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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Expected value of the sample mean is the population mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

2. Unstable return distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
If variance of the conditional distribution of u(i) is not constant
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Confidence set for two coefficients - two dimensional analog for the confidence interval

3. Two ways to calculate historical volatility
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Does not depend on a prior event or information
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

4. Mean(expected value)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

5. Non - parametric vs parametric calculation of VaR
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(x)

6. Test for statistical independence
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
Variance(x) + Variance(Y) + 2*covariance(XY)
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

7. Exponential distribution
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

8. Central Limit Theorem(CLT)
E(mean) = mean
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sampling distribution of sample means tend to be normal

9. Variance of aX + bY
P(X=x - Y=y) = P(X=x) * P(Y=y)
Confidence level
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))

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

11. Confidence ellipse
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Sample mean +/ - t*(stddev(s)/sqrt(n))
Confidence set for two coefficients - two dimensional analog for the confidence interval
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)

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

13. Sample covariance
Contains variables not explicit in model - Accounts for randomness
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
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

14. Sample correlation
Normal - Student's T - Chi - square - F distribution
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Rxy = Sxy/(Sx*Sy)
Expected value of the sample mean is the population mean

15. Cholesky factorization (decomposition)
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Concerned with a single random variable (ex. Roll of a die)
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

16. Efficiency
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Among all unbiased estimators - estimator with the smallest variance is efficient
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

17. ESS
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
P(Z>t)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

18. Bernouli Distribution
Distribution with only two possible outcomes
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

19. i.i.d.
Variance(x) + Variance(Y) + 2*covariance(XY)
Independently and Identically Distributed
Variance(X) + Variance(Y) - 2*covariance(XY)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

20. Implications of homoscedasticity
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Low Frequency - High Severity events
E(XY) - E(X)E(Y)

21. Panel data (longitudinal or micropanel)
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)
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
Special type of pooled data in which the cross sectional unit is surveyed over time

22. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Low Frequency - High Severity events

23. Implied standard deviation for options
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Has heavy tails
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sample mean +/ - t*(stddev(s)/sqrt(n))

24. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Z = (Y - meany)/(stddev(y)/sqrt(n))
When the sample size is large - the uncertainty about the value of the sample is very small
Statement of the error or precision of an estimate

25. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample mean +/ - t*(stddev(s)/sqrt(n))
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

26. Mean reversion
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

27. Pooled data
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
We reject a hypothesis that is actually true
For n>30 - sample mean is approximately normal
Returns over time for a combination of assets (combination of time series and cross - sectional data)

28. P - value
Regression can be non - linear in variables but must be linear in parameters
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
P(Z>t)

29. Tractable
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
Independently and Identically Distributed
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
Easy to manipulate

30. Sample variance
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Application of mathematical statistics to economic data to lend empirical support to models
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

31. Joint probability functions
Probability that the random variables take on certain values simultaneously
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Easy to manipulate

32. Continuous representation of the GBM
Regression can be non - linear in variables but must be linear in parameters
Distribution with only two possible outcomes
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)

33. Simplified standard (un - weighted) variance
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
(a^2)(variance(x)
Variance = (1/m) summation(u<n - i>^2)

34. Importance sampling technique
Price/return tends to run towards a long - run level
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Attempts to sample along more important paths
Summation((xi - mean)^k)/n

35. Type II Error
We accept a hypothesis that should have been rejected
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Distribution with only two possible outcomes
P(Z>t)

36. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Sample mean +/ - t*(stddev(s)/sqrt(n))
Least absolute deviations estimator - used when extreme outliers are not uncommon

37. Biggest (and only real) drawback of GARCH mode
Price/return tends to run towards a long - run level
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Nonlinearity
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

38. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
95% = 1.65 99% = 2.33 For one - tailed tests
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

39. What does the OLS minimize?
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
SSR

40. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Summation((xi - mean)^k)/n
Among all unbiased estimators - estimator with the smallest variance is efficient
SSR

41. Beta distribution
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Concerned with a single random variable (ex. Roll of a die)

42. Time series data
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Returns over time for an individual asset
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
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

43. Standard normal distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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!)
Transformed to a unit variable - Mean = 0 Variance = 1
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

44. 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
For n>30 - sample mean is approximately normal
If variance of the conditional distribution of u(i) is not constant

45. Variance of sample mean
Variance(y)/n = variance of sample Y
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Peaks over threshold - Collects dataset in excess of some threshold
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

46. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance = (1/m) summation(u<n - i>^2)
i = ln(Si/Si - 1)

47. Cross - sectional
Sample mean +/ - t*(stddev(s)/sqrt(n))
P(X=x - Y=y) = P(X=x) * P(Y=y)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Average return across assets on a given day

48. Simulating for VaR
Special type of pooled data in which the cross sectional unit is surveyed over time
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)
For n>30 - sample mean is approximately normal

49. WLS
Has heavy tails
Yi = B0 + B1Xi + ui
If variance of the conditional distribution of u(i) is not constant
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

50. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)