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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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Match each statement with the correct term.
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1. LFHS
SSR
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
Low Frequency - High Severity events
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

2. Priori (classical) probability
Statement of the error or precision of an estimate
Based on an equation - P(A) = # of A/total outcomes
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

3. Conditional probability functions
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Independently and Identically Distributed
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

4. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
More than one random variable
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

5. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Special type of pooled data in which the cross sectional unit is surveyed over time
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
Nonlinearity

6. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Statement of the error or precision of an estimate
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)
Low Frequency - High Severity events

7. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
More than one random variable
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
Regression can be non - linear in variables but must be linear in parameters

8. Covariance calculations using weight sums (lambda)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

9. Single variable (univariate) probability
For n>30 - sample mean is approximately normal
Variance(y)/n = variance of sample Y
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
Concerned with a single random variable (ex. Roll of a die)

10. Expected future variance rate (t periods forward)
Does not depend on a prior event or information
Distribution with only two possible outcomes
Based on an equation - P(A) = # of A/total outcomes
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

11. Confidence interval (from t)
Based on a dataset
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))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

12. Standard error for Monte Carlo replications
Confidence set for two coefficients - two dimensional analog for the confidence interval
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

13. Gamma distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

14. Exponential distribution
Attempts to sample along more important paths
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

15. Maximum likelihood method
Probability that the random variables take on certain values simultaneously
Choose parameters that maximize the likelihood of what observations occurring
We reject a hypothesis that is actually true
Has heavy tails

16. Two assumptions of square root rule
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
We reject a hypothesis that is actually true
Random walk (usually acceptable) - Constant volatility (unlikely)

17. Binomial distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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!)
Based on an equation - P(A) = # of A/total outcomes
(a^2)(variance(x)) + (b^2)(variance(y))

18. Bootstrap method
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

19. Econometrics
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Application of mathematical statistics to economic data to lend empirical support to models
Use historical simulation approach but use the EWMA weighting system
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

20. P - value
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
P(Z>t)
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
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

21. Variance of aX
Var(X) + Var(Y)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
(a^2)(variance(x)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

22. Type II Error
Concerned with a single random variable (ex. Roll of a die)
We accept a hypothesis that should have been rejected
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Rxy = Sxy/(Sx*Sy)

23. Continuous representation of the GBM
Has heavy tails
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)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance reverts to a long run level

24. Implications of homoscedasticity
Probability that the random variables take on certain values simultaneously
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
We accept a hypothesis that should have been rejected

25. Cholesky factorization (decomposition)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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

26. Variance of sample mean
Transformed to a unit variable - Mean = 0 Variance = 1
Low Frequency - High Severity events
Variance(y)/n = variance of sample Y
Least absolute deviations estimator - used when extreme outliers are not uncommon

27. WLS
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
Sampling distribution of sample means tend to be normal
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

28. Simulation models
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
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
Variance(X) + Variance(Y) - 2*covariance(XY)
Low Frequency - High Severity events

29. Multivariate probability
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
More than one random variable
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
Based on an equation - P(A) = # of A/total outcomes

30. Variance of weighted scheme
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
Variance(X) + Variance(Y) - 2*covariance(XY)
Var(X) + Var(Y)

31. Joint probability functions
Among all unbiased estimators - estimator with the smallest variance is efficient
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Probability that the random variables take on certain values simultaneously

32. Four sampling distributions

33. Economical(elegant)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Only requires two parameters = mean and variance
(a^2)(variance(x)

34. Continuously compounded return equation
i = ln(Si/Si - 1)
P(Z>t)
Variance(x) + Variance(Y) + 2*covariance(XY)
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. SER
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
[1/(n - 1)]*summation((Xi - X)(Yi - 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
Has heavy tails

36. LAD
Contains variables not explicit in model - Accounts for randomness
Least absolute deviations estimator - used when extreme outliers are not uncommon
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

37. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

38. Weibul distribution
We accept a hypothesis that should have been rejected
Has heavy tails
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

39. Hazard rate of exponentially distributed random variable
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
We accept a hypothesis that should have been rejected
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

40. Shortcomings of implied volatility
Regression can be non - linear in variables but must be linear in parameters
95% = 1.65 99% = 2.33 For one - tailed tests
Model dependent - Options with the same underlying assets may trade at different volatilities
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

41. POT
Concerned with a single random variable (ex. Roll of a die)
E(mean) = mean
Peaks over threshold - Collects dataset in excess of some threshold
Confidence level

42. Variance of sampling distribution of means when n<N
Rxy = Sxy/(Sx*Sy)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Based on a dataset

43. Importance sampling technique
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance = (1/m) summation(u<n - i>^2)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Attempts to sample along more important paths

44. Time series data
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)
Returns over time for an individual asset
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

45. Stochastic error term
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Contains variables not explicit in model - Accounts for randomness
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

46. Adjusted R^2
Z = (Y - meany)/(stddev(y)/sqrt(n))
Choose parameters that maximize the likelihood of what observations occurring
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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

47. Unconditional vs conditional distributions
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Special type of pooled data in which the cross sectional unit is surveyed over time

48. Mean reversion
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
We accept a hypothesis that should have been rejected
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

49. Hybrid method for conditional volatility
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
Use historical simulation approach but use the EWMA weighting system
Sample mean will near the population mean as the sample size increases
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

50. Lognormal
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Sample variance = (1/(k - 1))Summation(Yi - mean)^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