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.

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.

 

How to make 8 each apple and pear pies.

1.  Go to an orchard and buy 1/2 bushel each of pears and apples.  I paid $12.00 a half bushel. Better yet, find some wild trees and pick your own.  I was disappointed by what I bought at the orchard, compared to what I have found wild in years past.

applesEach half bushel will make 8 pies.  Here are the apple pie ingredients:

  • 4 packages of frozen deep dish pie crusts (2 crusts per package)
  • 1 bag sugar (5 pounds)
  • 1 1/2 cups flour
  • 2 sticks butter
  • 3 tablespoons cinnamon
  • 2 tablespoons vanilla

For the pear pies, ditch the cinnamon and vanilla and use 2 tablespoons or so of ginger.

2.  Peel, core, and slice the fruit. Get help from friends.  Use some lemon juice so the fruit doesn’t turn too brown.

13 quart roaster

3.  In your 13 quart aluminum roaster, mix all ingredients.  (did you just say 13 quart roaster?)

mixing all ingredients

4.  Arrange 2 pie crusts on a large aluminum cookie sheet.  At 2 cookie sheets per oven, you can do 4 pies per batch.

pies are in the oven

When I first tried this I somewhat over  optimistically thought that I could just put some aluminum foil on the oven racks, and squeeze 8 pies in the oven at a time.  Disaster.

5.  Cook in preheated 375 degree oven for 50 minutes or so.

finished pies

dishing out apple pie6.  Wrap and freeze pies.  Our party is in two weeks, and this is an easy thing to do early.  I love cooking big!

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.

Tons of Problems

68 more days until SOA/CAS Exam P/1.  Although I got up late today, I still managed to work tons of problems.  My goal until test time is to do at least 30 problems each day.

I have quickly realized that using spaced repetition for difficult problems is an entirely different process from using it for simple memorization items.  If I am having any trouble at all in remembering the flow of logic of the solution, I need to repeat the problem daily, or even twice daily, until I am really sure that I understand every little twist.  Then, I start rating the item so that it comes up on a spaced repetition schedule.  It is the old truism of SRS:  you can’t learn it until you understand it:

  1. Do not learn if you do not understandTrying to learn things you do not understand may seem like an utmost nonsense. Still, an amazing proportion of students commit the offence of learning without comprehension. Very often they have no other choice! The quality of many textbooks or lecture scripts is deplorable while examination deadlines are unmovable.

    If you are not a speaker of German, it is still possible to learn a history textbook in German. The book can be crammed word for word. However, the time needed for such “blind learning” is astronomical. Even more important: The value of such knowledge is negligible. If you cram a German book on history, you will still know nothing of history.

    The German history book example is an extreme. However, the materials you learn may often seem well structured and you may tend to blame yourself for lack of comprehension. Soon you may pollute your learning process with a great deal of useless material that treacherously makes you believe “it will be useful some day”.  

  2. Learn before you memorizeBefore you proceed with memorizing individual facts and rules, you need to build an overall picture of the learned knowledge. Only when individual pieces fit to build a single coherent structure, will you be able to dramatically reduce the learning time. This is closely related to the problem comprehension mentioned in Rule 1: Do not learn if you do not understand. A single separated piece of your picture is like a single German word in the textbook of history.

Taken from the SuperMemo website.

Using SRS for studying difficult problems violates several of the other principles of SRS, but it still a simple way to schedule the problems, and to monitor progress.

More tomorrow 🙂