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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. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Only requires two parameters = mean and variance
Random walk (usually acceptable) - Constant volatility (unlikely)
Model dependent - Options with the same underlying assets may trade at different volatilities

2. What does the OLS minimize?
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Peaks over threshold - Collects dataset in excess of some threshold
SSR

3. Key properties of linear regression
Confidence set for two coefficients - two dimensional analog for the confidence interval
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Regression can be non - linear in variables but must be linear in parameters
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

4. Lognormal
Low Frequency - High Severity events
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
Nonlinearity
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

5. GEV
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

6. Hybrid method for conditional volatility
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
Var(X) + Var(Y)
Use historical simulation approach but use the EWMA weighting system

7. Cholesky factorization (decomposition)
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Does not depend on a prior event or information

8. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
i = ln(Si/Si - 1)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

9. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
E(mean) = mean
Contains variables not explicit in model - Accounts for randomness

10. Sample variance
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

11. Expected future variance rate (t periods forward)
Variance = (1/m) summation(u<n - i>^2)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Sample mean +/ - t*(stddev(s)/sqrt(n))

12. Standard error for Monte Carlo replications
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E(mean) = mean
Low Frequency - High Severity events
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

13. Confidence interval for sample mean
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
For n>30 - sample mean is approximately normal
Z = (Y - meany)/(stddev(y)/sqrt(n))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

14. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Sample mean will near the population mean as the sample size increases
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

15. Confidence interval (from t)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Sample mean +/ - t*(stddev(s)/sqrt(n))

16. Continuously compounded return equation
i = ln(Si/Si - 1)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Special type of pooled data in which the cross sectional unit is surveyed over time
Among all unbiased estimators - estimator with the smallest variance is efficient

17. Variance of aX + bY
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
(a^2)(variance(x)) + (b^2)(variance(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)
Random walk (usually acceptable) - Constant volatility (unlikely)

18. Chi - squared distribution
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Based on an equation - P(A) = # of A/total outcomes

19. Bootstrap method
Least absolute deviations estimator - used when extreme outliers are not uncommon
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Confidence level
P(Z>t)

20. Discrete representation of the GBM
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

21. Hazard rate of exponentially distributed random variable
Contains variables not explicit in model - Accounts for randomness
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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

22. Implications of homoscedasticity
If variance of the conditional distribution of u(i) is not constant
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

23. Type II Error
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
We accept a hypothesis that should have been rejected

24. ESS
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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

25. Skewness
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Among all unbiased estimators - estimator with the smallest variance is efficient
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
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

26. SER
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

27. Potential reasons for fat tails in return distributions
Variance(y)/n = variance of sample Y
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Contains variables not explicit in model - Accounts for randomness
We reject a hypothesis that is actually true

28. Inverse transform method
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Regression can be non - linear in variables but must be linear in parameters
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

29. F distribution
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

30. Mean(expected value)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
P - value

31. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
More than one random variable
Mean of sampling distribution is the population mean
i = ln(Si/Si - 1)

32. Variance of X+Y
Variance(x)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Var(X) + Var(Y)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

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

34. Two assumptions of square root rule
P(Z>t)
Regression can be non - linear in variables but must be linear in parameters
Among all unbiased estimators - estimator with the smallest variance is efficient
Random walk (usually acceptable) - Constant volatility (unlikely)

35. Antithetic variable technique
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

36. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Model dependent - Options with the same underlying assets may trade at different volatilities
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

37. WLS
Sampling distribution of sample means tend to be normal
Variance = (1/m) summation(u<n - i>^2)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

38. Simulation models
Based on a dataset
Does not depend on a prior event or information
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
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

39. Pooled data
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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 a combination of assets (combination of time series and cross - sectional data)

40. Shortcomings of implied volatility
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))
Variance(X) + Variance(Y) - 2*covariance(XY)
Sample mean will near the population mean as the sample size increases

41. Tractable
Easy to manipulate
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Confidence set for two coefficients - two dimensional analog for the confidence interval

42. SER
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
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
Nonlinearity
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

43. Weibul distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Rxy = Sxy/(Sx*Sy)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

44. Continuous representation of the GBM
i = ln(Si/Si - 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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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)

45. Covariance calculations using weight sums (lambda)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Probability that the random variables take on certain values simultaneously

46. Economical(elegant)
Contains variables not explicit in model - Accounts for randomness
Only requires two parameters = mean and variance
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

47. Block maxima
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.
Price/return tends to run towards a long - run level
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

48. POT
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Peaks over threshold - Collects dataset in excess of some threshold
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

49. Sample covariance
Variance(x) + Variance(Y) + 2*covariance(XY)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Peaks over threshold - Collects dataset in excess of some threshold
Average return across assets on a given day

50. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)