)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

40. Weibul distribution
Has heavy tails
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Yi = B0 + B1Xi + ui

41. Extreme Value Theory
Confidence level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
When the sample size is large - the uncertainty about the value of the sample is very small
Only requires two parameters = mean and variance

42. Law of Large Numbers
Returns over time for an individual asset
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Sample mean will near the population mean as the sample size increases
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

43. Result of combination of two normal with same means
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)
Returns over time for an individual asset
Regression can be non - linear in variables but must be linear in parameters

44. Variance - covariance approach for VaR of a portfolio
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

45. Skewness
Statement of the error or precision of an estimate
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
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!)
Expected value of the sample mean is the population mean

46. Mean reversion in variance
Variance(x)
We accept a hypothesis that should have been rejected
Variance reverts to a long run level
P(Z>t)

47. Square root rule
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Distribution with only two possible outcomes
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

48. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
95% = 1.65 99% = 2.33 For one - tailed tests
Z = (Y - meany)/(stddev(y)/sqrt(n))

49. Gamma distribution
Summation((xi - mean)^k)/n
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

50. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Average return across assets on a given day