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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Continuously compounded return equation
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
i = ln(Si/Si - 1)
Population denominator = n - Sample denominator = n - 1
Sample mean will near the population mean as the sample size increases

2. SER
Variance reverts to a long run level
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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)

3. Skewness
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Easy to manipulate
Based on an equation - P(A) = # of A/total outcomes
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

4. Persistence
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
95% = 1.65 99% = 2.33 For one - tailed tests
Variance reverts to a long run level

5. Test for unbiasedness
E(mean) = mean
Among all unbiased estimators - estimator with the smallest variance is efficient
Z = (Y - meany)/(stddev(y)/sqrt(n))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

6. Regime - switching volatility model
Confidence set for two coefficients - two dimensional analog for the confidence interval
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
P - value

7. Maximum likelihood method
Easy to manipulate
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Choose parameters that maximize the likelihood of what observations occurring

8. Poisson Distribution
Normal - Student's T - Chi - square - F distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Yi = B0 + B1Xi + ui
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

9. Adjusted R^2
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

10. Mean reversion
Has heavy tails
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

11. Simulation models
Low Frequency - High Severity events
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under 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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

12. Least squares estimator(m)
Regression can be non - linear in variables but must be linear in parameters
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Only requires two parameters = mean and variance
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

13. Implied standard deviation for options
Returns over time for an individual asset
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
We accept a hypothesis that should have been rejected

14. Statistical (or empirical) model
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Yi = B0 + B1Xi + ui
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

15. Lognormal
Variance = (1/m) summation(u<n - i>^2)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Average return across assets on a given day

16. Continuous representation of the GBM
(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)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

17. Multivariate probability
More than one random variable
P - value
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Based on a dataset

18. Binomial distribution equations for mean variance and std dev
Statement of the error or precision of an estimate
Average return across assets on a given day
Mean = np - Variance = npq - Std dev = sqrt(npq)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

19. Priori (classical) probability
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Based on an equation - P(A) = # of A/total outcomes
Easy to manipulate

20. Covariance
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
E(XY) - E(X)E(Y)
Variance = (1/m) summation(u<n - i>^2)
Attempts to sample along more important paths

21. Extreme Value Theory
Combine to form distribution with leptokurtosis (heavy tails)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

22. Tractable
Easy to manipulate
Confidence level
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

23. Normal distribution
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Low Frequency - High Severity events
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
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

24. GARCH
Population denominator = n - Sample denominator = n - 1
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(sample y) = (variance(y)/n)*(N - n/N - 1)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

25. Potential reasons for fat tails in return distributions
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
E(XY) - E(X)E(Y)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

26. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
When the sample size is large - the uncertainty about the value of the sample is very small

27. Homoskedastic only F - stat
Z = (Y - meany)/(stddev(y)/sqrt(n))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
(a^2)(variance(x)) + (b^2)(variance(y))
P(X=x - Y=y) = P(X=x) * P(Y=y)

28. Variance of X+Y assuming dependence
Independently and Identically Distributed
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Only requires two parameters = mean and variance
Variance(x) + Variance(Y) + 2*covariance(XY)

29. SER
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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

30. Continuous random variable
Based on an equation - P(A) = # of A/total outcomes
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

31. Sample covariance
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Sample mean will near the population mean as the sample size increases
95% = 1.65 99% = 2.33 For one - tailed tests

32. Implications of homoscedasticity
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Price/return tends to run towards a long - run level
Nonlinearity
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

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

34. Heteroskedastic
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Rxy = Sxy/(Sx*Sy)
Concerned with a single random variable (ex. Roll of a die)
If variance of the conditional distribution of u(i) is not constant

35. T distribution
Confidence level
If variance of the conditional distribution of u(i) is not constant
Variance(x) + Variance(Y) + 2*covariance(XY)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

36. Consistent
Special type of pooled data in which the cross sectional unit is surveyed over time
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
When the sample size is large - the uncertainty about the value of the sample is very small

37. Logistic distribution
i = ln(Si/Si - 1)
Has heavy tails
Price/return tends to run towards a long - run level
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

38. Unbiased
Based on an equation - P(A) = # of A/total outcomes
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Mean of sampling distribution is the population mean
Confidence level

39. Exact significance level
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
P - value
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

40. Efficiency
P(X=x - Y=y) = P(X=x) * P(Y=y)
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(x)
Among all unbiased estimators - estimator with the smallest variance is efficient

41. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
P - value
Based on a dataset
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

42. Sample correlation
P(Z>t)
Variance = (1/m) summation(u<n - i>^2)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Rxy = Sxy/(Sx*Sy)

43. Joint probability functions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Probability that the random variables take on certain values simultaneously
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

44. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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)
Easy to manipulate
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

45. Cross - sectional
Variance reverts to a long run level
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Summation((xi - mean)^k)/n
Average return across assets on a given day

46. Central Limit Theorem
When one regressor is a perfect linear function of the other regressors
For n>30 - sample mean is approximately normal
Probability that the random variables take on certain values simultaneously
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

47. Antithetic variable technique
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

48. Beta distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

49. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
i = ln(Si/Si - 1)
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

50. F distribution
Average return across assets on a given day
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
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