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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. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(y)/n = variance of sample Y
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance(x)

2. EWMA
Choose parameters that maximize the likelihood of what observations occurring
Probability that the random variables take on certain values simultaneously
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

3. K - th moment
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)
Summation((xi - mean)^k)/n
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

4. Critical z values
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

5. Continuous representation of the GBM
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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)
Sample mean will near the population mean as the sample size increases
95% = 1.65 99% = 2.33 For one - tailed tests

6. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
More than one random variable
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

7. Persistence
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

8. Hybrid method for conditional volatility
Normal - Student's T - Chi - square - F distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Use historical simulation approach but use the EWMA weighting system

9. Logistic distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Has heavy tails
Based on a dataset
Normal - Student's T - Chi - square - F distribution

10. 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
Does not depend on a prior event or information
Least absolute deviations estimator - used when extreme outliers are not uncommon
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

11. Standard error for Monte Carlo replications
P - value
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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

12. Covariance
P(X=x - Y=y) = P(X=x) * P(Y=y)
Regression can be non - linear in variables but must be linear in parameters
Z = (Y - meany)/(stddev(y)/sqrt(n))
E(XY) - E(X)E(Y)

13. Sample covariance
Statement of the error or precision of an estimate
Distribution with only two possible outcomes
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Normal - Student's T - Chi - square - F distribution

14. LFHS
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
For n>30 - sample mean is approximately normal
Low Frequency - High Severity events
Easy to manipulate

15. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Normal - Student's T - Chi - square - F distribution

16. Simulation models
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean of sampling distribution is the population mean

17. Variance of weighted scheme
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
i = ln(Si/Si - 1)
Attempts to sample along more important paths

18. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Easy to manipulate
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

19. Gamma distribution
Variance(X) + Variance(Y) - 2*covariance(XY)
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 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
Transformed to a unit variable - Mean = 0 Variance = 1

20. Biggest (and only real) drawback of GARCH mode
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
If variance of the conditional distribution of u(i) is not constant
Nonlinearity
Application of mathematical statistics to economic data to lend empirical support to models

21. Time series data
Returns over time for an individual asset
Variance(y)/n = variance of sample Y
Special type of pooled data in which the cross sectional unit is surveyed over time
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

22. Statistical (or empirical) model
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Yi = B0 + B1Xi + ui
More than one random variable
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

23. Simulating for VaR
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Application of mathematical statistics to economic data to lend empirical support to 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

24. Multivariate probability
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
More than one random variable
Peaks over threshold - Collects dataset in excess of some threshold

25. Direction of OVB
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Does not depend on a prior event or information
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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

26. Adjusted R^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P - value
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

27. Cholesky factorization (decomposition)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Yi = B0 + B1Xi + ui
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

28. Central Limit Theorem
For n>30 - sample mean is approximately normal
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Variance(y)/n = variance of sample Y

29. Discrete representation of the GBM
Confidence level
Population denominator = n - Sample denominator = n - 1
P(X=x - Y=y) = P(X=x) * P(Y=y)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

30. Non - parametric vs parametric calculation of VaR
We reject a hypothesis that is actually true
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

31. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Transformed to a unit variable - Mean = 0 Variance = 1

32. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
More than one random variable
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
(a^2)(variance(x)

33. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
E(XY) - E(X)E(Y)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

34. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/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
Random walk (usually acceptable) - Constant volatility (unlikely)
Sample mean will near the population mean as the sample size increases

35. Type II Error
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
We accept a hypothesis that should have been rejected
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

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

37. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Z = (Y - meany)/(stddev(y)/sqrt(n))
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

38. Mean reversion
E(XY) - E(X)E(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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

39. ESS
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Confidence level

40. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
If variance of the conditional distribution of u(i) is not constant

41. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
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())
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)

42. Multivariate Density Estimation (MDE)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Does not depend on a prior event or information
Transformed to a unit variable - Mean = 0 Variance = 1

43. Sample correlation
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)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Rxy = Sxy/(Sx*Sy)

44. Type I error
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
We reject a hypothesis that is actually true
E(XY) - E(X)E(Y)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

45. Mean reversion in variance
Variance reverts to a long run level
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
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)

46. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Distribution with only two possible outcomes
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

47. Empirical frequency
Attempts to sample along more important paths
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
For n>30 - sample mean is approximately normal
Based on a dataset

48. Test for statistical independence
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

49. Economical(elegant)
Only requires two parameters = mean and variance
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Choose parameters that maximize the likelihood of what observations occurring
Independently and Identically Distributed

50. Two ways to calculate historical volatility
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Has heavy tails
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications