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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. Tractable
(a^2)(variance(x)) + (b^2)(variance(y))
Easy to manipulate
When one regressor is a perfect linear function of the other regressors
Variance reverts to a long run level

2. Mean reversion in variance
Variance reverts to a long run level
Sampling distribution of sample means tend to be normal
Combine to form distribution with leptokurtosis (heavy tails)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

3. Beta distribution
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

4. R^2
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Combine to form distribution with leptokurtosis (heavy tails)
Does not depend on a prior event or information

5. Overall F - statistic
Variance = (1/m) summation(u<n - i>^2)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance reverts to a long run level
Summation((xi - mean)^k)/n

6. Panel data (longitudinal or micropanel)
Yi = B0 + B1Xi + ui
Sampling distribution of sample means tend to be normal
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
Special type of pooled data in which the cross sectional unit is surveyed over time

7. Unbiased
Mean of sampling distribution is the population mean
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(x) + Variance(Y) + 2*covariance(XY)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

8. Expected future variance rate (t periods forward)
Expected value of the sample mean is the population mean
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
95% = 1.65 99% = 2.33 For one - tailed tests

9. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(X) + Variance(Y) - 2*covariance(XY)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

10. Confidence interval for sample mean
Only requires two parameters = mean and variance
Based on an equation - P(A) = # of A/total outcomes
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Expected value of the sample mean is the population mean

11. Implications of homoscedasticity
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Easy to manipulate

12. Binomial distribution equations for mean variance and std dev
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
More than one random variable
Mean = np - Variance = npq - Std dev = sqrt(npq)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

13. Variance - covariance approach for VaR of a portfolio
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
i = ln(Si/Si - 1)

14. Unconditional vs conditional distributions
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
P(X=x - Y=y) = P(X=x) * P(Y=y)

15. Continuous representation of the GBM
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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)
When the sample size is large - the uncertainty about the value of the sample is very small
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

16. Central Limit Theorem(CLT)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sampling distribution of sample means tend to be normal
Variance = (1/m) summation(u<n - i>^2)

17. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Independently and Identically Distributed
If variance of the conditional distribution of u(i) is not constant
Confidence level

18. Control variates technique
Mean = np - Variance = npq - Std dev = sqrt(npq)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
E(XY) - E(X)E(Y)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

19. Multivariate probability
More than one random variable
(a^2)(variance(x)) + (b^2)(variance(y))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Probability that the random variables take on certain values simultaneously

20. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Z = (Y - meany)/(stddev(y)/sqrt(n))
Least absolute deviations estimator - used when extreme outliers are not uncommon
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

21. Law of Large Numbers
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Sample mean will near the population mean as the sample size increases
Price/return tends to run towards a long - run level
Model dependent - Options with the same underlying assets may trade at different volatilities

22. Chi - squared distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Variance reverts to a long run level
Least absolute deviations estimator - used when extreme outliers are not uncommon

23. Test for statistical independence
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
P(X=x - Y=y) = P(X=x) * P(Y=y)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Normal - Student's T - Chi - square - F distribution

24. Variance of X - Y assuming dependence
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
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance(x)
Distribution with only two possible outcomes

25. Limitations of R^2 (what an increase doesn't necessarily imply)

26. 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
Use historical simulation approach but use the EWMA weighting system
SSR
When one regressor is a perfect linear function of the other regressors

27. Logistic distribution
More than one random variable
P - value
Has heavy tails
Variance(X) + Variance(Y) - 2*covariance(XY)

28. Confidence ellipse
P - value
Confidence set for two coefficients - two dimensional analog for the confidence interval
Combine to form distribution with leptokurtosis (heavy tails)
Sample mean will near the population mean as the sample size increases

29. Simulation models
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
We reject a hypothesis that is actually true
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

30. Bernouli Distribution
Distribution with only two possible outcomes
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

31. Poisson Distribution
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Var(X) + Var(Y)
Regression can be non - linear in variables but must be linear in parameters
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

32. Homoskedastic
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

33. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
E(mean) = mean
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Mean = np - Variance = npq - Std dev = sqrt(npq)

34. Lognormal
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance = (1/m) summation(u<n - i>^2)

35. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

36. Non - parametric vs parametric calculation of VaR
Peaks over threshold - Collects dataset in excess of some threshold
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
Mean of sampling distribution is the population mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

37. P - value
P(Z>t)
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

38. Key properties of linear regression
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

39. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Z = (Y - meany)/(stddev(y)/sqrt(n))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

40. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
E(mean) = mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

41. Maximum likelihood method
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Low Frequency - High Severity events
Choose parameters that maximize the likelihood of what observations occurring
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

42. i.i.d.
When the sample size is large - the uncertainty about the value of the sample is very small
Low Frequency - High Severity events
Based on a dataset
Independently and Identically Distributed

43. Exact significance level
P - value
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
Based on an equation - P(A) = # of A/total outcomes
Variance reverts to a long run level

44. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 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)
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

45. Skewness
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)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Low Frequency - High Severity events

46. BLUE
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Random walk (usually acceptable) - Constant volatility (unlikely)

47. Cholesky factorization (decomposition)
Among all unbiased estimators - estimator with the smallest variance is efficient
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

48. Persistence
Rxy = Sxy/(Sx*Sy)
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
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

49. Sample covariance
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Independently and Identically Distributed
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

50. Unstable return distribution
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
We reject a hypothesis that is actually true