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 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.

Some Easy Time Value Money Exercises for Exam FM

Mathematics texts of a certain level contain lots of difficult problems, yet seldom any easy exercises.  For me, creating simple practice problems is an essential part of learning.  I have plenty of difficult material to learn.  Practicing the easy stuff is a great way of mastering the fundamentals.

I have had this same philosophy in anything I have learned in life.  Each thing that you learn, now becomes something that you can practice.  As you expand your forward knowledge, you gain an entirely different understanding of the things that you learned in the past.

After studying calculus for one semester, the definition of the derivative which you learned in week 3 seems too cumbersome and inefficient to bother with.  Then, a couple of semesters later, you encounter functions for which your quick and easy rules of derivatives no longer apply.  Then, you will be glad that you memorized the definition of derivative back in high school:

f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

During the 25 years of my life that I was an avid juggler, I had a rule of working forward and backward each day.  When my new work (say juggling five or seven balls), would get too difficult, I would retreat for a while to easier stuff, then move back to the new.

So here are some easy problems to practice.

  1. With compound interest i = 0.05, what is present value of 10,000 dollars in 40 years?
  2. With compound interest i = 0.03, what is present value of 2,000 dollars in 9 years?
  3. With compound interest i = 0.01, what is present value of 10,000 dollars in 5 years?
  4. i = 5%.  v = ?
  5. i = 9%.  v = ?
  6. d = 0.055.  v = ?
  7. d = 0.0025.  v = ?
  8. v = 0.96. d = ?
  9. v = 0.85. d = ?
  10. The present value of $50000 payable in 30 years at an effective annual discount rate of 5%.
  11. The present value of $1000 payable in 15 years at an effective annual discount rate of 4%.
  12. Rate is 2% per quarter. Effective rate i = ?
  13. Rate is 1% per month. Annual Nominal rate = i(12)= ?
  14. Rate is 0.25% per month. Effective rate i = ?
  15. Rate is 0.25% per month. Annual Nominal rate = i(12)
  16. Nominal Annual rate is i(2) = 6%. Effective rate i = ?
  17. Nominal Annual rate is i(12)= 12%. Monthly rate = ?
  18. i = 0.09. v = ?
  19. i = 0.06. d = ?
  20. Effective Yearly Interest Rate =  6.1679%.
    i(12) = ?
  21. Effective Yearly Interest Rate = 2.27543%, Compounded daily.
    Nominal Annual Rate ?
  22. d = 0.06. i = ?
  23. $10000 today yields $100000 in 40 years.
    Interest Rate?
  24. $10 today yields $20 in 7 years.
    Interest Rate?

Solutions:

  1.  1420.46
    PV = \frac{FV}{(1+i)^{n}} = \frac {10,000}{1.05^{40}}
  2. 1532.83
    = \frac {2,000}{1.03^{9}}
    PV = \frac{FV}{(1+i)^{n}}
  3. 9514.66
    = \frac {10,000}{1.01^{5}}
    PV = \frac{FV}{(1+i)^{n}}
  4. 0.9524
    v = (1+i)^{-1}
  5. 0.9174
    v = (1+i)^{-1}
  6. 0.945
    v = 1-d
  7. 0.9975
    v = 1-d
  8. 0.04
    d = 1-v
    v = 1-d
    both sides are present value of 1 paid at end of period
  9. 0.15
    d = 1-v
    v = 1-d
    both sides are present value of 1 paid at end of period
  10. 50000 (1-0.05)30= 10731.94
    PV=FV vn
  11. 1000 (1-0.04)15= 542.09
    PV=FV vn
  12. 1.02^4 -1 =0.0824
    (1+ \frac {0.08}{4} ) ^{4} -1 = 0.082
    = (1+ \frac {i^{(m)}}{m} ) ^{m} -1
  13. 12%
  14. 1.0025^{12 }-1 = 0.0304
    (1+ \frac {0.03}{12} ) ^{12} -1 = 0.0304
    = (1+ \frac {i^{(m)}}{m} ) ^{m} -1
  15. 3%
  16. (1+ \frac {0.06}{2} ) ^{2} -1 = 0.0609
    = (1+ \frac {i^{(m)}}{m} ) ^{m} -1
  17. 1%
  18. 0.9174
    v = \frac{1}{1+i}
  19. 0.05660
    d = \frac{i}{(1+i)}
  20. 6%
    =12[(1.061679)^{\frac{1}{12}}-1]
    i^{(m)} = m[(1+i)^{\frac{1}{m}}-1]
  21. 2.25%
    = 365[(1.0227543)^{\frac{1}{365}}-1]
    i^{(m)} = m[(1+i)^{\frac{1}{m}}-1 ]
  22. 0.0638
    i = \frac{d}{(1-d)}
  23. 5.9%
    i = (\frac{FV}{PV})^{\frac{1}{n}}-1
  24. 10.4%
    i = (\frac{FV}{PV})^{\frac{1}{n}}-1

Oh, by the way.  If you have been reading these posts for a while, I am sure that you realize that all of these exercises, plus about 200 more, are part of my Anki deck for exam FM.

 

What worked, What didn’t

Studying the Easy Stuff

I didn’t spend enough time with the “easy” stuff. On the exam, the problems surrounding the algebra of events were real chestnuts. Likewise, the first time that I took the exam I encountered an “easy” normal curve problem with a clever twist of algebra. This time, there were two normal distribution problems with the same difficult twist. Study the easy stuff.

One and a Half Years of Study

Actually, two winters worth of studying. My first winter of studying suffered from a lack of study materials. The only book that I had was the free Finan book, plus whatever statistics books were on my shelf (Hogg and Tanis, Casella and Berger). I mostly worked problems from the Online Math Tests Home Page. This was not nearly enough of a variety of problems. Also, my first crack at studying required much reviewing of calculus topics.

The second winter worth of studying was more varied, and more productive. For me, one and a half years was just the right amount of study time. (During this same period, I also bought and renovated a dilapidated house, worked lots of 55 hour weeks at my job, had major surgery, and was reasonably available for my friends, family, pets, and spouse.) For exam FM, I am planning 1 year.

Actex Manual (Broverman)

The exercises in this book are difficult and varied. I will get the Actex manual for FM.

Guo Book

This book provided me with some variety of material when I needed it, and got me thinking about highly efficient solutions. I may get the book for FM.

Finan Book

This book was a good start for me. There are no worked solutions for the exercises, which can become frustrating. I am now started on the book for FM.

Coaching Actuaries

This was a great help at the last minute, in getting a feel for the real test. For the next exam, I am going to get a one month subscription, two months before my exam.

Having Some Background with the Material

Thinking that I had some sort of familiarity with probability was a handicap for the first exam. It encouraged me to not be as thorough with the “easy” stuff as I should have been.

Fortunately, for exam FM/2, I will not have the same problem. I have no background with the material, so I am starting everything from the beginning.

Memorization

My memory work deserves some real thought, and a post of its own, to follow later.

In short, I am glad that I put as much time as I did into memory work, although it is hard to place its role in passing the exam. The biggest payoff is in the work that I performed in turning complex topics into little digestible examples. As a result, the elementary distributions now have a permanent place in my consciousness.

Early Mornings

Early mornings work for me. I stuck with a wake up time of 3:30 am, or at latest 4:30 am, for more than 6 months. I had time each mornings to complete the most important study tasks, so any additional study time after work was bonus study time. I could study in the afternoons in the coffee shop, and not worry if I lost some study time in socializing