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.

# 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.

# The Importance of Meta Learning

I tend to use a variety of meta-learning techniques.  The first one is simply to monitor my daily work and progress.  I have notebooks from 30 years ago filled with my daily juggling notes, notebooks from 20 years ago full of daily banjo progress, and notebooks going back 10 years full of mathematical success and failure.  Other notebooks on my shelf document medical progress.  If I look, I could find the note from 1988 when I first was strong enough to drive a stick shift, a year after my 1987 automobile accident.

Here is a photo of what some of my notebooks look like at this very moment:

If you are curious, here is a random page from a random notebook:

December 15, 2003

New Things I’ve done in the last 5 days

1. Flown in an airplane
2. Seen the Pacific Ocean
3. Been in California
4. Seen the Bush Man
6. Rode on a cable car
7. Been in San Francisco
8. Been in Hyde Park
9. Been in the Redwood forests
11. Ate Sushi
13. Been in a telecommunications switching facility
14. Saw Alcatraz
15. Crossed the Golden Gate Bridge
16. Flown over Salt Lake City

When working on a computer, there are other simple ways to document ones progress.  Here is a picture of the current graph of my progress on the 153 SOA sample problems:

The graph goes back about 70 days, and simply shows how many of the problems I solved each day (about 15, on average).    Those problems are only part of my daily studying.

By the way, 31 days left!!

# Sorry, More Technical Stuff

Currently, I am coming to grips with the law of total variance.

In words, the variance of X equals the variance of the expected value of X, given Y, plus the expected value of the variance of X, given Y.

In symbols, Var(X) = Var(E(X|Y)) + E(Var(X|Y).

By reading the verbal definition, one can see that the logic is convoluted. Given two probability distributions, it can be tricky to see how to apply the law. Once the law is applied, there are additional tricky steps of logic involving the independence of variables.

From the SOA/CAS sample problems:

A motorist makes 3 driving errors, each independently resulting in an accident with probability 0.25.  Each accident results in a loss that is exponentially distributed with mean 0.80.  Losses are mutually independent and independent of the number of accidents.  The motorists insurer reimburses 70% of each loss.  Find the variance of the total unreimbursed loss.

The above item illustrates the technical problem that I have been having when using Spaced Repetition Software to schedule repetitions of complex material.

When I encounter a difficult item, I spend time exploring the given solution, and alternate solutions. Eventually, I move on to another item. Most likely, I am not entirely comfortable with the material, and I would like to see the same material every day for a while, to approach it with different solutions. The way that the defaults on SRS software are set up, as soon as you start rating an item anything other than the most difficult setting, the item starts getting pushed off way into the future. After a week or two weeks, I have entirely forgotten many of the finer details of the item, and it is almost as if I am starting from scratch.  For difficult material, it is beneficial to see the same items every day, or every 2nd or 3rd day.

The solution to this problem is to reset the forgetting index. In Anki, I have now set my forgetting index to 3%. This task is done by downloading the shared “Forgetting Index” plugin. In the “File” menu, select “Download Shared Plugin”, and find the forgetting index plugin.

The result is that I can now look at problems on a nearly daily basis before they start whizzing into the future. Each time I look at the problem with fresh eyes, I observe new things.

In the last week, I finally have my Spaced Repetition Software working to its utmost. I use it to schedule repetitions of difficult exam problems, and of memorization items. These are two fundamentally different tasks, so it is important to set up Anki (the software that I am now using), in two different ways.  After spending time every day for a week with the above “unreimbursed loss” problem, I look at it, and think “piece of cake.”  Now, when I encounter other law of total variance situations, I have several good comparison problems stashed away in my head.