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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. Skewness
Application of mathematical statistics to economic data to lend empirical support to models
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
We accept a hypothesis that should have been rejected
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

2. Variance of X+b
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance(x)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

3. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
i = ln(Si/Si - 1)
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
Confidence level

4. F distribution
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Has heavy tails
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
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

5. Antithetic variable technique
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

6. Expected future variance rate (t periods forward)
Only requires two parameters = mean and variance
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

7. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P - value
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

8. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Application of mathematical statistics to economic data to lend empirical support to models

9. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(X) + Variance(Y) - 2*covariance(XY)
When one regressor is a perfect linear function of the other regressors

10. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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)
(a^2)(variance(x)

11. Stochastic error term
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Based on a dataset
Contains variables not explicit in model - Accounts for randomness
More than one random variable

12. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Random walk (usually acceptable) - Constant volatility (unlikely)
Independently and Identically Distributed
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

13. Biggest (and only real) drawback of GARCH mode
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Nonlinearity
Among all unbiased estimators - estimator with the smallest variance is efficient

14. Marginal unconditional probability function
Does not depend on a prior event or information
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
Nonlinearity

15. Variance of X+Y assuming dependence
Variance(y)/n = variance of sample Y
Var(X) + Var(Y)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(x) + Variance(Y) + 2*covariance(XY)

16. Continuous representation of the GBM
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)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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

17. Maximum likelihood method
Normal - Student's T - Chi - square - F distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Choose parameters that maximize the likelihood of what observations occurring
We reject a hypothesis that is actually true

18. Binomial distribution equations for mean variance and std dev
Average return across assets on a given day
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(y)/n = variance of sample Y

19. Economical(elegant)
Only requires two parameters = mean and variance
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Application of mathematical statistics to economic data to lend empirical support to models

20. Continuous random variable
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Var(X) + Var(Y)

21. Central Limit Theorem
For n>30 - sample mean is approximately normal
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
(a^2)(variance(x)) + (b^2)(variance(y))

22. Least squares estimator(m)
Transformed to a unit variable - Mean = 0 Variance = 1
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Sample mean +/ - t*(stddev(s)/sqrt(n))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

23. LAD
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
i = ln(Si/Si - 1)
Least absolute deviations estimator - used when extreme outliers are not uncommon

24. Homoskedastic only F - stat
Variance(x)
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)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

25. Adjusted R^2
Does not depend on a prior event or information
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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. Perfect multicollinearity
Sample mean will near the population mean as the sample size increases
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
When one regressor is a perfect linear function of the other regressors
Based on an equation - P(A) = # of A/total outcomes

27. Implications of homoscedasticity
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

28. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

29. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean will near the population mean as the sample size increases

30. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
95% = 1.65 99% = 2.33 For one - tailed tests
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
E(mean) = mean

31. Historical std dev
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(X) + Variance(Y) - 2*covariance(XY)
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

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

33. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Expected value of the sample mean is the population mean
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

34. Standard normal distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

35. Binomial distribution
(a^2)(variance(x)) + (b^2)(variance(y))
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!)
Has heavy tails
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

36. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Rxy = Sxy/(Sx*Sy)
Variance(y)/n = variance of sample Y

37. Unconditional vs conditional distributions
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance reverts to a long run level
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

38. Mean reversion in variance
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance reverts to a long run level
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Attempts to sample along more important paths

39. Kurtosis
Mean = np - Variance = npq - Std dev = sqrt(npq)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Easy to manipulate
Z = (Y - meany)/(stddev(y)/sqrt(n))

40. Joint probability functions
Concerned with a single random variable (ex. Roll of a die)
Probability that the random variables take on certain values simultaneously
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

41. Two drawbacks of moving average series
Does not depend on a prior event or information
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Concerned with a single random variable (ex. Roll of a die)
Sampling distribution of sample means tend to be normal

42. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Sampling distribution of sample means tend to be normal
Special type of pooled data in which the cross sectional unit is surveyed over time

43. Non - parametric vs parametric calculation of VaR
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
E(XY) - E(X)E(Y)

44. Type I error
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
We reject a hypothesis that is actually true
E(XY) - E(X)E(Y)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

45. Bernouli Distribution
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Distribution with only two possible outcomes
Concerned with a single random variable (ex. Roll of a die)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

46. Confidence interval for sample mean
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

47. Cholesky factorization (decomposition)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
When the sample size is large - the uncertainty about the value of the sample is very small
Regression can be non - linear in variables but must be linear in parameters

48. Two ways to calculate historical volatility
We reject a hypothesis that is actually true
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Mean of sampling distribution is the population mean

49. BLUE
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Mean = np - Variance = npq - Std dev = sqrt(npq)

50. 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)
Variance = (1/m) summation(u<n - i>^2)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE