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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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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. Priori (classical) probability
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Nonlinearity
When the sample size is large - the uncertainty about the value of the sample is very small
Based on an equation - P(A) = # of A/total outcomes

2. SER
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

3. Type II Error
We accept a hypothesis that should have been rejected
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Least absolute deviations estimator - used when extreme outliers are not uncommon
(a^2)(variance(x)

4. Marginal unconditional probability function
Does not depend on a prior event or information
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
We reject a hypothesis that is actually true

5. Deterministic Simulation
Special type of pooled data in which the cross sectional unit is surveyed over time
Does not depend on a prior event or information
Sample mean +/ - t*(stddev(s)/sqrt(n))
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

6. Consistent
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
When the sample size is large - the uncertainty about the value of the sample is very small
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

7. Covariance
(a^2)(variance(x)
Statement of the error or precision of an estimate
E(XY) - E(X)E(Y)
Average return across assets on a given day

8. Standard variable for non - normal distributions
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!)
Z = (Y - meany)/(stddev(y)/sqrt(n))
When one regressor is a perfect linear function of the other regressors
If variance of the conditional distribution of u(i) is not constant

9. Overall F - statistic
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
E(XY) - E(X)E(Y)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

10. Time series data
E(mean) = mean
Low Frequency - High Severity events
Returns over time for an individual asset
Nonlinearity

11. BLUE
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Regression can be non - linear in variables but must be linear in parameters
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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!)

12. Potential reasons for fat tails in return distributions
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

13. Homoskedastic
Average return across assets on a given day
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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

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

15. Hybrid method for conditional volatility
Random walk (usually acceptable) - Constant volatility (unlikely)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Use historical simulation approach but use the EWMA weighting system
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

16. Homoskedastic only F - stat
Based on an equation - P(A) = # of A/total outcomes
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

17. Type I error
Returns over time for a combination of assets (combination of time series and cross - sectional data)
We reject a hypothesis that is actually true
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
If variance of the conditional distribution of u(i) is not constant

18. Importance sampling technique
P - value
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Attempts to sample along more important paths
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

19. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
E(mean) = mean
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

20. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance reverts to a long run level
Var(X) + Var(Y)
E(mean) = mean

21. Continuous representation of the GBM
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
We reject a hypothesis that is actually true
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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)

22. WLS
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

23. Variance of weighted scheme
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
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
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

24. Direction of OVB
95% = 1.65 99% = 2.33 For one - tailed tests
When the sample size is large - the uncertainty about the value of the sample is very small
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

25. GPD
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

26. Sample correlation
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Among all unbiased estimators - estimator with the smallest variance is efficient
Rxy = Sxy/(Sx*Sy)

27. Implications of homoscedasticity
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
Independently and Identically Distributed

28. Monte Carlo Simulations
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
E(mean) = mean
95% = 1.65 99% = 2.33 For one - tailed tests

29. 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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Attempts to sample along more important paths

30. Mean reversion in asset dynamics
Confidence level
Price/return tends to run towards a long - run level
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
When the sample size is large - the uncertainty about the value of the sample is very small

31. Two assumptions of square root rule
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Random walk (usually acceptable) - Constant volatility (unlikely)

32. Mean reversion in variance
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance reverts to a long run level
E(XY) - E(X)E(Y)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

33. Implied standard deviation for options
Rxy = Sxy/(Sx*Sy)
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
Only requires two parameters = mean and variance
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

34. Sample mean
Only requires two parameters = mean and variance
Does not depend on a prior event or information
Expected value of the sample mean is the population mean
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

35. Variance of sampling distribution of means when n<N
Var(X) + Var(Y)
Based on a dataset
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Distribution with only two possible outcomes

36. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Probability that the random variables take on certain values simultaneously
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

37. Four sampling distributions

38. Pooled data
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Based on an equation - P(A) = # of A/total outcomes

39. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Contains variables not explicit in model - Accounts for randomness
Variance reverts to a long run level
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

40. Historical std dev
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Model dependent - Options with the same underlying assets may trade at different volatilities
Among all unbiased estimators - estimator with the smallest variance is efficient

41. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
We reject a hypothesis that is actually true
Based on a dataset
When one regressor is a perfect linear function of the other regressors

42. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

43. Sample variance
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance reverts to a long run level
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

44. Test for statistical independence
Returns over time for an individual asset
P(X=x - Y=y) = P(X=x) * P(Y=y)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

45. Exact significance level
Sample mean will near the population mean as the sample size increases
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
P - value

46. Difference between population and sample variance
Statement of the error or precision of an estimate
Population denominator = n - Sample denominator = n - 1
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
If variance of the conditional distribution of u(i) is not constant

47. Significance =1
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Model dependent - Options with the same underlying assets may trade at different volatilities
Confidence level

48. Binomial distribution equations for mean variance and std dev
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Mean = np - Variance = npq - Std dev = sqrt(npq)
Normal - Student's T - Chi - square - F distribution

49. Shortcomings of implied volatility
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance(X) + Variance(Y) - 2*covariance(XY)

50. Normal distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Nonlinearity
Sample mean +/ - t*(stddev(s)/sqrt(n))