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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. P - value
P(Z>t)
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

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

3. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

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

5. Two requirements of OVB
P(X=x - Y=y) = P(X=x) * P(Y=y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

6. POT
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)
Peaks over threshold - Collects dataset in excess of some threshold
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Easy to manipulate

7. GEV
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

8. 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
When one regressor is a perfect linear function of the other regressors
(a^2)(variance(x)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

9. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Average return across assets on a given day
If variance of the conditional distribution of u(i) is not constant

10. Perfect multicollinearity
Does not depend on a prior event or information
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
When one regressor is a perfect linear function of the other regressors
Rxy = Sxy/(Sx*Sy)

11. Variance of X+Y
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Var(X) + Var(Y)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(X) + Variance(Y) - 2*covariance(XY)

12. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(x) + Variance(Y) + 2*covariance(XY)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

13. Gamma distribution
Variance(x)
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
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
Sample mean will near the population mean as the sample size increases

14. Variance of X+b
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Application of mathematical statistics to economic data to lend empirical support to models
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance(x)

15. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

16. Marginal unconditional probability function
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
P - value
Does not depend on a prior event or information

17. Sample variance
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Application of mathematical statistics to economic data to lend empirical support to models

18. Law of Large Numbers
i = ln(Si/Si - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Confidence level
Sample mean will near the population mean as the sample size increases

19. Reliability
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
Statement of the error or precision of an estimate
Mean of sampling distribution is the population mean

20. Block maxima
Regression can be non - linear in variables but must be linear in parameters
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Rxy = Sxy/(Sx*Sy)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

21. Variance of sampling distribution of means when n<N
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance(sample y) = (variance(y)/n)*(N - n/N - 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)
Application of mathematical statistics to economic data to lend empirical support to models

22. Beta distribution
Does not depend on a prior event or information
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

23. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
If variance of the conditional distribution of u(i) is not constant
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
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

24. Bootstrap method
Sample mean +/ - t*(stddev(s)/sqrt(n))
P(Z>t)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

25. LFHS
For n>30 - sample mean is approximately normal
Low Frequency - High Severity events
SSR
Statement of the error or precision of an estimate

26. Significance =1
If variance of the conditional distribution of u(i) is not constant
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
Confidence level
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

27. Weibul distribution
SSR
Only requires two parameters = mean and variance
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

28. Overall F - statistic
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Among all unbiased estimators - estimator with the smallest variance is efficient
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

29. Implied standard deviation for options
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

30. Bernouli Distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Confidence level
Independently and Identically Distributed
Distribution with only two possible outcomes

31. Standard variable for non - normal distributions
Price/return tends to run towards a long - run level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

32. ESS
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Use historical simulation approach but use the EWMA weighting system

33. 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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Has heavy tails

34. Unbiased
Mean of sampling distribution is the population mean
Combine to form distribution with leptokurtosis (heavy tails)
Independently and Identically Distributed
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

35. Central Limit Theorem(CLT)
Sample mean will near the population mean as the sample size increases
Sampling distribution of sample means tend to be normal
Z = (Y - meany)/(stddev(y)/sqrt(n))
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)

36. Four sampling distributions

37. Variance of sample mean
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Mean of sampling distribution is the population mean
Variance(y)/n = variance of sample Y
Yi = B0 + B1Xi + ui

38. Variance of aX + bY
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
Mean of sampling distribution is the population mean
(a^2)(variance(x)) + (b^2)(variance(y))

39. Simulation models
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

40. Variance(discrete)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Peaks over threshold - Collects dataset in excess of some threshold
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

41. Mean reversion in variance
Variance reverts to a long run level
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Confidence level

42. Homoskedastic only F - stat
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)
Has heavy tails
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

43. Difference between population and sample 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
Population denominator = n - Sample denominator = n - 1
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Nonlinearity

44. Two ways to calculate historical volatility
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Mean = np - Variance = npq - Std dev = sqrt(npq)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

45. Inverse transform method
When the sample size is large - the uncertainty about the value of the sample is very small
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

46. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))

47. Type I error
If variance of the conditional distribution of u(i) is not constant
We reject a hypothesis that is actually true
Population denominator = n - Sample denominator = n - 1
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

48. Continuous random variable
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Peaks over threshold - Collects dataset in excess of some threshold

49. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Variance(y)/n = variance of sample Y
Peaks over threshold - Collects dataset in excess of some threshold
Yi = B0 + B1Xi + ui

50. Tractable
Average return across assets on a given day
Easy to manipulate
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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