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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. Deterministic Simulation
Average return across assets on a given day
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Nonlinearity

2. Square root rule
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Among all unbiased estimators - estimator with the smallest variance is efficient
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

3. Importance sampling technique
Attempts to sample along more important paths
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
We accept a hypothesis that should have been rejected

4. Poisson Distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Special type of pooled data in which the cross sectional unit is surveyed over time
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

5. SER
Has heavy tails
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

6. Bootstrap method
Peaks over threshold - Collects dataset in excess of some threshold
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

7. Cholesky factorization (decomposition)
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Distribution with only two possible outcomes

8. Standard normal distribution
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Statement of the error or precision of an estimate
Transformed to a unit variable - Mean = 0 Variance = 1

9. Variance of weighted scheme
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

10. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Sample mean will near the population mean as the sample size increases
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

11. Continuous random variable
Use historical simulation approach but use the EWMA weighting system
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Expected value of the sample mean is the population mean

12. Priori (classical) probability
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
Sample mean will near the population mean as the sample size increases
Based on an equation - P(A) = # of A/total outcomes
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

13. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Concerned with a single random variable (ex. Roll of a die)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
More than one random variable

14. SER
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
Statement of the error or precision of an estimate
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

15. Hybrid method for conditional volatility
Nonlinearity
Use historical simulation approach but use the EWMA weighting system
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

16. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Confidence level
Mean = np - Variance = npq - Std dev = sqrt(npq)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

17. Binomial distribution
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!)
Var(X) + Var(Y)
Confidence level
Returns over time for a combination of assets (combination of time series and cross - sectional data)

18. i.i.d.
Variance reverts to a long run level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
Independently and Identically Distributed

19. Normal distribution
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Does not depend on a prior event or information
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

20. Four sampling distributions

21. Difference between population and sample variance
Model dependent - Options with the same underlying assets may trade at different volatilities
E(mean) = mean
Use historical simulation approach but use the EWMA weighting system
Population denominator = n - Sample denominator = n - 1

22. Implied standard deviation for options
Use historical simulation approach but use the EWMA weighting system
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance(x)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

23. Chi - squared distribution
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)
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
Normal - Student's T - Chi - square - F distribution

24. Variance of aX + bY
Normal - Student's T - Chi - square - F distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
(a^2)(variance(x)) + (b^2)(variance(y))
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!)

25. GARCH
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(X) + Variance(Y) - 2*covariance(XY)

26. Result of combination of two normal with same means
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 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
We accept a hypothesis that should have been rejected
Combine to form distribution with leptokurtosis (heavy tails)

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

28. Expected future variance rate (t periods forward)
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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

29. Inverse transform method
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Easy to manipulate
Variance(y)/n = variance of sample Y

30. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Least absolute deviations estimator - used when extreme outliers are not uncommon
(a^2)(variance(x)) + (b^2)(variance(y))

31. Variance of sample mean
For n>30 - sample mean is approximately normal
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)
Variance(y)/n = variance of sample Y
P(Z>t)

32. F distribution
Choose parameters that maximize the likelihood of what observations occurring
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
Concerned with a single random variable (ex. Roll of a die)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

33. Two requirements of OVB
When one regressor is a perfect linear function of the other regressors
P - value
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Based on a dataset

34. Variance of X+Y assuming dependence
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance(x) + Variance(Y) + 2*covariance(XY)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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

35. What does the OLS minimize?
When the sample size is large - the uncertainty about the value of the sample is very small
SSR
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)
Contains variables not explicit in model - Accounts for randomness

36. Law of Large Numbers
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
Sample mean will near the population mean as the sample size increases
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Distribution with only two possible outcomes

37. GEV
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

38. Mean reversion
Among all unbiased estimators - estimator with the smallest variance is efficient
Returns over time for an individual asset
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
P(Z>t)

39. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
95% = 1.65 99% = 2.33 For one - tailed tests
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

40. Mean(expected value)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

41. BLUE
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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

42. Time series data
Returns over time for an individual asset
More than one random variable
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

43. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

44. Exponential distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

45. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance reverts to a long run level
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

46. Sample correlation
Attempts to sample along more important paths
Does not depend on a prior event or information
For n>30 - sample mean is approximately normal
Rxy = Sxy/(Sx*Sy)

47. Simulation models
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

48. Biggest (and only real) drawback of GARCH mode
P(X=x - Y=y) = P(X=x) * P(Y=y)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Random walk (usually acceptable) - Constant volatility (unlikely)
Nonlinearity

49. Tractable
Transformed to a unit variable - Mean = 0 Variance = 1
Based on an equation - P(A) = # of A/total outcomes
Easy to manipulate
Nonlinearity

50. Sample mean
Summation((xi - mean)^k)/n
When the sample size is large - the uncertainty about the value of the sample is very small
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Expected value of the sample mean is the population mean