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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. LAD
Expected value of the sample mean is the population mean
Least absolute deviations estimator - used when extreme outliers are not uncommon
Low Frequency - High Severity events
Application of mathematical statistics to economic data to lend empirical support to models

2. Two drawbacks of moving average series
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

3. Two requirements of OVB
Variance reverts to a long run level
Regression can be non - linear in variables but must be linear in parameters
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Based on a dataset

4. Continuous representation of the GBM
P(X=x - Y=y) = P(X=x) * P(Y=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)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Distribution with only two possible outcomes

5. Tractable
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Distribution with only two possible outcomes
Easy to manipulate
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)

6. Lognormal
For n>30 - sample mean is approximately normal
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

7. POT
Peaks over threshold - Collects dataset in excess of some threshold
Price/return tends to run towards a long - run level
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

8. Central Limit Theorem
P(Z>t)
For n>30 - sample mean is approximately normal
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

9. Simulation models
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

10. R^2
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
(a^2)(variance(x)) + (b^2)(variance(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

11. Regime - switching volatility model
Variance reverts to a long run level
Variance(y)/n = variance of sample Y
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

12. Kurtosis
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Random walk (usually acceptable) - Constant volatility (unlikely)

13. Unbiased
Mean of sampling distribution is the population mean
Concerned with a single random variable (ex. Roll of a die)
Among all unbiased estimators - estimator with the smallest variance is efficient
Returns over time for an individual asset

14. Binomial distribution equations for mean variance and std dev
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

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

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16. 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)
Rxy = Sxy/(Sx*Sy)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

17. Cross - sectional
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
Variance(X) + Variance(Y) - 2*covariance(XY)
E(XY) - E(X)E(Y)
Average return across assets on a given day

18. Weibul distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Sampling distribution of sample means tend to be normal
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

19. Historical std dev
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

20. GARCH
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)
Variance(y)/n = variance of sample Y
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

21. Adjusted R^2
Application of mathematical statistics to economic data to lend empirical support to models
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
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

22. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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!)
If variance of the conditional distribution of u(i) is not constant
i = ln(Si/Si - 1)

23. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Regression can be non - linear in variables but must be linear in parameters
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

24. Law of Large Numbers
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
If variance of the conditional distribution of u(i) is not constant
Sample mean will near the population mean as the sample size increases
Variance(x) + Variance(Y) + 2*covariance(XY)

25. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
For n>30 - sample mean is approximately normal
If variance of the conditional distribution of u(i) is not constant

26. ESS
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance = (1/m) summation(u<n - i>^2)

27. Unstable return distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Confidence set for two coefficients - two dimensional analog for the confidence interval

28. Bernouli Distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Distribution with only two possible outcomes
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

29. Exact significance level
P - value
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
(a^2)(variance(x)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

30. What does the OLS minimize?
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Model dependent - Options with the same underlying assets may trade at different volatilities
Among all unbiased estimators - estimator with the smallest variance is efficient
SSR

31. Variance of aX
(a^2)(variance(x)
For n>30 - sample mean is approximately normal
95% = 1.65 99% = 2.33 For one - tailed tests
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

32. K - th moment
Sampling distribution of sample means tend to be normal
Summation((xi - mean)^k)/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 increase accuracy by reducing sample variance instead of increasing sample size

33. Time series data
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Returns over time for an individual asset
Average return across assets on a given day

34. Continuous 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)
Yi = B0 + B1Xi + ui
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

35. Sample correlation
If variance of the conditional distribution of u(i) is not constant
Rxy = Sxy/(Sx*Sy)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

36. Heteroskedastic
We reject a hypothesis that is actually true
If variance of the conditional distribution of u(i) is not constant
Combine to form distribution with leptokurtosis (heavy tails)
95% = 1.65 99% = 2.33 For one - tailed tests

37. Test for unbiasedness
E(mean) = mean
Statement of the error or precision of an estimate
Regression can be non - linear in variables but must be linear in parameters
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

38. Hybrid method for conditional volatility
Contains variables not explicit in model - Accounts for randomness
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Statement of the error or precision of an estimate
Use historical simulation approach but use the EWMA weighting system

39. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

40. Covariance calculations using weight sums (lambda)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Peaks over threshold - Collects dataset in excess of some threshold
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Application of mathematical statistics to economic data to lend empirical support to models

41. F distribution
Choose parameters that maximize the likelihood of what observations occurring
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

42. Variance of X+Y
Var(X) + Var(Y)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Normal - Student's T - Chi - square - F distribution

43. Importance sampling technique
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Attempts to sample along more important paths
Based on an equation - P(A) = # of A/total outcomes
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

44. Variance of sampling distribution of means when n<N
Average return across assets on a given day
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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

45. Sample covariance
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Z = (Y - meany)/(stddev(y)/sqrt(n))

46. Binomial distribution
Peaks over threshold - Collects dataset in excess of some threshold
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
When the sample size is large - the uncertainty about the value of the sample is very small
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!)

47. Difference between population and sample variance
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance = (1/m) summation(u<n - i>^2)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Population denominator = n - Sample denominator = n - 1

48. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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!)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

49. Chi - squared distribution
Application of mathematical statistics to economic data to lend empirical support to models
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

50. Variance of X - Y assuming dependence
When one regressor is a perfect linear function of the other regressors
Variance = (1/m) summation(u<n - i>^2)
Low Frequency - High Severity events
Variance(X) + Variance(Y) - 2*covariance(XY)