5 ms·

Actuarial notation

78 points by azalemeth 5y ago

deleted 5y ago
[deleted]
hbrav 5y ago
I really want to say that I hate this, but I'm sure it was invented for a reason. And as someone who used to write half a dozen indices on tensors in general relativity problems, I should probably not be so quick to judge.
- _Microft 5y ago
  I had commented here before but removed the comment for this reason. It said "And I thought I had seen it all while still doing physics and math". I think we just feel like people who are unfamiliar with math notation when they see a physics paper. It works for physicists - and well - but that does not mean that it is not absolutely inscrutable to someone who comes from a different field. Now for once we are the ones coming from a different field.
  - azalemeth 5y ago
    That is largely why I posted it! ;-). I came across a question on the TeX stack exchange about typesetting the -| subscript, and wondered where on earth it came from...
    - _Microft 5y ago
      Funny that you mention TeX because I had wondered whether I could have rendered that correctly in my mind if I had only seen the math mode source of it.
- genedan 5y ago
  It's the same reason other notations get invented, it makes calculating things and communicating concepts regarding insurance products faster and easier. One pain the ass I have though is needing to install the actuarial symbol LaTeX package for various software I use and write. I got it working with stuff like Sphinx and Anki. Things like communicating over bulletin boards is still very challenging if the site does not support custom notation: https://ctan.math.illinois.edu/macros/latex/contrib/actuarialsymbol/actuarialsymbol.pdf https://ctan.math.illinois.edu/macros/latex/contrib/actuaria...
- AQuantized 5y ago
  Some seems a bit like Dirac notation with more horizontal lines. Seems overly complex for what it represents, however. They're not representing complex conjugates or integrals.
  - hbrav 5y ago
    Well, interest rates have a bit of complication. There's at least 4 things they're a function of: * Time you're looking to borrow / lend at. * Duration of the loan. * Period that the rate is in. * Compounding frequency. e.g. you'd want to specify something like "The 10 year interest rate on 2021-09-22, in percent per year, continuously compounded".
genedan 5y ago
Actuary here. There's a subject called mathematical interest theory that explores time value of money beyond simple and compund interest that you learn in grade school. It makes use of this notation and is part of the US curriculum for actuaries. For example, there's a concept of continually compounding interest as the limit of compounding frequency approaches infinity. Other growth patterns such as polynomial growth are theoretically possible. Anyone with basic calculus and algebra can learn it and it's a good subject for enhancing one's financial literacy. I'm surprised it's not taught in general finance.
- hbrav 5y ago
  Does polynomial growth actually have applications in finance? I would be surprised to see one, but I like to be surprised :-)
- azalemeth 5y ago
  I think that is interesting (particularly the polynomial part) and I've never heard of it before. Do you have any good 'easy' introductory references?
  - genedan 5y ago
    Yeah, these are some books on the syllabus: https://bookstore.ams.org/text-57 https://bookstore.ams.org/text-57 https://www.goodreads.com/book/show/1715689.Theory_of_Interest https://www.goodreads.com/book/show/1715689.Theory_of_Intere...
- mbg721 5y ago
  I took a life-contingencies math course in college that used the life table notation. The instructor's background was actually in medical clinical trials, where as with life insurance, you start with a population, some of them die (or get another outcome the trial is testing for), and some withdraw or don't report back.
jonrx 5y ago
This makes me weirdly happy to see on hacker news. :) Actual notation was the reason I got deep into LaTeX during my undergrad degree (actuarial science). I don't recall the library I used at the time, but if I were to re-do it today, I'd use this one: https://vigou3.gitlab.io/actuarialsymbol/ https://vigou3.gitlab.io/actuarialsymbol/. Vincent Goulet (from Laval University in Quebec) has done a lot to bring actuaries into open source and free software. Once you internalize the notation (you have to for actuarial exams), it's actually quite powerful. Even though I have not practised in half a decade, I still am able to read 90% of the symbols and get the underlying mathematical formula. It also makes a tremendous amount of sense when you use it in the context of a mortality table (see https://upload.wikimedia.org/wikipedia/commons/4/47/Excerpt_from_CDC_2003_Table_1.pdf https://upload.wikimedia.org/wikipedia/commons/4/47/Excerpt_... as an example).
aseerdbnarng 5y ago
Once you get the hang of it, AN reads beautifully. The concept of Life Tables and the ways the math can be abstracted away using these mystical symbols is incredibly elegant. Side comment, when you get into the weeds with actuarial exams you quickly learn to respect older actuaries who had to solve these complex problems using clever heuristics instead of computing power. Once those heuristics are internalized they are able to give incredibly accurate off-the-cuff answers using their intuition. I remember my boss and I spending a week valuing a new product and presenting it to the wise old chief actuary who immediately felt something was off in the numbers and could point out where we likely made an error. It took me all afternoon to find the error, but of course he was right.
_moof 5y ago
Interesting. Couldn't help but be reminded of station models in meteorology: https://en.m.wikipedia.org/wiki/Station_model#/media/File%3AStation_model.gif https://en.m.wikipedia.org/wiki/Station_model#/media/File%3A...
klelatti 5y ago
One of the points that I don’t think is fully clear from Wikipedia is that this is not just about notation - it’s about creating a limited set of tables that combine mortality and compound interest and which can then be used to calculate the desired discounted cashflow value by hand. As such I think the use of the tables and notation have reduced significantly since PCs first appeared on the scene. Today almost all life actuarial computations are done using either spreadsheets or dedicated (proprietary) software packages from actuarial consultancies which can do much more sophisticated cashflow projections - for example using stochastic investment return assumptions. Might be too far to say this is a historical curiosity by now but not by much.
DesTeddy 5y ago
Reluctant actuary here. It’s not as bad as Wikipedia makes it look, you guys. Not that much harder than some advanced algebra courses.
- wheelinsupial 5y ago
  Why are you a reluctant actuary? Since there is professional certifications for actuaries, how much of the work is highly regulated?
lucas_membrane 5y ago
I used to do programming for a company that did loan calculation software. We would get specs from insurance company actuaries for how to calculate consumer loans that were made with credit life insurance attached. The ##### actuaries would always send us formulas that looked like the annuity formulas in the article: annuity due = (1 - v^n) / d, annuity immediate = (1 - v^n) / i. Except, they did not notice that the formula fails for zero interest loans ( zero divided by zero), which were a popular sales gimmick. There are probably a million or more of those loans out there still getting paid off with payments calculated using some arbitrarily small interest rate selected by the programmer to eliminate the zerodivide error when the actual interest rate is zero. Some of the actuaries calculated the insurance charges using really fancy formulas that were not zero divided by zero when the interest rate was zero, but zero divided by zero divided by zero. Limited-precision calculations of these functions were very badly behaved in quite a large neighborhood of interest rate zero, so, despite all the professional refinement of the actuaries, there is no telling how erroneous the payment number on your loan documents came out.

78 points by azalemeth 5y ago

deleted 5y ago

[deleted]

I really want to say that I hate this, but I'm sure it was invented for a reason. And as someone who used to write half a dozen indices on tensors in general relativity problems, I should probably not be so quick to judge.

_Microft 5y ago
I had commented here before but removed the comment for this reason. It said "And I thought I had seen it all while still doing physics and math". I think we just feel like people who are unfamiliar with math notation when they see a physics paper. It works for physicists - and well - but that does not mean that it is not absolutely inscrutable to someone who comes from a different field. Now for once we are the ones coming from a different field.
- azalemeth 5y ago
  That is largely why I posted it! ;-). I came across a question on the TeX stack exchange about typesetting the -| subscript, and wondered where on earth it came from...
  - _Microft 5y ago
    Funny that you mention TeX because I had wondered whether I could have rendered that correctly in my mind if I had only seen the math mode source of it.
genedan 5y ago
It's the same reason other notations get invented, it makes calculating things and communicating concepts regarding insurance products faster and easier. One pain the ass I have though is needing to install the actuarial symbol LaTeX package for various software I use and write. I got it working with stuff like Sphinx and Anki. Things like communicating over bulletin boards is still very challenging if the site does not support custom notation: https://ctan.math.illinois.edu/macros/latex/contrib/actuarialsymbol/actuarialsymbol.pdf https://ctan.math.illinois.edu/macros/latex/contrib/actuaria...
AQuantized 5y ago
Some seems a bit like Dirac notation with more horizontal lines. Seems overly complex for what it represents, however. They're not representing complex conjugates or integrals.
- hbrav 5y ago
  Well, interest rates have a bit of complication. There's at least 4 things they're a function of: * Time you're looking to borrow / lend at. * Duration of the loan. * Period that the rate is in. * Compounding frequency. e.g. you'd want to specify something like "The 10 year interest rate on 2021-09-22, in percent per year, continuously compounded".

genedan 5y ago

Actuary here. There's a subject called mathematical interest theory that explores time value of money beyond simple and compund interest that you learn in grade school. It makes use of this notation and is part of the US curriculum for actuaries. For example, there's a concept of continually compounding interest as the limit of compounding frequency approaches infinity. Other growth patterns such as polynomial growth are theoretically possible. Anyone with basic calculus and algebra can learn it and it's a good subject for enhancing one's financial literacy. I'm surprised it's not taught in general finance.

hbrav 5y ago
Does polynomial growth actually have applications in finance? I would be surprised to see one, but I like to be surprised :-)
azalemeth 5y ago
I think that is interesting (particularly the polynomial part) and I've never heard of it before. Do you have any good 'easy' introductory references?
- genedan 5y ago
  Yeah, these are some books on the syllabus: https://bookstore.ams.org/text-57 https://bookstore.ams.org/text-57 https://www.goodreads.com/book/show/1715689.Theory_of_Interest https://www.goodreads.com/book/show/1715689.Theory_of_Intere...
mbg721 5y ago
I took a life-contingencies math course in college that used the life table notation. The instructor's background was actually in medical clinical trials, where as with life insurance, you start with a population, some of them die (or get another outcome the trial is testing for), and some withdraw or don't report back.

jonrx 5y ago

This makes me weirdly happy to see on hacker news. :) Actual notation was the reason I got deep into LaTeX during my undergrad degree (actuarial science). I don't recall the library I used at the time, but if I were to re-do it today, I'd use this one: https://vigou3.gitlab.io/actuarialsymbol/ https://vigou3.gitlab.io/actuarialsymbol/. Vincent Goulet (from Laval University in Quebec) has done a lot to bring actuaries into open source and free software. Once you internalize the notation (you have to for actuarial exams), it's actually quite powerful. Even though I have not practised in half a decade, I still am able to read 90% of the symbols and get the underlying mathematical formula. It also makes a tremendous amount of sense when you use it in the context of a mortality table (see https://upload.wikimedia.org/wikipedia/commons/4/47/Excerpt_from_CDC_2003_Table_1.pdf https://upload.wikimedia.org/wikipedia/commons/4/47/Excerpt_... as an example).

aseerdbnarng 5y ago

Once you get the hang of it, AN reads beautifully. The concept of Life Tables and the ways the math can be abstracted away using these mystical symbols is incredibly elegant. Side comment, when you get into the weeds with actuarial exams you quickly learn to respect older actuaries who had to solve these complex problems using clever heuristics instead of computing power. Once those heuristics are internalized they are able to give incredibly accurate off-the-cuff answers using their intuition. I remember my boss and I spending a week valuing a new product and presenting it to the wise old chief actuary who immediately felt something was off in the numbers and could point out where we likely made an error. It took me all afternoon to find the error, but of course he was right.

_moof 5y ago

Interesting. Couldn't help but be reminded of station models in meteorology: https://en.m.wikipedia.org/wiki/Station_model#/media/File%3AStation_model.gif https://en.m.wikipedia.org/wiki/Station_model#/media/File%3A...

klelatti 5y ago

One of the points that I don’t think is fully clear from Wikipedia is that this is not just about notation - it’s about creating a limited set of tables that combine mortality and compound interest and which can then be used to calculate the desired discounted cashflow value by hand. As such I think the use of the tables and notation have reduced significantly since PCs first appeared on the scene. Today almost all life actuarial computations are done using either spreadsheets or dedicated (proprietary) software packages from actuarial consultancies which can do much more sophisticated cashflow projections - for example using stochastic investment return assumptions. Might be too far to say this is a historical curiosity by now but not by much.

DesTeddy 5y ago

Reluctant actuary here. It’s not as bad as Wikipedia makes it look, you guys. Not that much harder than some advanced algebra courses.

wheelinsupial 5y ago
Why are you a reluctant actuary? Since there is professional certifications for actuaries, how much of the work is highly regulated?

lucas_membrane 5y ago

I used to do programming for a company that did loan calculation software. We would get specs from insurance company actuaries for how to calculate consumer loans that were made with credit life insurance attached. The ##### actuaries would always send us formulas that looked like the annuity formulas in the article: annuity due = (1 - v^n) / d, annuity immediate = (1 - v^n) / i. Except, they did not notice that the formula fails for zero interest loans ( zero divided by zero), which were a popular sales gimmick. There are probably a million or more of those loans out there still getting paid off with payments calculated using some arbitrarily small interest rate selected by the programmer to eliminate the zerodivide error when the actual interest rate is zero. Some of the actuaries calculated the insurance charges using really fancy formulas that were not zero divided by zero when the interest rate was zero, but zero divided by zero divided by zero. Limited-precision calculations of these functions were very badly behaved in quite a large neighborhood of interest rate zero, so, despite all the professional refinement of the actuaries, there is no telling how erroneous the payment number on your loan documents came out.