# 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

# Nirvana

My Study Helper

I’m so excited about studying right now. I still have some derivative markets stuff to take on, but the essentials (calls and puts, purchased and written) are becoming fairly intuitive. A couple of weeks ago, I was feeling more negatively. The basic options material looked like a giant heap of meaningless graphs and formulas. Now, I have formed some associations for all of it.

Normally, I study with a certain amount of distraction. There may be a cat sitting on my lap, or chasing the cursor on the screen. There may be a dog that wants to play, or some issue to discuss with my spouse. I may be distracted by tomorrow’s weather forecast, or by the latest Boing Boing post. These distractions are acceptable: I know how to deal with them, and I am motivated enough to regain my focus. At a certain point, however, there is nothing like a single-minded absolute focus on the work in front of me. And to achieve this, there is nothing comparable to taking a test.

I assume that, among actuary students, I am typical in my love of taking exams (I suspect that I am also typical in having been truly humbled by the actuary examinations  🙂 ). Taking tests, and often out-thinking tests, is what got me through grade school, when I did absolutely no school work or studying. My love of standardized tests, and resulting scores, is what eventually led me to be placed in classrooms appropriate to my ability. So, if you want to get my absolute attention, put an exam in front of me.

With this knowledge in mind, I have registered with Coaching Actuaries again. I took my first exam a couple of evenings ago, and I was immediately transported into test nirvana. For three hours, my mind didn’t blink. The fact that the scores are measured, and that there is a leader board to work myself onto, is extremely motivational to me. I can’t wait to take another test this afternoon.

# The Tribulations of Calculations

81 days until exam FM.  I am spending most of my time plowing head-first into derivatives  markets, but I am also working random interest theory problems each day, as well as finishing up a thorough study of duration, volatility, and convexity.  Most of these problems present a calculation challenge.  For instance, as soon as a solution requires calculating an increasing annuity, trouble is near:

$\displaystyle (Ia)_{\overline{n}\lvert }=\frac{\ddot a_{\overline{n}\lvert }-nv^n}{i}=\frac{\frac{1-v^n}{i}(1+i)-nv^n}{i}$

This looks even prettier once you put some numbers into it (suppose that n=20 and i=0.09):

$\displaystyle (Ia)_{\overline{20}\lvert }=\frac{\frac{1-(1.09)^{-20}}{0.09}(1.09)-20(1.09)^{-20}}{0.09}$

For the duration of a bond, this calculation is just a small piece of the whole numerical birds-nest.  You might know just what needs to be calculated for a solution, but it is trouble to coax the right number out of the other end of a calculator.  It seems silly to get wrong answers for this reason, so I have been putting a little thought into the best ways to arrive at numerically correct answers.  My initial guess was that the best technique is to store a bunch of intermediate values in the calculator.

Let’s look the duration of a bond.  This quantity is

$\displaystyle \frac{\sum tv^tCF_t}{\sum v^tCF_t}=\frac{Fr(Ia)_{\overline{n}\lvert }+Cnv^n}{Fra_{\overline{n}\lvert }+Cv^n}$

The way I figure it, you are better off using the definition on the left for a bond with only a few coupon periods. Lets take a 4 year par value 1000 bond with 8% coupons and a 7% yield rate.  The left hand formula produces:

 t vt CF 1 1.07-1 80 2 1.07-2 80 3 1.07-3 80 4 1.07-4 1080

Which yields: (technique #1)

$\displaystyle \frac{1(1.07)^{-1}(80)+2(1.07)^{-2}(80)+3(1.07)^{-3}(80)+2(1.07)^{-4}(1080)}{(1.07)^{-1}(80)+(1.07)^{-2}(80)+(1.07)^{-3}(80)+(1.07)^{-4}(1080)}$

That looks a little hairy.  But our alternative is this:  (technique #2)

$\displaystyle \frac {80 (\frac{\frac{1-1.07^{-4}}{0.07}(1.07)-4(1.07)^{-4}}{0.07})+1000(4)(1.07)^{-4}} {80 (\frac{1-1.07^{-4}}{0.07})+1000(1.07)^{-4}}$

That is much worse.

The moral of the story is: just because your TI-30XS multiview calculator allows you to create nice looking nesting fractions on the screen, doesn’t mean that you should always actually make those fractions.  To find durations, or convexities, using technique #1, we simply make a chart on paper, then multiply across the rows.  Using technique #2, we create a monstrous mess that is nearly impossible to trouble-shoot.  If we absolutely need to compute using technique #2, we need to store some intermediate values in calculator memories.  I usually store 1+i, a angle n, and v^n.  It is still treacherous.  I think I will post some practice exercises tomorrow.

# Eighty Three Days Until Exam FM

I scheduled my exam yesterday, for exactly eighty-three days from now.  I am well over the hump with studying.  I can work exam level problems on all of the interest theory material, but I have a long way to go with derivatives markets.  So, my goal until the end of December is to keep working interest theory exam problems, filling in missing bits when I find them, while I pour over the derivatives markets material.  By January I will be taking practice tests every other day, while I review solutions on the off days.

Most recently, I am working with duration and convexity.  This is good, meaty stuff, with many different levels of understanding.  I will be doing a post on it tomorrow.  I will also be posting on how to best arrive at a correct answer on these computationally complex problems, as well as posting some computation exercises.

# 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:

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?

Things 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.