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| it's only so fitting to be there, where so many moments were spent: fighting, laughing, thinking, throwing sand and splashing water.
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| i was thinking a while back about how to set up a DB plan in a fairly DC-based way, to benefit with the concept of pooling risks. today, i had a meeting, and we discussed how this issue of if a DB plan looks too much like a DC plan, it will falter and ergo fall in accordance to DC rules.
with law firms, this is important because a DB plan is deeply beneficial for highly compensated persons. that is, the tax-deductible contribution limit for a DC plan is currently 49k (only 16.5k in a 401(k), whereas for a DB plan, it is as much as the present value of one-tenth of the maximum 415 AB limit. this can, depending on how close you are to retirement, which can be hundreds of thousands of dollars pre-tax. and so this gray issue never came to the forefront of the mind of anybody until one particular law firm was being audited by the IRS, and the IRS decided to bring up this issue to the public. this issue that has come up is attributable to poor investment losses and how law firms are countering such a loss by allocating individual losses to partners participating in a DB plan (which makes a DB plan look just a little more like a DC plan). usually, when investment gains/losses are mild, then this issue isn't worth considering unless it was out of the IRS's sheer boredom.
but whatever re the issue is with law firms, but what about for regular joes. or even more, the underprivileged?
it's essentially impossible for me set up a DB plan for people who want a conservative and well-defined retirement vehicle unless i employ people, or if i act as a participating employer to a bunch of employers who have come together to form a multiple employer or multiemployer plan. but what do i know about running a business? i'm not a management consultant. i'm just an actuarial analyst. i guess i can learn. and then my goal can be to run a company to employ people (oh the headache).
sometimes i wonder if i'm settling, like if my ambitions or my skillsets are supposed to be better utilized than just this, or if i'm supposed to make the most of what bores me more often than i care for. or is this is it? is this my lot in life? if it is, i'll take it. and i'll live out my days focused on my family - my wife and kids... and the community around me. it's probable that my ambitions are nauseating to God, since it may be so self-directed, that these deep passions stem from my humble upbringing where i had to prove to the world i'm worth more than just another reject mouth to feed from a free summer lunch program. or maybe God had designed me to crave more and pursue more, because the world wants people who care more about bigger things. but what do i know of humanity, and what do i know of God besides His love for us? and His capabilities? and His mystery. it's almost as if we all mainly (or only) know His mystery, and the rest is mere speculation and faith. i don't know why i often feel so distant from knowing Him, yet close enough. well, i actually know exactly why - i don't interact with Him much. i only glib unthoughtful letters to Him, not really seeing how He's doing and what He's up to. but it's mainly just babbling about my measly circumstance. my priorities are just whack.
we're moving out of this office on friday, and i'll miss the beautiful view from 35 stories up, the highest point from here to san jose. and i'll miss the larger office environment, and how it carries a degree of independence and anonymity. but i'll appreciate the small space, how we'll be forced to learn to deal with each other (otherwise, we'll just have to leave each other).
i guess i'll go home now.
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| eugene scratched his rims today. he's sad. and a little tantrummy. little angry asian man.
now he's laughing. stop, he says. but no. i will not. eugene then responded "oh, it's one of these posts, where you type whatever's happening" he doesn't realize "and where you delete...", it's more than that "bob lob law". it's deeper. it's more multi-layered than he can ever imagine, because the simple in this world complicate things for those who seek out more. BS, he says. that's what my degree says. but no. BA. BA...
scoff. as he usually does.
ben's reading a book about basketball. i think the book is supposed to, in 700 pages, encapsulate the history of basketball.
eugene and ben are having a little squabble about this post. "it's not a squabble" he squabbles.
and now, eugene is on his way to clean the rims of his car - the terribly scratched rims. he's going to cry about it. it's more devastating to him than that motorcyclist candice saw got hit badly by an all-too-chill-sip-on-my-coffee-as-cyclist-is-on-the-ground little asian woman. ok i lied, i'm not sure if she's asian... or if she's little for that matter. and because i'm not a man of statistics, i will recant any insinuations about little asian women having a higher likelihood of getting into accidents. you know what? it's as true as it is not true... but that is the world we live in. where the things that affect us are much more magnified than the rest of the world. for our worlds are microcosms of the perceptions we live out - about each other and about ourselves. and happily, these perceptions, in all its diversity, unified only by the fact that they are lived out by each of us, form the social/antisocial fabric we call society. though we all fall short in our understanding in some fashion or a n o t h e r, we still got something to contribute and to live for. cuz we're that valuable, i think. but i'll leave commentaries on sociology up to the experts. .like franky. oh he can get into some in-depth analysis on how i'm a social deviant, a pariah who fools himself into thinking he's just a maverick, a term less heavy on the negative stenches of sulfur and body odor. yes, like palin. out for fame. and money. and jesus.
candice and i watched away we go today. i liked it a lot. it had a familiar tone about it, something that had a depth i feel with good films. i later found out some collaborators include sam mendes (of american beauty), and dave eggers (of, well, ahwosg and wtwta, and what is the what). bill (my coworker, not dave's brother), reminded me last week that dave eggers is not a god. i know. but if he has nothing better to do than take care of his wife (who also co-wrote the film we saw) and two kids (familial themes of the film still lingering in my mental faculties right now), he and i can hang, sip on some hot apple cider or something more refined. like granulated sugar.
but whatever the case, i'm either just subconsciously honing down on anything related to dave eggers, or i'm chancing upon his work and liking it, and recognizing who to give credit to. i'd want to say i was him in another life, but i guess we'd touch upon some deep philosophy on personal identity, reincarnation, and the clash between two currently existing lives being part of one person... how it'd mess with the time-space continuum. only one johnny from modesto with his such distinct experiences can exist at a time. or ever. how meaningful these distinct experiences are to anyone else is a matter of subjectivity anyway. well hey, it means a lot to me. because the special thread used to weave together the fabric of my history, and likely colored by me memory, is shaping who i am today. a child of the Lord Most High, an actuary, a son, a person with four syllables in his name (if you count his middle name), braces, embraces, traces of crazy, and brazen tragedy.
no, are you still writing? yes. i am.
i will go to bed at midnight... well i'll at least close this laptop and attempt to get some rest. yes.
ben gave me two thumbs up. eugene gives me a smile. well. maybe he didn't give it to me. uh oh. crazy eyes. you know what that means. well if you do, let me know, cuz i sure as the heezy don't. i'd guess it's just his way of dramatizing his joy. sometimes it's used to hide the deep pains from within. either way, it exists for a purpose. for lies and for truth. deuce.
clenched fist. inaction. well besides his foot touching his little down comforter. i think he has a napoleon complex. he's magnified on the concept of little. now ben is going to play scramble. the three of us are on my stinky bed. it's like a (how do you spell managueta? oh yes, menage a trois (absent accent marks))
my test is tomorrow. i realized my notes for it were posted publicly. but i didn't care enough to privatize it.
i hope i pass. but if i don't (which i won't not), i will count every hour of study valuable. oh how it produced fruit. the perseverance i've endured. countless hours with cricks in my neck and green tea and cafe mak. well, the truth is, if i failed, i'd regret it. but that's the truth and the beauty of life. in america.
now he's just giving me weird looks, and this entry is getting a little epic, so i will end now... well soon. four minutes.
oh snap. i didn't type anything for two minutes, now i have less than two minutes left to find a conclusion. one minute. well. i'm scrambling. aiyah.
i love autumn. and november. and leaves you can crunch with your feet or hands or butt or head or knees or tongue (if you're weird)...
::edit:: eugene said something offensive to me, and i wanted to document it "grubby"
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| 1. when they give you a value that yields the median, then you plug that into the distributoin WITHOUT application of the deductible. if they ask for the mean excess loss (with deductible) then you add the mean excess loss with the deductible.
2. when given a poisson/gamma, the posterior density follows the gamma with new parameters. if they ask for the mode of the posterior, you are fine. if they ask for some posterior probability, you'd have to account for the marginal "constant" factor. if they ask for predictive expectation, then you'd have a negative binomial density... the mean of the posterior is the same as the mean of the predictive for this conjugate prior.
3. when asked to calculate the bootstrap approximation, your given sample is your "true" sample. when you take the variance of this, you do not divide by n-1. however, when calculating the MSE, and your recombinating your sample with the 256 total combinations, you can sometimes look at your sample as binomial distribution. this gives your the weights for each mean square error (for each different recombinant sample).
4. when asked to calculate the variance of this loss with franchise deductible (if given an ogive), you can break down the group below the deductible to yield no expected cost. you can do the same when calculating the variance. you can calculate the variance in one of two methods... calculate the second moment and subtract by the square of the first. or, more easily, take the square of the size of the interval and divide by 12 for each interval. you weight according to the number of losses/members fall into each interval. you can eventually get this just fine.  .
if they asked for the variance of the loss per payment, you completely ignore the interval between 0 and the deductible. this should increase your overall loss as you're truncating the interval that contributes no loss.
5. when given information for data in which participants are observed to die or surrender, where all entries, deaths, and surrenders occur uniformly, and given that you have single-decrement probabilities of surrender and death, then there is no built in independence where we can just subtract qs and qd from 1 to get the probability of survival. to calculate your survival for one type of decrement versus another, you must adjust the risk set so that that it accounts for the new entrants and decrements (of the other type(s)) before applying the kaplan-meier method.
6. if you have a linear confidence interval around S(x), then you can calculate the variance that corresponds with our S(x). now if you were asked to take some power (such as a cube root) of S(x), how will the confidence interval be adjusted? well you proceed as usual, calculating what S(x) is, and what the corresponding variance is. then let your transforming function be g(x) = x^k, where k is the power you apply to x. well we apply that power to our S(x). then the variance will be the original variance of S(x) times the square of the first derivative of g(x) evaluated at S(x). then we evaluate our new confidence interval of S(x)^k + or - our symmetric p value times the square root of our new variance. if they asked you to take the inverse power to back into our adjusted interval, we then do that. yay.
7. if you're given a normal (theta,v)/normal (u,a) conjugate prior, you can evaluate this by either buhlmann or bayesian. note that for the buhlmann approach, or "v", or expected process variance, is v, and our "a", or variance of the hypothetical mean, is a, or the variance of the prior density. it is probably easier to use the buhlmann approach, but it really depends on the data.
8. if given N follows poisson with a given mean, and claim sizes with an empirical distribution, and you're asked to calculate the survival probability or cdf at some value, you'll just have to evaluate your "g" or aggregrate functions by brute force. it's easy.
9. if you're given a bernoulli with a prior that is piecewise continuous, n years of experience with k insureds, and that the credibility factor is the same for buhlmann approach and limited fluctuation credibility approach, then you can find the expected value of the prior based on first principles (integral of x*f(x) or S(x), whichever is easier to evaluate), then you can calculate the buhlmann AND the limited fluctuation credibility. if they ask to solve for a variable, you just do algebra. you can do this with any model or prior, so long as you have sufficient information. your k insureds is the numerator. your n years is the n for buhlmann. you can usually calculate v, a, and u, etc. sometimes you may be asked to relate bayesian credibility with buhlmann. in which case you have to look at what info you have. for bayesian, we usually need data to evaulate the posterior.
10. note that for a likelihood function, we must remember that our upper bound for the uniform distribution, say theta, must be at least as great as the max observation. so we maximize our theta as small as possible, namely setting our theta to be equal to our max observation. so if you're asked to use the likelihood ratio test, then know that our likelihood function multiplies 1/theta for each observation (unless they're censored). so our loglikelihood is n times the natural log of the maximum observed value of a given population. and the ratio likelihood "test statistic" is twice the difference between the loglikelihood of the alternate and null hypotheses. the degrees of freedom is the number of free parameters in the alternate less the number of free parameters available in the null hypothesis.
11. if you're given some values out of 100 generated values, and asked to evaluate the variance of your TVaR, then your VaR at the pth percentile is the pth generated value. the TVaR is the mean of the observations beyond the pth percentile. then the variance of the TVaR (we'll need to unbias by multiplying the variance by n/(n-1) because we're dealing with a sample of values) is basically how we take the variance of empirical data. but additionally, we then add p*(TVaR - VaR) and divide it all by the number of observations. yeah... another formula motivated by the conditional variance formula.
12. when given an equally weighted mixture of two poissons with a mean and variance provided, you can then do some algebra to solve for each of the distributions... then we can back into the median.
13. if given non-uniform exposures, identify what your mijs and xijs are... if asked to calculated the credibility factor for one of the groups, your number of members is n. you can calculate Z and you can also calculate the premium where Z is weighted on the mean of the group of interest, and the (1-Z) is applied to the aggregrate mean.
14. if given a coverage subject to a deductible, given payments of various sizes, and given that you're fitting a distribution with 1 or 2 parameters (if they want you to sift through lots of algebra, they can give you a hundred parameters). you then apply MLE to get your parameter, doing the whole derivative loglikelihood thing.
15. sometimes you can have a set of equations to solve for things algebraically. know that Var(S) = v + a, and that buhlmann Z = n / (n + k), where k = v/a.
16. chi square statistic. for a pareto, note that we adjust our pdf and cdf as this: F(Xadj) = 1 - ((theta + d)/(theta plus x))^alpha.
17. equilibrium distribution:
the expected value of the equilibrium distribution is: E(X^2)/(2E(X))
the pdf is S(x)/E(X), where x is nonnegative.
the survival function: integral of pdf from x to infinite.
the hazard rate is the eq. pdf over eq. survival function
18. the buhlmann credibility factor, Z is equal to Cov(X,Y)/Var(X) if X is a payment and Y is the bayesian posterior. not that Cov(X,Y) = a, and Var(X) = v + a not that this is related to bulhmann in that Z = a / (a + v) --> 1 / (1 + k), where k = v/a. we calculate the expected value, which should be a simple data average calculation. then we calculate the variance and covariance. we then get our Z. we can back into our k (we may not need to know our v and a), then we recalculate the buhlmann credibility with n = 4 instead of 1. the covariance is the sum of probability times (pmt size less mean) times (posterior expectation of next pmt given pmt size less mean) for each payment.
19. if you are asked to determine the simulated buhlmann credibility premium, you are usually given some u's. first you calculate the buhlmann credibility factor, which means you need n observeds, and information to calculate the the overall mean, v, and a. we then calculate the bulhmann credibility factor. then we will simulate the claims and take the average over n simulations (same n as the n observeds used to calculated the credibility factor). then the buhlmann credibility premium is the sum of the simulated mean and the overall mean by the factor of (Z) and (1-Z) respecitively.
20. if given means and variances of number of claims and size of claims for various groups, then we can calculate our overall mean and overall variance. this can be used to calculate the limited fluctuation credibility if we're given the number of claims observed... if course we can switch up the algebra if they give you the factor but not the number of observations. this data lends itself to calculate the buhlmann credibility stuff too.
21. say you have losses following a compound distribution, where both frequency and severity have discrete distributions. if we are asked to calculate the probability of aggregrate losses being exactly something, or a cdf/survival or something, then we plug and chug to obtained an aggregrate value of s. if you're given probability generating functions, you take the kth derivative evaluated at 0 to calculate the probability of k claims or of losses of size k. THEN you'll try to find the various combinations needed to solve the pdf, cdf, or survival function.
22. if you're given stop loss expected loss with dedicutibles, you can take the difference b/t two expecteds to get the integral from lower deductible to higher deductible to evaluate survival funtions at deductible points... we might hae to do some algebra and hand waving to back into the things we need.
23. if asked to calculate an expected value (or limited expected value) based on applying the kaplan meier method to calculate probabilities, then we just calculate discretely the expected (or lev) values.
24. note that (1 - x/n)^n goes to e^-x as n goes to infinite.
25. similar to the other problem where we evaluate the , u, v, a, k, n, Z, and then get the sample mean and evaluate the buhlmann credibility premium.
26. when comparing a percentage of expected earnings with some amount plus expected losses. a sales advance can be interpreted as a deductible... so what you end up having is a franchise deductible, where the adjusted percentage is applied only to the random variable.
27. algebra.
28. if we are fitting tPx = (Px)^t, then deaths that occur in the MOY yields (1-Px^.5). if we have withdrawals, they are censors and we treat them as we do with censors in MLE. if someone dies between a specified interval, we give it Px^.5 times (1-Px^.5).
29. the way the problem is set up is you have new entrants entering each year, but in reality, you care about the time a settlement is made with respect to when a claim enters. so don't get confused. the censors are the claims that don't have data beyond years after settlements.
30. it's as if you have overlapping ogives when you're asked to calculate an LEV for distribution of losses following a uniform kernal.
31. if you have a unif/single pareto prior, apply the bayesian estimation with continuous prior, where the lowest "b" value is 2 for the uniform.
32. percentile matching for lognormal is easy.
33. when given that a mean of a distribution is estimated by n data points, and the estimator include a constant c to the sample mean, then you just do some algebra and find the c that minimizes the MSE. recall that bias is equal to the expected of the estimator less the true parameter. and the MSE is equal to the variance of the estimator plus the square of the bias.
34. given a binomial with a uniform prior, you just apply the predictive probability.
35. same as 32.
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| so here is a summary of what i've been doing.
bayesian estimates with continuous priors:
you have a model with some observations.
you have a prior distribution that the model distribution is dependent on in some fashion.
you take the product of the two aforementioned to get a joint distribution.
you take the integral of the joint function to get the marginal distribution.
the posterior distribution is the joint function divided by the marginal distribution.
the expected posterior is the joint function times the a variable we call x of the model distribution... all divided by the marginal distribution.
the predictive posterior is the integral of the product of some pdf, cdf, or survival function and the joint distribution... all divided by the marginal distribution.
the predictive expectation or bayesian estimate is the integral of the product of the joint function and the expected value of the model distribution... then you divide all this by the marginal.
there are these things called conjugate priors. this means that the posterior has the same distribution as the prior, but with different parameters.
for a poisson/gamma prior, you have a marginal that follows a negative binomial distribution that is equal to the gamma's parameters. the posterior is gamma with adjusted parameters. the predictive is negative binomial with the posterior gamma's parameters.
other conjugate priors include the inv. expo/gamma prior, the exp/inverse gamma prior, the normal/normal, binomial/beta, and single pareto/ unif combo. the conjugate priors match a specific "k" in buhlmann credibility.
poission/gamma gives 1/theta, bin/beta = a+b, normal/normal is v/a, and the single pareto one is (alpha - 1) i think.
the uniform distribution is beta if the beginning interval is zero. if that condition holds, then you have a = 1, b = 1, and theta = 1.
uncategorized notes:
if you take the expected value or limited expected value of a single pareto with alpha = 1, you'd have to do it by integrals, because the appendices do not provide an ev or lev for it. when you do this, remember that for any values below the theta parameter yields a survival probability of 1. so you always take the integral from 0 to either a a point of interest or infinite. the integral from 0 to the point of interest (usually the deductible) yields the value at the point of interest... don't forget to add that point of interest value.
goodness to fit tests:
p-plot... (x,y) = (smoothed empirical distribution Fn(x), fitted parameter distribution F*(x))
D-plot... (x,F*(x))
K-S or Kolmogorov Smirnov test. works for individual data only, critical value depends on sample size, the test statistic determines, like any other test with significance testing, at what point by which to "reject" the null hypothesis.
A-D or Anderson-Darling test. this is a method of approximating the integral for each interval. there's a continuous version and a discrete version... the formula is messy... it looks like this
-nF*(d) +n((1-Fn(yi))^2 times the summation (form 0 to k) of the natural log of (1-F*(yi) / 1-F*(yi+1) + F*(yi) times the summation (from 1 to k)of the natural log of (F*(yi+1)/F*(yi)))... talk about rediculosity!
the continuous version is n times the integral (from t to u) of: the square of the difference b/t Fn(x) - F*(x) divided by (F*x)(1-F*x) times f*(x) dx.
there's also the chi-square distribution.
it's motivated by the poission distribution, unless specified with another. we calculate the "error" for each group of observeds as it compares to the expecteds. if there is a spike difference in one or more intervals (difference b/t observeds and expecteds in a group), then you really drive up the chi square statistic. and in doing so, we become less likely to reject the null hypothesis.
one thing about the chi-square is that if you're deaing with data from several periods, the way in which you count your degrees of freedom may be different... basically the degrees of freedom are the total number of "variables". if they specify a total number of exposures, then you usually have n-1-r degrees of freedom. the last interval of the n intervals would be forced... if there are r parameters estimated based on the data, that will lower your DOF.
there's also a likelihood ratio test, which is 2 times the natural log of (H1 - H0), or alternative hypothesis less null hypothesis. the degrees of freedom here are the number of free parameters in the alternative less the number of free parameters in the null hypothesis. and you consult the chi square table to determine at which point to "reject" and "accept" your null hypothesis.
schwarz bayesian criterion... it's the loglikelihood of some model less r/2times natural log of n, where n is the sample size. the best choice is the one that produces the largest loglikelihood (values are usually negative, so we get the smallest negative)
look at how many formulas there are. i guess the whole point of these goodness to fit tests is to find different ways in determining how well an alternative hypothesis fits in any situation... and it fits better than the null, we'd be more inclined to "reject" it.
i don't like statistics.
there is this whole idea of bias, which is E(theta hat given theta) less theta.
ok... i'm tired.... there's so much information!
and i've done the worst job of learning the way i should, and that's by way of theory and foundation. then extra details related to application. i'm so sad.
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