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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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1. Reliability
Yi = B0 + B1Xi + ui
Rxy = Sxy/(Sx*Sy)
Statement of the error or precision of an estimate
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

2. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

3. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sampling distribution of sample means tend to be normal
E(XY) - E(X)E(Y)
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)

4. Stochastic error term
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Contains variables not explicit in model - Accounts for randomness
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
95% = 1.65 99% = 2.33 For one - tailed tests

5. Heteroskedastic
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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!)
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

6. Normal distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
(a^2)(variance(x)) + (b^2)(variance(y))
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

7. Discrete representation of the GBM
i = ln(Si/Si - 1)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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)

8. Significance =1
Confidence level
Variance reverts to a long run level
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Attempts to sample along more important paths

9. Time series data
(a^2)(variance(x)
Returns over time for an individual asset
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Sample mean +/ - t*(stddev(s)/sqrt(n))

10. Simulation models
When one regressor is a perfect linear function of the other regressors
Yi = B0 + B1Xi + ui
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

11. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
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 reject a hypothesis that is actually true

12. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

13. Beta distribution
Price/return tends to run towards a long - run level
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

14. Type I error
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Expected value of the sample mean is the population mean
We reject a hypothesis that is actually true

15. LFHS
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Low Frequency - High Severity events

16. BLUE
Based on an equation - P(A) = # of A/total outcomes
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Statement of the error or precision of an estimate
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)

17. ESS
Sample mean will near the population mean as the sample size increases
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(y)/n = variance of sample Y

18. WLS
Only requires two parameters = mean and variance
Sample mean +/ - t*(stddev(s)/sqrt(n))
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

19. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Statement of the error or precision of an estimate
Variance = (1/m) summation(u<n - i>^2)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

20. Test for statistical independence
Based on an equation - P(A) = # of A/total outcomes
P(X=x - Y=y) = P(X=x) * P(Y=y)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Distribution with only two possible outcomes

21. Efficiency
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Peaks over threshold - Collects dataset in excess of some threshold

22. Sample variance
Variance reverts to a long run level
Variance(x)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

23. Mean reversion
Variance reverts to a long run level
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

24. Covariance
E(XY) - E(X)E(Y)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Confidence level

25. What does the OLS minimize?
Price/return tends to run towards a long - run level
SSR
Choose parameters that maximize the likelihood of what observations occurring
For n>30 - sample mean is approximately normal

26. Exponential distribution
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Contains variables not explicit in model - Accounts for randomness

27. Multivariate probability
More than one random variable
Variance = (1/m) summation(u<n - i>^2)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

28. Direction of OVB
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

29. Antithetic variable technique
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

30. Standard error
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Probability that the random variables take on certain values simultaneously
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

31. Standard normal distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Independently and Identically Distributed
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Transformed to a unit variable - Mean = 0 Variance = 1

32. Critical z values
Confidence set for two coefficients - two dimensional analog for the confidence interval
Sample mean will near the population mean as the sample size increases
Contains variables not explicit in model - Accounts for randomness
95% = 1.65 99% = 2.33 For one - tailed tests

33. Standard error for Monte Carlo replications
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
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Combine to form distribution with leptokurtosis (heavy tails)

34. Empirical frequency
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
Based on a dataset
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

35. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Low Frequency - High Severity events
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
Variance(y)/n = variance of sample Y

36. Non - parametric vs parametric calculation of VaR
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
P(Z>t)

37. Least squares estimator(m)
Model dependent - Options with the same underlying assets may trade at different volatilities
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Average return across assets on a given day

38. Variance of X+b
If variance of the conditional distribution of u(i) is not constant
E(XY) - E(X)E(Y)
i = ln(Si/Si - 1)
Variance(x)

39. Skewness
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Combine to form distribution with leptokurtosis (heavy tails)
Sample mean +/ - t*(stddev(s)/sqrt(n))

40. Unconditional vs conditional distributions
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

41. Bernouli Distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Summation((xi - mean)^k)/n
Distribution with only two possible outcomes
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

42. Variance of X+Y assuming dependence
Choose parameters that maximize the likelihood of what observations occurring
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Price/return tends to run towards a long - run level
Variance(x) + Variance(Y) + 2*covariance(XY)

43. Mean reversion in variance
Only requires two parameters = mean and variance
Independently and Identically Distributed
Variance reverts to a long run level
Among all unbiased estimators - estimator with the smallest variance is efficient

44. Difference between population and sample variance
Confidence set for two coefficients - two dimensional analog for the confidence interval
Population denominator = n - Sample denominator = n - 1
Based on a dataset
(a^2)(variance(x)

45. Central Limit Theorem
For n>30 - sample mean is approximately normal
Model dependent - Options with the same underlying assets may trade at different volatilities
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

46. Confidence ellipse
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
Returns over time for an individual asset
Special type of pooled data in which the cross sectional unit is surveyed over time
Confidence set for two coefficients - two dimensional analog for the confidence interval

47. Weibul distribution
Variance(x) + Variance(Y) + 2*covariance(XY)
Var(X) + Var(Y)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Returns over time for an individual asset

48. Exact significance level
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Attempts to sample along more important paths
P - value
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

49. Inverse transform method
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Based on an equation - P(A) = # of A/total outcomes
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Sample mean +/ - t*(stddev(s)/sqrt(n))

50. Statistical (or empirical) model
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Mean of sampling distribution is the population mean
Random walk (usually acceptable) - Constant volatility (unlikely)
Yi = B0 + B1Xi + ui