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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
Sample mean will near the population mean as the sample size increases
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Special type of pooled data in which the cross sectional unit is surveyed over time

2. Variance of aX
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
(a^2)(variance(x)
Distribution with only two possible outcomes
Only requires two parameters = mean and variance

3. POT
Sample mean will near the population mean as the sample size increases
Peaks over threshold - Collects dataset in excess of some threshold
Population denominator = n - Sample denominator = n - 1
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

4. Multivariate Density Estimation (MDE)
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Variance(x)
Concerned with a single random variable (ex. Roll of a die)

5. Unbiased
Mean of sampling distribution is the population mean
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Use historical simulation approach but use the EWMA weighting system

6. Inverse transform method
Mean of sampling distribution is the population mean
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance = (1/m) summation(u<n - i>^2)
Transformed to a unit variable - Mean = 0 Variance = 1

7. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
When one regressor is a perfect linear function of the other regressors
Easy to manipulate
(a^2)(variance(x)) + (b^2)(variance(y))

8. Kurtosis
Choose parameters that maximize the likelihood of what observations occurring
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

9. Square root rule
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
When the sample size is large - the uncertainty about the value of the sample is very small
Expected value of the sample mean is the population mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

10. Control variates technique
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Statement of the error or precision of an estimate
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

11. Regime - switching volatility model
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
SSR

12. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

13. Covariance calculations using weight sums (lambda)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Based on an equation - P(A) = # of A/total outcomes
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

14. Homoskedastic only F - stat
Choose parameters that maximize the likelihood of what observations occurring
Model dependent - Options with the same underlying assets may trade at different volatilities
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

15. Direction of OVB
P - value
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
(a^2)(variance(x)) + (b^2)(variance(y))

16. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

17. Result of combination of two normal with same means
95% = 1.65 99% = 2.33 For one - tailed tests
Combine to form distribution with leptokurtosis (heavy tails)
Sampling distribution of sample means tend to be normal
For n>30 - sample mean is approximately normal

18. Implied standard deviation for options
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
E(mean) = mean
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

19. Multivariate probability
(a^2)(variance(x)) + (b^2)(variance(y))
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
More than one random variable

20. Variance of X+b
Variance(x)
Mean of sampling distribution is the population mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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

21. Mean reversion in variance
(a^2)(variance(x)
Variance reverts to a long run level
95% = 1.65 99% = 2.33 For one - tailed tests
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

22. Confidence interval for sample mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
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

23. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Special type of pooled data in which the cross sectional unit is surveyed over time
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
P(X=x - Y=y) = P(X=x) * P(Y=y)

24. Biggest (and only real) drawback of GARCH mode
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Choose parameters that maximize the likelihood of what observations occurring
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Nonlinearity

25. Variance of sample mean
Variance(y)/n = variance of sample Y
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

26. Logistic distribution
Has heavy tails
SSR
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
95% = 1.65 99% = 2.33 For one - tailed tests

27. Time series data
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Returns over time for an individual asset
Easy to manipulate

28. Variance of sampling distribution of means when n<N
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Independently and Identically Distributed

29. Variance of X+Y assuming dependence
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Rxy = Sxy/(Sx*Sy)
Mean of sampling distribution is the population mean
Variance(x) + Variance(Y) + 2*covariance(XY)

30. LFHS
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Population denominator = n - Sample denominator = n - 1
Low Frequency - High Severity events
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

31. Block maxima
Confidence level
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
More than one random variable

32. Cross - sectional
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
P - value
Average return across assets on a given day

33. Lognormal
P(X=x - Y=y) = P(X=x) * P(Y=y)
If variance of the conditional distribution of u(i) is not constant
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
Statement of the error or precision of an estimate

34. What does the OLS minimize?
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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)
SSR
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

35. Statistical (or empirical) model
E(XY) - E(X)E(Y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Yi = B0 + B1Xi + ui

36. EWMA
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
More than one random variable

37. Monte Carlo Simulations
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
Nonlinearity
Variance(x) + Variance(Y) + 2*covariance(XY)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

38. Normal distribution
E(mean) = mean
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance(x)

39. Econometrics
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Application of mathematical statistics to economic data to lend empirical support to models

40. Adjusted R^2
Contains variables not explicit in model - Accounts for randomness
Population denominator = n - Sample denominator = n - 1
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

41. Stochastic error term
Confidence set for two coefficients - two dimensional analog for the confidence interval
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Contains variables not explicit in model - Accounts for randomness
Least absolute deviations estimator - used when extreme outliers are not uncommon

42. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance(x)
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

43. Mean(expected value)
Use historical simulation approach but use the EWMA weighting system
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)
Summation((xi - mean)^k)/n
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

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

45. Single variable (univariate) probability
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Concerned with a single random variable (ex. Roll of a die)
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

46. Standard error for Monte Carlo replications
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E(XY) - E(X)E(Y)

47. Hazard rate of exponentially distributed random variable
P - value
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

48. Expected future variance rate (t periods forward)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

49. Sample mean
Contains variables not explicit in model - Accounts for randomness
Has heavy tails
Confidence level
Expected value of the sample mean is the population mean

50. Skewness
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
Distribution with only two possible outcomes
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled