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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. Law of Large Numbers
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Sample mean will near the population mean as the sample size increases
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
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

2. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Sampling distribution of sample means tend to be normal
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

3. 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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
More than one random variable
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

4. Covariance calculations using weight sums (lambda)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Sample mean will near the population mean as the sample size increases
Peaks over threshold - Collects dataset in excess of some threshold
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

5. SER
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Returns over time for a combination of assets (combination of time series and cross - sectional data)
More than one random variable

6. LAD
i = ln(Si/Si - 1)
Peaks over threshold - Collects dataset in excess of some threshold
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Least absolute deviations estimator - used when extreme outliers are not uncommon

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

8. GPD
E(XY) - E(X)E(Y)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Statement of the error or precision of an estimate

9. Sample covariance
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Peaks over threshold - Collects dataset in excess of some threshold
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

10. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
Returns over time for an individual asset
Returns over time for a combination of assets (combination of time series and cross - sectional data)

11. Binomial distribution equations for mean variance and std dev
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
When the sample size is large - the uncertainty about the value of the sample is very small
Variance reverts to a long run level
Mean = np - Variance = npq - Std dev = sqrt(npq)

12. Homoskedastic only F - stat
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Use historical simulation approach but use the EWMA weighting system
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

13. Cholesky factorization (decomposition)
Z = (Y - meany)/(stddev(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
For n>30 - sample mean is approximately normal
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

14. GARCH
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

15. Marginal unconditional probability function
P - value
Does not depend on a prior event or information
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
When one regressor is a perfect linear function of the other regressors

16. Two assumptions of square root rule
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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 +/ - t*(stddev(s)/sqrt(n))
Random walk (usually acceptable) - Constant volatility (unlikely)

17. Binomial distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Normal - Student's T - Chi - square - F 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!)

18. LFHS
Model dependent - Options with the same underlying assets may trade at different volatilities
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Low Frequency - High Severity events
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

19. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Nonlinearity

20. Perfect multicollinearity
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
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
When one regressor is a perfect linear function of the other regressors

21. SER
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Price/return tends to run towards a long - run level
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
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

22. Weibul distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Mean of sampling distribution is the population mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

23. Continuous representation of the GBM
Peaks over threshold - Collects dataset in excess of some threshold
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)
Choose parameters that maximize the likelihood of what observations occurring
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

24. Test for unbiasedness
E(mean) = mean
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Average return across assets on a given day
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

25. Implications of homoscedasticity
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
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
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
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!)

26. Variance of X+b
Concerned with a single random variable (ex. Roll of a die)
Transformed to a unit variable - Mean = 0 Variance = 1
Normal - Student's T - Chi - square - F distribution
Variance(x)

27. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
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
Summation((xi - mean)^k)/n

28. Standard normal distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Normal - Student's T - Chi - square - F distribution
Transformed to a unit variable - Mean = 0 Variance = 1
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

29. Importance sampling technique
Among all unbiased estimators - estimator with the smallest variance is efficient
Returns over time for an individual asset
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Attempts to sample along more important paths

30. Bootstrap method
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Only requires two parameters = mean and variance
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

31. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
P(Z>t)
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
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

32. Priori (classical) probability
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
If variance of the conditional distribution of u(i) is not constant
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Based on an equation - P(A) = # of A/total outcomes

33. Hazard rate of exponentially distributed random variable
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

34. Tractable
Easy to manipulate
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Independently and Identically Distributed

35. K - th moment
Summation((xi - mean)^k)/n
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
We reject a hypothesis that is actually true
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

36. Pooled data
P - value
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Random walk (usually acceptable) - Constant volatility (unlikely)

37. Confidence ellipse
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Summation((xi - mean)^k)/n
Sampling distribution of sample means tend to be normal

38. Two ways to calculate historical volatility
Average return across assets on a given day
Summation((xi - mean)^k)/n
Among all unbiased estimators - estimator with the smallest variance is efficient
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

39. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Price/return tends to run towards a long - run level
P - value
Summation((xi - mean)^k)/n

40. Poisson Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance(x) + Variance(Y) + 2*covariance(XY)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

41. Simulation models
Independently and Identically Distributed
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
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

42. Exact significance level
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
P - value
If variance of the conditional distribution of u(i) is not constant
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

43. Standard variable for non - normal distributions
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Use historical simulation approach but use the EWMA weighting system
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Z = (Y - meany)/(stddev(y)/sqrt(n))

44. Antithetic variable technique
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
i = ln(Si/Si - 1)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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

45. Potential reasons for fat tails in return distributions
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Price/return tends to run towards a long - run level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

46. Joint probability functions
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Attempts to sample along more important paths
P(X=x - Y=y) = P(X=x) * P(Y=y)
Probability that the random variables take on certain values simultaneously

47. Efficiency
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
Population denominator = n - Sample denominator = n - 1
Among all unbiased estimators - estimator with the smallest variance is efficient
Summation((xi - mean)^k)/n

48. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Variance reverts to a long run level
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

49. Variance of aX
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
(a^2)(variance(x)
Nonlinearity
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

50. Extreme Value Theory
Variance(x)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
(a^2)(variance(x)) + (b^2)(variance(y))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)