Update: the Annuity Symbol on WordPress

A few months back, I was trying to get the annuity symbol to come out right in Latex. I got it working pretty good, but the hack that I used did not work on these blog posts.

At the time, I did not really put enough work into trying to get the annuity symbol to appear correctly just using standard ams symbols. But here it is:

a_{\overline{n}\lvert }
a_{\overline{n}\lvert }

\ddot{a}_{\overline{n}\lvert }
\ddot{a}_{\overline{n}\lvert }

There you have it.

The Cards

Cutting them up:

cutter small

I have owned this paper cutter for 25 years.  At that time, I did not own much but the clothes on my back.  How would I have survived without a paper cutter to make little books and cards and such?

finished cards smallThings that I gained by making these cards:

  1. A strengthened knowledge of FM fundamentals.
  2. Increased fluency with LaTeX.
  3. A creative project which kept me focused on the subject at hand.
  4. A nice little pile of finished cards.
  5. Time to think about the pluses and minuses of virtual products versus actual.

 

 

Exam FM, Chapter 1 Flashcards

I just posted my nearly final version of the first chapter of flashcards for exam FM.  I added some questions on geometric progressions, and some on force of interest, so now there are 226 cards.  I know that there are still a few holes, but I will worry about that later.  I could probably come up with 300 cards for the first chapter very easily.  But, having put together several sets of study cards, I find that I add fewer cards for subsequent topics.  In almost any subject, the most vital information to commit permanently to memory is at the beginning.

I tried to use a sampling of the different terminologies.  For instance, I used the accumulation function a(t) as well as the FV, PV terminology.  When I look at the solutions for the exam sample questions, I see both types used, so I suppose that it is possible for either to appear in an examination question.

For most formulas, I give several types of numerical examples.  It is important to see the relationships at work.  To see the animals in their native habitat.

In a later post, I will describe how to use these cards to best effect.  I do intend them as a means of permanently learning the material.

 

Finite Geometric Progressions

If you look over in the right sidebar, under Memes For You, you will find the expression If not now, when? Here is how I apply this expression to mathematics: During the course of work, if I encounter a mathematical expression, algorithm, or symbol which I don’t fully understand, I take some time and learn it right now. Probably I will run into this mathematical truth again. Perhaps I should have learned this mathematical truth in High School, or College. Plausibly I am missing out on a beautiful gem of mathematics. A few examples: the triangle inequality (simple nearly to the point of triviality, yet beautiful, and vital in certain proofs), the quadratic equation (immensely practical, yet also historical (you can’t use a formula to solve equations of any higher degree), why did I not memorize it in high school?), the definition of real numbers (my favorite definition, centuries in the making).

That leaves us right now with Finite Geometric Progressions. Somehow, they avoided me, or I avoided them, for all of these years. But, here they are, at the root of financial mathematics. I may be able to learn financial mathematics without them, but why should I miss out on an opportunity to become friends with these cute little critters? Here we go.

Starting a month from today, you are going to deposit one dollar each month into an account that pays 0.25 % interest per month. How much money will be in the account in 6 months, at the time of the last payment?

Let’s work backwards. You have made 6 payments.

  • The 6th payment has accumulated no interest, so is still worth 1.
  • The 5th payment has accumulated one period of interest, so is worth 1 (1.0025)
  • The 4th payment has accumulated two months of interest, so it is worth 1 (1.0025)^2
  • The 3rd payment has accumulated three months of interest, so it is worth 1 (1.0025)^3
  • The 2nd payment has accumulated four months of interest, so it is worth 1 (1.0025)^4
  • The 1st payment has accumulated five months of interest, so it is worth 1 (1.0025)^5

The sum of all the deposits, plus the interest, is hence 1 + (1.0025) + (1.0025)^2 + (1.0025)^3 + 1.0025)^4 + (1.0025)^5

We might have written this as t_1 + t_1 r + t_1 r^2 + t_1 r^3 + t_1 r^4 + t_1 r^5 or the sum of the first n terms of a geometric progression with common ratio r. In high school, we should have learned that this sum is equal to:

t_1 \frac{r^n -1}{r-1}

In our case, r = 1.0025, n = 6.

1 \frac{1.0025^5 -1}{1.0025-1}= 6.0376

We might also write this as \frac {(1+i)^n -1}{(1+i)-1}

Which clearly equals \frac {(1+i)^n -1}{i}

Which mysteriously is also s angle n, or the accumulated value of the annuity immediate.

Another Update on Exam FM Flashcards

I updated my Financial Mathematics cards again this morning. The link from a couple of days ago reflects the updates. There are now 190 cards.

The cards are in the form of a PDF file, to be printed on letter size paper (8.5 x 11 in). Many printers do not print the front and back of each page in good alignment, so I made sure not to place text too near the borders of the back of the cards.  Simply print the cards out, cut a 1cm border from around the page, then cut the cards out.  Whoops!  I have mixed inches and centimeters.  I will straighten out the unit issue on the next version, with a 1/2 inch border.

I am trying to really fill out most of the fundamentals.  This makes the cards useful not just for someone studying for the actuary exams, but also for anyone learning the basics of finance.

Let me know how the cards work for you, or if you find any errors.

Nearly done With First Chapter of Cards

I have just updated the first chapter of my Financial Mathematics cards.  (link is on post from a couple of days ago).  The deck is up to about 160 cards.  I am looking at the exam FM syllabus, plus some books, and filling in topics.  Next week the first chapter will be finalized, and I will post a permanent link on the sidebar.

Later, I will post some advice on how these cards are intended to be used.  In essence, these cards reinforce the fundamental pieces that go into solving difficult problems.  They certainly do not replace solving tough problems, or replace learning the concepts.  Often, they may suggest new ways of solving complex problems.  I intend them to be used along with a pencil, paper, and calculator.  Setting up the solution is more important than a numerical answer, however.  The numerical cards are intended more as ways of recognizing and reinforcing the fundamental equations and relationships.

Lots of people have downloaded the deck already.  Thanks!

I can’t wait to finally post my Exam P cheat sheet.  LaTeX is great 🙂

Downloadable Financial Mathematics Flashcards

I have been working hard today.  Here is a PDF of the first draft of my exam FM flashcards.  So far, they just cover “chapter 1” stuff like Interest Theory and Lump Sums.  I think that there are about 110 cards so far.  I will soon be fleshing out the missing Interest Theory topics, then I will add the Annuity cards that I have made.

I am delighted to have used LaTeX to design the cards.  They are 2 cards by 5 cards on a letter size page.  This is a standard card size, which you can buy paper for at the store.  I will add some cut marks when I figure out how.

More tomorrow.

Update on Card Formatting

Yesterday, I posted this flash card as an example:

When I did my repetitions this morning, I realized that this card, and several similar ones that I created yesterday, are too complicated.  Cards should only test one very small item of knowledge. (Read SuperMemo 20 Rules of Formatting Knowledge)  The cards from yesterday test not only the function of the “a angle n” function, but also require a calculation with this function.  A much better card is as so:

The wording of the card now makes it obvious that I am not looking for a numeric answer.  The purpose of this card is simply to help the mind to recognize and recall a common relationship that involves the present value of the annuity immediate.

 

The Procrastination Equation

A few days ago, I discovered the Procrastination Equation, by P. Steel.
\displaystyle Motivation= \frac{Expectancy*Value}{Impulsiveness*Delay}

I like how this equation accounts for the factors that go into motivation.  To increase motivation (and lessen procrastination),  we use strategies that increase our expectancy of success, and increase the value of the reward for success.  At the same time, we use techniques to reduce impulsiveness and distractions, and reduce delay until beginning.  Any motivational methods can be inserted somewhere onto this equation.  When I break things into small tasks, I am increasing my expectancy.  When I get up very early in the morning, I am decreasing impulsiveness and decreasing delay.  When I perform long and short-term goal setting, I am increasing value.  You can read an article that relates to this here.

The Annuity Symbol in LaTeX and Anki

As you know, one of my favorite creative projects is to create Anki flashcards using the LaTeX markup language.  I have written about some of the process on this blog.  Originally, I needed to install MiKTeX on my system.  Now, Anki can interpret the language when it sees it.

My recent problem has been to format the symbols for annuities.  That looks like so:

Which is read as “a angle n”, and:

Which is read as “a double dot angle n.”

It gets really annoying to write and interpret things like “a double dot angle n;”  that is where the proper symbols come in.

First, I discovered that there is an \actuarialangle command which has been recently added to the MiKTeX library.  Open the Package Manager on MiKTeX, and you may install it.  To activate it in Anki, you need to use some code, in the usepackage line in the preamble of your cards.  See below.

I also discovered the lifecon package on the web.  This contains all sorts of actuarial symbols, but you need to know how to add a package to MiKTeX directly with a .sty file.  I had a little trouble with this task.

The alternative to both of the above solutions is to add some code into the preamble of your document, that defines a new LaTeX command.  There are several versions of this floating around.  The following is what I used:

\DeclareRobustCommand{\lcroof}[1]{
\hbox{\vtop{\vbox{%
\hrule\kern 1pt\hbox{%
$\scriptstyle #1$%
\kern 1pt}}\kern1pt}%
\vrule\kern1pt}}
\DeclareRobustCommand{\angle}[1]{
_{\lcroof{#1}}}

In Anki:

  1. Hit the button to add a note.
  2. Hit the button to choose the note type.
  3. Select “Options”
  4. Cut and paste the above code into the header, just under “\usepackage{amssymb,amsmath}”
  5. Now you may use the \angle command in your cards.

Here is an example:

I put things on cards that I figure I should be able to solve nearly instantaneously.  The notes on the bottom of the card are to refresh my mind, if I have forgotten.  The numerical answer is not really very important, the real thing is reading the problem, and visualizing a solution.  Real test problems, and real life problems, are much more complex than the ones I put on my cards.  But the component parts need to be solved quickly, with confidence.

I am so happy now that I have annuity symbols appearing correctly.  I think that I will go make some flashcards.