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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. Perfect multicollinearity
Price/return tends to run towards a long - run level
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Does not depend on a prior event or information
When one regressor is a perfect linear function of the other regressors

2. Stochastic error term
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
Contains variables not explicit in model - Accounts for randomness
When one regressor is a perfect linear function of the other regressors

3. Historical std dev
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Does not depend on a prior event or information
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

4. Mean reversion in variance
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
Variance reverts to a long run level
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!)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

5. Test for statistical independence
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!)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
P(X=x - Y=y) = P(X=x) * P(Y=y)
Random walk (usually acceptable) - Constant volatility (unlikely)

6. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Only requires two parameters = mean and variance

7. Continuous random variable
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Only requires two parameters = mean and variance
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

8. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Special type of pooled data in which the cross sectional unit is surveyed over time
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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

9. Empirical frequency
Use historical simulation approach but use the EWMA weighting system
Based on a dataset
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sample mean +/ - t*(stddev(s)/sqrt(n))

10. Bernouli Distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Distribution with only two possible outcomes

11. Homoskedastic only F - stat
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
E(mean) = mean
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

12. Shortcomings of implied volatility
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Model dependent - Options with the same underlying assets may trade at different volatilities
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

13. Binomial distribution equations for mean variance and std dev
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
P(X=x - Y=y) = P(X=x) * P(Y=y)
Mean = np - Variance = npq - Std dev = sqrt(npq)

14. Variance of sampling distribution of means when n<N
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
(a^2)(variance(x)
Sampling distribution of sample means tend to be normal

15. Cross - sectional
Average return across assets on a given day
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!)
Variance(y)/n = variance of sample Y
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

16. Exact significance level
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
P - value
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)

17. Variance of X - Y assuming dependence
SSR
Variance(X) + Variance(Y) - 2*covariance(XY)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

18. Sample mean
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
Expected value of the sample mean is the population mean
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Low Frequency - High Severity events

19. Binomial distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
95% = 1.65 99% = 2.33 For one - tailed tests
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!)
Least absolute deviations estimator - used when extreme outliers are not uncommon

20. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

21. GEV
More than one random variable
Population denominator = n - Sample denominator = n - 1
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Var(X) + Var(Y)

22. Confidence interval for sample mean
Returns over time for a combination of assets (combination of time series and cross - sectional data)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

23. Standard normal distribution
Var(X) + Var(Y)
Transformed to a unit variable - Mean = 0 Variance = 1
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

24. 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)
Choose parameters that maximize the likelihood of what observations occurring
When one regressor is a perfect linear function of the other regressors
E(XY) - E(X)E(Y)

25. What does the OLS minimize?
SSR
Yi = B0 + B1Xi + ui
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

26. Discrete representation of the GBM
E(XY) - E(X)E(Y)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

27. Control variates technique
Concerned with a single random variable (ex. Roll of a die)
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)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

28. Block maxima
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 = (1/m) summation(u<n - i>^2)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

29. Importance sampling technique
E(XY) - E(X)E(Y)
Sample mean +/ - t*(stddev(s)/sqrt(n))
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)
Attempts to sample along more important paths

30. Gamma distribution
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance(x)
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
SSR

31. Square root rule
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
E(XY) - E(X)E(Y)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

32. Conditional probability functions
Does not depend on a prior event or information
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

33. BLUE
Easy to manipulate
Choose parameters that maximize the likelihood of what observations occurring
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

34. Tractable
Expected value of the sample mean is the population mean
Easy to manipulate
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

35. Variance of sample mean
Variance(y)/n = variance of sample Y
When one regressor is a perfect linear function of the other regressors
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

36. Two ways to calculate historical volatility
Statement of the error or precision of an estimate
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
For n>30 - sample mean is approximately normal

37. Mean(expected value)
E(mean) = mean
Variance(y)/n = variance of sample Y
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

38. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Peaks over threshold - Collects dataset in excess of some threshold
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Least absolute deviations estimator - used when extreme outliers are not uncommon

39. Inverse transform method
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Statement of the error or precision of an estimate
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

40. Expected future variance rate (t periods forward)
Confidence level
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Expected value of the sample mean is the population mean
(a^2)(variance(x)) + (b^2)(variance(y))

41. Simulation models
Attempts to sample along more important paths
For n>30 - sample mean is approximately normal
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
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

42. Continuously compounded return equation
i = ln(Si/Si - 1)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

43. Marginal unconditional probability function
Does not depend on a prior event or information
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Expected value of the sample mean is the population mean
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

44. Weibul distribution
Average return across assets on a given day
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
E(XY) - E(X)E(Y)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

45. Central Limit Theorem(CLT)
Random walk (usually acceptable) - Constant volatility (unlikely)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sampling distribution of sample means tend to be normal

46. WLS
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
If variance of the conditional distribution of u(i) is not constant
Variance(y)/n = variance of sample Y
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

47. Two assumptions of square root rule
Rxy = Sxy/(Sx*Sy)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Random walk (usually acceptable) - Constant volatility (unlikely)
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!)

48. LAD
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Least absolute deviations estimator - used when extreme outliers are not uncommon
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Rxy = Sxy/(Sx*Sy)

49. Antithetic variable technique
Variance reverts to a long run level
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

50. Statistical (or empirical) model
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Only requires two parameters = mean and variance
Yi = B0 + B1Xi + ui