New Study Materials for Exam FM

You may have noticed that I have been adding some study materials to my blog, as a side project. If you click on the “Easy FM Quizes” tab, you will find some interactive quiz problems. These are dumped over from my easy practice problems I made when I was studying for the exam. I have more than 1000 more exercises that I will transfer over in the coming weeks.
I am even more excited about the exercises that I am putting into nice PDF problem sets.  If you have heard me talk about math, you know that I learned algebra and calculus from early 20th century text books.  I have a sentimental attachment to the problem sets at the ends of chapters.  It has given me great satisfaction to write at simple LaTeX class that duplicates the feel of these textbook problems.  When I am done, they will be a single document.
The links for these problem sets will be posted on the right side bar.


Some Study Materials For Exam MFE

It looks like I will take MFE, Models For Financial Economics, in March, 2015. The best source for the information is the SOA page for the test. It has links for the current Syllabus, the sample problems, and video solutions from the Wisconsin School of Business. The great thing about this exam is that the only recommended study resource is the MacDonald book, which we all still have from exam FM.

G. Stolyarov II has a very nice free study guide. At first glance I like it, and I like his studying hints.

Beyond that, I will be creating study cards again, reviewing FM material, and probably using Coaching Actuaries. Once again I will be studying to instrumental jazz music.  My goal is to get to the point of taking timed exams a few weeks before the actual exam.  So, my countdown on the sidebar will be to February 20, but I will be taking the actual exam somewhere around the ides of March.

Wish me luck.

Onward to Exam MFE

The title indicates that I passed FM. Here is how it happened.

I went easy on studying for the two days before the exam. I did some light reviewing of formulas, shoveled snow, walked the dog, and made candy. Weather was a concern. We have been getting blasted by winter each week, along with most of the rest of you in the Northeast. My spouse had generously agreed to come with me to the testing site (40 minutes away, in Lancaster), but I know that winter driving is very stressful for her, and that it was a pretty big thing for her to agree to come with me at all. I also knew that I would really like to have her emotional support after the exam, pass or fail.

We are buried under snow. I am glad that I checked online, because sometime in the last couple of years the Fruiteville Pike Prometric location moved down the street to the next plaza. I might have had trouble finding it, under the twenty-foot tall snow piles.

So, the test itself. I am so glad that I was over prepared. I had plenty of negative thoughts during the test. But I have been doing these problems under all sorts of conditions for months. It is like having a bad day at work: you know that you can still get your job done well, even on a bad day.

When I was all done with the exam, I still had about 21 minutes to go. This is about how I have been timing my sample exams. In those few minutes, I found errors on three marked questions. These were questions that I had reached the end of, and then found that my answer was not on the list. So frustrating. A little time away from the problem, though, and the mistake is often obvious. In these problems, it is often some vital misreading at the front end or back end of the problem.

Anyways, I worked right up to the end. I filled the survey at the end out in a big hurry, because I couldn’t wait to see the result. I was so surprised to see a “pass.”

That’s it. Now that I am done, it is like this big thing that has been dominating my life since early fall is just gone.

When I got home, I pulled out the exam MFE syllabus.

Forward Contracts

The thing about math at a certain level is that there are no more easy exercises.  I have learned to make a habit of creating simple exercises.  These are for the forward price, which is the contracted price to buy an asset at time T in the future; and the prepaid forward price, which is the price paid now for an asset that will be delivered at time T.  In these problems, r is the continuous interest rate, and delta is the continuous dividend rate.
F^P_{0, T} = S_o = S_o -PV(divs) = S_0 e^{-\delta} T
F_{0, T} = S_0 e^{rT} = S_oE^{rT} - AV(divs) = S_0e^{(r-\delta)}T

  1. S_0 =1000, r=0.04, \delta = 0.01\quad F^P_{0, 6m}?
  2. S_0 =800, r=0.02, \delta = 0\quad F_{0, 2m}?
  3. S_0 =800, r=0.02, \delta = 0\quad F^P_{0, 2yr}?
  4. S_0 =500, r=0.04, \delta = 0\quad F_{0, 2yr}?
  5. S_0 =500, r=0.04, \delta = 0\quad F_{0, 6m}?
  6. S_0 =500, r=0.04, \delta = 0\quad F^P_{0, 1yr}?
  7. S_0 =100, r=0.03, \delta = 0.01\quad F^P_{0, 2yr}?
  8. S_0 =100, r=0.03, \delta = 0.01\quad F^P_{0, 3m}?
  9. S_0 =1000, r=0.04, \delta = 0.01\quad F_{0, 1yr}?
  10. S_0 =1000, r=0.04, \delta = 0.01\quad F_{0, 6m}?
  11. S_0 =800, r=0.02, \delta = 0\quad F^P_{0, 5m}?
  12. S_0 =100, r=0.03, \delta = 0.01\quad F_{0, 1yr}?
  13. S_0 =100, r=0.03, \delta = 0.01\quad F_{0, 9m}?
  14. S_0 =500, r=0.04, \delta = 0\quad F^P_{0, 9m}?
  15. S_0 =1000, r=0.04, \delta = 0.01\quad F^P_{0, 1yr}?
  16. S_0 =800, r=0.02, \delta = 0\quad F_{0, 1yr}?


  1. 1000e^{-0.01*0.5}=995.01
  2. 800e^{0.02*(\frac 1 6)}=802.67
  3. 800
  4. 500e^{.04*2}=541.64
  5. 500e^{0.04*.5}=510.10
  6. 500
  7. 100e^{-0.01 *2}=98.02
  8. 100e^{-0.01*0.25}=99.75
  9. 1000e^{0.04-0.01}=1030.45
  10. 1000e^{(0.04-0.01)*0.5} =1015.11
  11. 800
  12. 100e^{0.03-0.01}=102.02
  13. 100e^{(0.03-0.01)*0.75}=101.51
  14. 500
  15. 1000e^{-0.01}=990.05
  16. 800e^{0.02}=816.16

Later today, I will post some tougher ones that require a little thinking.

My Soundtrack

I have been listening to music while I study for this exam. Partly because I am studying in a room where there is some coming and going of people and pets, partly because the music keeps my body wiggling while I study, and partly because I am beginning to immerse myself in bebop jazz. I can’t study to most music with words, and I also have come to have a certain like for some current ambient and electronic music. So here’s my Financial Mathematics study music list:

Classic Bop Stuff:

  • John Coltrane – Giant Steps
  • Miles Davis – Kind of Blue
  • Charles Minus – Ah Um
  • Thelonious Monk
  • Dave Brubeck – Time Out



  • Christian McBride – People Music
  • The Crystal Method
  • The Teddy Bears
  • Disparition

This is the first time in my life that I have studied to music.  Strangely, it all started because I listen to podcasts, and I found that I can’t concentrate on math while I listen to podcasts, so I switched to music.  Now, I am hooked.

Today, I am going to listen to classic Nirvana, even though it has words.  What do you think of studying to music?

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


Flop-Ear Cat

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 Cards

Cutting them up:

cutter small

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?

finished cards smallThings 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.