Eight Secret Tricks to Passing (Exam FM)

1.  Be over-prepared: The syllabus is so extensive, and there are so many twists that may be put on standard material, that there are always going to be questions that look foreign. The more that you are prepared, the more that you will be able to handle some of these. Being over prepared also helps to deal with emotions. There were points during the exam when a little voice in my head said “There is no way that you are going to pass this.” Yet, that same voice was in my head during many sample exams which I passed with flying colors. Experience and over-preparation reveals that you can succeed even when you are not at your best. If you walk into the exam center with worries, or a cold,or the flu, or pain in your back, you know that you can still succeed.

2.  Fill in the tough spots: It can hurt your head to learn new things. Identify the areas with which you struggle, and try to get your head around them. Two weeks before the exam, I still had some head-pain topics. Things like duration matching, convexity, interest rate swaps, and put-call parity. But, they were on my daily study schedule, and I forced myself to confront them each day. It paid off.

3.  Fill in the fundamentals: Once you gain proficiency in a topic, the fundamentals look different. In the month before the test, I went back and reviewed some of the basics of interest theory. At this point, the basics were easy, and I picked up on some of the finer points that I missed the first time around. Amortization schedules, for instance.  After I learned the basic formulas, it was easy to ignore the schedules they are based on. Yet many problems are made much easier by using schedules, rather than relying on formulas.

4.  Memorization is easy: Once you understand things thoroughly, you don’t really need to do much memory work for these exams. After months of working problems, most things will be anchored in your head. And yet suppose that you find yourself trapped in a dark alley by two annuities, one of which is twice as long as the other. Then you will be glad that you memorized \frac{a_{\overline{2n}\lvert }}{a_{\overline{n}\lvert }}=1-v^n I have never encountered this fact in practice, but if I did, recognizing this identity might save me a couple of minutes. Spend a little time on memory work, it is easy.

5.  Test symbolic solutions with numbers: If you have a problem with a purely symbolic solution, and you can narrow it down to a couple of possible solutions, you can frequently replace the variables with reasonable numbers, and see if the result is true. This is my favorite new solution technique I learned while studying for this exam.

6.  Read MacDonald: Actually read it. After you learn the math, go back and read it. If you look at the notes from the sample problems, you will notice that the exam writers have made it pretty clear that the exam will be based on the book. Those non-numeric multiple choice questions can be very challenging. Read the book.

7.  Number your Scrap Paper: I learned this trick when taking practice exams.  If you have time to review problems at the end of the exam, you will need to find the scratch-work easily.

8.  Practice with dull number two pencils: Two weeks before the exam, I put all of my mechanical pencils away, and worked only with old fangled sharpenable pencils. Two hours into the exam, you will be working with dull stubs, so you need to be prepared.

Don’t Let Yourself be Thrown By Easy Calculations

You have to calculate quickly on this exam. I didn’t really realize that until took my first practice examination.  I was cruising along on problem number 25, with forty minutes left to go.  No problem!

Then I looked at the upper corner of my screen and found that I still had ten problems to do!  There are 35 problems on this exam!  That is only a smidgen more than five minutes per problem!  Aaaahhhhhhhhh!!!!!!!!!!

Practice your calculations.  You need your time for problem solving: you don’t have time to recalculate when you get to a solution that is not one of the choices.  Practice your calculations.

  1. (Ia) _{\overline{15}\lvert 0.05}
  2. (Ia)_{\overline{5}\lvert 0.01}
  3. (Ia)_{\overline{12}\lvert 0.08}
  4. (Ia)_{\overline{20}\lvert 0.005}
  5. (Ia)_{\overline{10}\lvert 0.0425}
  6. (Ia)_{\overline{40}\lvert 0.1225}
  1. 73.67
  2. 14.46
  3. 42.17
  4. 196.22
  5. 41.32
  6. 70.86

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.

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.

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.

Basic Annuities

A Nice Little Chart for Basic Annuities:

annuities immediate and annuities dueI used the image occlusion feature on my memorization software to make cards like so:

basic annuity sample cardThere is a simple, subtle, and significant difference between annuities immediate and annuities due, and these cards help to illustrate the difference.  Really the “immediate” and “due” terms are not important.  The important thing is to recognize there you are measuring from, in relation to the payments.