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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Mean reversion in asset dynamics
Attempts to sample along more important paths
Price/return tends to run towards a long - run level
i = ln(Si/Si - 1)
Summation((xi - mean)^k)/n

2. Variance of X+Y
Sample mean will near the population mean as the sample size increases
Var(X) + Var(Y)
Contains variables not explicit in model - Accounts for randomness
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

3. Weibul distribution
For n>30 - sample mean is approximately normal
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Sample mean will near the population mean as the sample size increases

4. Unconditional vs conditional distributions
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

5. Type I error
We reject a hypothesis that is actually true
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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 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

6. Poisson distribution equations for mean variance and std deviation
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
If variance of the conditional distribution of u(i) is not constant
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

7. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
P(Z>t)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

8. SER
Application of mathematical statistics to economic data to lend empirical support to models
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Among all unbiased estimators - estimator with the smallest variance is efficient
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

9. Covariance
E(XY) - E(X)E(Y)
Among all unbiased estimators - estimator with the smallest variance is efficient
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Model dependent - Options with the same underlying assets may trade at different volatilities

10. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Does not depend on a prior event or information
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

11. Overall F - statistic
For n>30 - sample mean is approximately normal
Nonlinearity
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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

12. Single variable (univariate) probability
P(X=x - Y=y) = P(X=x) * P(Y=y)
Concerned with a single random variable (ex. Roll of a die)
Only requires two parameters = mean and variance
Variance(X) + Variance(Y) - 2*covariance(XY)

13. Persistence
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
(a^2)(variance(x)
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

14. Multivariate Density Estimation (MDE)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

15. Variance - covariance approach for VaR of a portfolio
P - value
Regression can be non - linear in variables but must be linear in parameters
Combine to form distribution with leptokurtosis (heavy tails)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

16. Potential reasons for fat tails in return distributions
i = ln(Si/Si - 1)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Model dependent - Options with the same underlying assets may trade at different volatilities

17. Continuous representation of the GBM
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Z = (Y - meany)/(stddev(y)/sqrt(n))
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)
Confidence level

18. Unbiased
Confidence level
Mean of sampling distribution is the population mean
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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

19. Difference between population and sample variance
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Population denominator = n - Sample denominator = n - 1
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
(a^2)(variance(x)

20. Sample variance
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
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
Based on an equation - P(A) = # of A/total outcomes
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

21. Mean(expected value)
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)
Distribution with only two possible outcomes
Choose parameters that maximize the likelihood of what observations occurring

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

23. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Returns over time for an individual asset
Variance(x) + Variance(Y) + 2*covariance(XY)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

24. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Returns over time for an individual asset
P(X=x - Y=y) = P(X=x) * P(Y=y)

25. Critical z values
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
95% = 1.65 99% = 2.33 For one - tailed tests

26. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Based on an equation - P(A) = # of A/total outcomes
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
Variance reverts to a long run level

27. Logistic distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Confidence set for two coefficients - two dimensional analog for the confidence interval
Has heavy tails
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)

28. Central Limit Theorem
Easy to manipulate
Variance reverts to a long run level
For n>30 - sample mean is approximately normal
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

29. Exponential distribution
Variance(y)/n = variance of sample Y
Variance(x)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
P(X=x - Y=y) = P(X=x) * P(Y=y)

30. Exact significance level
Distribution with only two possible outcomes
P(Z>t)
P - value
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

31. GARCH
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Confidence level
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)

32. Law of Large Numbers
When one regressor is a perfect linear function of the other regressors
Sample mean will near the population mean as the sample size increases
Confidence set for two coefficients - two dimensional analog for the confidence interval
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

33. Bernouli Distribution
E(XY) - E(X)E(Y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Contains variables not explicit in model - Accounts for randomness
Distribution with only two possible outcomes

34. Pooled data
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!)
We reject a hypothesis that is actually true
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)

35. Maximum likelihood method
Confidence set for two coefficients - two dimensional analog for the confidence interval
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Choose parameters that maximize the likelihood of what observations occurring
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

36. Two ways to calculate historical volatility
Expected value of the sample mean is the population mean
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When one regressor is a perfect linear function of the other regressors
Easy to manipulate

37. Continuously compounded return equation
i = ln(Si/Si - 1)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E(mean) = mean
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

38. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Easy to manipulate
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
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)

39. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Price/return tends to run towards a long - run level
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

40. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

41. Standard error for Monte Carlo replications
Mean of sampling distribution is the population mean
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Sampling distribution of sample means tend to be normal

42. Test for unbiasedness
Var(X) + Var(Y)
Sample mean +/ - t*(stddev(s)/sqrt(n))
E(mean) = mean
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

43. Kurtosis
Returns over time for an individual asset
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Contains variables not explicit in model - Accounts for randomness
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

44. Implications of homoscedasticity
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Only requires two parameters = mean and variance
(a^2)(variance(x)
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. Multivariate probability
Variance = (1/m) summation(u<n - i>^2)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
More than one random variable
(a^2)(variance(x)

46. SER
Use historical simulation approach but use the EWMA weighting system
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

47. K - th moment
Summation((xi - mean)^k)/n
Concerned with a single random variable (ex. Roll of a die)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

48. Confidence interval (from t)
Sampling distribution of sample means tend to be normal
Sample mean +/ - t*(stddev(s)/sqrt(n))
Transformed to a unit variable - Mean = 0 Variance = 1
Choose parameters that maximize the likelihood of what observations occurring

49. Skewness
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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!)
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

50. Sample mean
Yi = B0 + B1Xi + ui
Concerned with a single random variable (ex. Roll of a die)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Expected value of the sample mean is the population mean