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FRM Foundations Of Risk Management Quantitative Methods

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1. Sample correlation
Rxy = Sxy/(Sx*Sy)
Does not depend on a prior event or information
Nonlinearity
E(XY) - E(X)E(Y)

2. Stochastic error term
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
When one regressor is a perfect linear function of the other regressors
Contains variables not explicit in model - Accounts for randomness
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

3. Single variable (univariate) probability
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)
Concerned with a single random variable (ex. Roll of a die)
Among all unbiased estimators - estimator with the smallest variance is efficient
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

4. Pooled data
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Based on a dataset

5. Extreme Value Theory
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Transformed to a unit variable - Mean = 0 Variance = 1

6. Conditional probability functions
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)
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
Special type of pooled data in which the cross sectional unit is surveyed over time

7. Continuously compounded return equation
Var(X) + Var(Y)
i = ln(Si/Si - 1)
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

8. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Concerned with a single random variable (ex. Roll of a die)
SSR

9. Mean reversion
Yi = B0 + B1Xi + ui
Sample mean will near the population mean as the sample size increases
95% = 1.65 99% = 2.33 For one - tailed tests
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

10. 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)
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

11. Standard variable for non - normal distributions
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
Easy to manipulate

12. SER
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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

13. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Distribution with only two possible outcomes
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

14. Key properties of linear regression
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Regression can be non - linear in variables but must be linear in parameters
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

15. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Distribution with only two possible outcomes
Returns over time for a combination of assets (combination of time series and cross - sectional data)
(a^2)(variance(x)) + (b^2)(variance(y))

16. Economical(elegant)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Only requires two parameters = mean and variance

17. Normal distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Distribution with only two possible outcomes
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

18. Overall F - statistic
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
We accept a hypothesis that should have been rejected
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

19. Significance =1
Var(X) + Var(Y)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Expected value of the sample mean is the population mean
Confidence level

20. Central Limit Theorem
For n>30 - sample mean is approximately normal
Sample mean will near the population mean as the sample size increases
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

21. Logistic distribution
Z = (Y - meany)/(stddev(y)/sqrt(n))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Has heavy tails

22. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Least absolute deviations estimator - used when extreme outliers are not uncommon
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

23. Mean(expected value)
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Combine to form distribution with leptokurtosis (heavy tails)

24. SER
Variance(X) + Variance(Y) - 2*covariance(XY)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Sampling distribution of sample means tend to be normal

25. Simulation models
Sample mean +/ - t*(stddev(s)/sqrt(n))
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)
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

26. Variance of aX + bY
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Model dependent - Options with the same underlying assets may trade at different volatilities
(a^2)(variance(x)) + (b^2)(variance(y))
Does not depend on a prior event or information

27. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
(a^2)(variance(x)) + (b^2)(variance(y))
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

28. Variance of sampling distribution of means when n<N
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

29. Multivariate probability
More than one random variable
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Probability that the random variables take on certain values simultaneously

30. F distribution
Rxy = Sxy/(Sx*Sy)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

31. Variance of aX
Peaks over threshold - Collects dataset in excess of some threshold
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
(a^2)(variance(x)

32. Joint probability functions
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Probability that the random variables take on certain values simultaneously
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

33. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Random walk (usually acceptable) - Constant volatility (unlikely)
Has heavy tails

34. Empirical frequency
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E(XY) - E(X)E(Y)
Based on a dataset
Independently and Identically Distributed

35. Poisson distribution equations for mean variance and std deviation
Variance = (1/m) summation(u<n - i>^2)
Transformed to a unit variable - Mean = 0 Variance = 1
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Var(X) + Var(Y)

36. Difference between population and sample variance
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Population denominator = n - Sample denominator = n - 1

37. LFHS
i = ln(Si/Si - 1)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Low Frequency - High Severity events
Var(X) + Var(Y)

38. Unbiased
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 of sampling distribution is the population mean
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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

39. Sample mean
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Expected value of the sample mean is the population mean
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

40. Implications of homoscedasticity
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!)
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
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

41. Mean reversion in variance
Variance reverts to a long run level
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
More than one random variable
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

42. Binomial distribution
We accept a hypothesis that should have been rejected
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!)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Least absolute deviations estimator - used when extreme outliers are not uncommon

43. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Price/return tends to run towards a long - run level

44. Central Limit Theorem(CLT)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Based on a dataset
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

45. Deterministic Simulation
Confidence level
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)
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

46. Time series data
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
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

47. Econometrics
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 = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Application of mathematical statistics to economic data to lend empirical support to models

48. Discrete random variable
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Combine to form distribution with leptokurtosis (heavy tails)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

49. Bernouli Distribution
P(Z>t)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Average return across assets on a given day
Distribution with only two possible outcomes

50. Variance - covariance approach for VaR of a portfolio
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
(a^2)(variance(x)) + (b^2)(variance(y))

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