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.


Eight Secret Tricks to Passing (Exam FM)

1.  Be over-prepared: The syllabus is so extensive, and there are so many twists that may be put on standard material, that there are always going to be questions that look foreign. The more that you are prepared, the more that you will be able to handle some of these. Being over prepared also helps to deal with emotions. There were points during the exam when a little voice in my head said “There is no way that you are going to pass this.” Yet, that same voice was in my head during many sample exams which I passed with flying colors. Experience and over-preparation reveals that you can succeed even when you are not at your best. If you walk into the exam center with worries, or a cold,or the flu, or pain in your back, you know that you can still succeed.

2.  Fill in the tough spots: It can hurt your head to learn new things. Identify the areas with which you struggle, and try to get your head around them. Two weeks before the exam, I still had some head-pain topics. Things like duration matching, convexity, interest rate swaps, and put-call parity. But, they were on my daily study schedule, and I forced myself to confront them each day. It paid off.

3.  Fill in the fundamentals: Once you gain proficiency in a topic, the fundamentals look different. In the month before the test, I went back and reviewed some of the basics of interest theory. At this point, the basics were easy, and I picked up on some of the finer points that I missed the first time around. Amortization schedules, for instance.  After I learned the basic formulas, it was easy to ignore the schedules they are based on. Yet many problems are made much easier by using schedules, rather than relying on formulas.

4.  Memorization is easy: Once you understand things thoroughly, you don’t really need to do much memory work for these exams. After months of working problems, most things will be anchored in your head. And yet suppose that you find yourself trapped in a dark alley by two annuities, one of which is twice as long as the other. Then you will be glad that you memorized \frac{a_{\overline{2n}\lvert }}{a_{\overline{n}\lvert }}=1-v^n I have never encountered this fact in practice, but if I did, recognizing this identity might save me a couple of minutes. Spend a little time on memory work, it is easy.

5.  Test symbolic solutions with numbers: If you have a problem with a purely symbolic solution, and you can narrow it down to a couple of possible solutions, you can frequently replace the variables with reasonable numbers, and see if the result is true. This is my favorite new solution technique I learned while studying for this exam.

6.  Read MacDonald: Actually read it. After you learn the math, go back and read it. If you look at the notes from the sample problems, you will notice that the exam writers have made it pretty clear that the exam will be based on the book. Those non-numeric multiple choice questions can be very challenging. Read the book.

7.  Number your Scrap Paper: I learned this trick when taking practice exams.  If you have time to review problems at the end of the exam, you will need to find the scratch-work easily.

8.  Practice with dull number two pencils: Two weeks before the exam, I put all of my mechanical pencils away, and worked only with old fangled sharpenable pencils. Two hours into the exam, you will be working with dull stubs, so you need to be prepared.

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.

Problems in the Queue

A week left until the financial mathematics exam. The best thing that I can say is that I can honestly not have worked any harder at studying. In the last three months, I have studied at least three hours a day, and frequently I have done eight to ten hours a day. Much of that time has been absolutely focused. The question is: have I studied as smartly as possible? I won’t know the answer to that until after the examination.

I have done about two hours of problems today. I see that there are 19 problems remaining in the queue. These are problems that I have seen before, and that I still wish to spend some time with. I spend enough time with each problem that I feel I am confident that I understand several ways to get to a solution, or until I get sick of looking at it. If I get sick of looking at it, it comes up on the queue again later today or tomorrow. If I am fluent with it, I don’t look at it for a few days. Eventually, it disappears from the queue.

Earlier today, I already did some easy numerical exercises. Later today, I will probably do an exam, which is a whole other kind of problem solving, because of time pressure (5 minutes per problem.)

What I am going to do here is type in problems as they come up on my screen, then write up solutions. If I don’t understand the problem, you will get to watch me struggle.

Person X enters into a long forward contract. If the spot price at expiration were S, the payoff would be -20. If the spot price at expiration were 1.2S, the payoff would be X.

Person Y enters into a short forward contract. If the spot price at expiration were 0.8S, the payoff would be 40. If the spot price at expiration were 1.1S, the payoff would be Y.

The forward price on each contract is the same.

What is X+Y?

We aren’t going to do a diagram for this one. I tried, on paper, and since we don’t know the values of 0.8S, 1.1S, or 1.2S, it is hard to know where to place them in relation to the forward price. So, let’s go with straight algebra, and see if that leads us to a reasonable solution.

Payoff on Long Forward = S – F

Payoff on Short Forward = F – S

From Givens:

S-F = -20 \\    1.2S=S=X \\    F-0.8S=40 \\    F-1.1S = Y

Add equations 1 and 3 to get S=100. Then solve for F and get F=120.

Using equations 2 and 4:

X+Y = 1.2(100)-120+120-1.1(100) = 10

That wasn’t too bad. These derivative problems can be a little intimidating at first.

A 15 year bond with semiannual coupons has a redemption value of $100. It is purchased at a discount to yield 10% compounded semiannually. If the amount for accumulation of discount in the 27th payment is $2.25, which of the following is closest to the total amount of discount in the original purchase price?

We have all sorts of information here, so it should be easy to get to an answer. First, since the coupons are semiannual, we can just think entirely in 6 month terms. Since the bond is sold at a discount, we know that the coupon rate is less than the interest rate. We know that we can solve this problem by finding the original purchase price, and subtracting it from the redemption price. First, we need the coupon, which we can find using the premium discount formula.

Let’s start with that. Assuming that F=C, the amount of discount in the kth payment is:


Here are our givens:

n=30 \\    F=C=100 \\    i=0.1 \\    F(i-r)v^{n-1+k} = 2.25 \\    100 (0.05-r)1.05^{-(31-27)} = 2.25 \\    r=0.02265 \\

Now solve for the original sale price. We might as well continue with the premium discount formula:

P = C+(Fr-Ci) a_{\overline{n}\lvert } \\    P=100-0.2735\frac{1-1.05^{-30}}{0.05} \\    P=57.96

Discount in original price = 42.04.

Let’s just do one more.

To accumulate 8000 at the end of 3n years, deposits are made at the end of the each of the first n years and 196 at the end of each of the next 2n years.  (1+i)^n = 2.

What is n?

These problems are much simplified by visualizing them in the right way. The easiest way to think of it is payments of 98 from 1 to 3n years, plus payments of 98 in years 2n+1 through 3n.

In math:

98s_{\overline{3n}\lvert }+98_{\overline{2n}\lvert }=8000 \\    \frac{(1+i)^{3n}}{i} + \frac{(1+i)^{2n}}{i}=81.633 \\    \frac{2^3-1}{i} + \frac{2^2-1}{i} = 81.633 \\    i= 12.25%

I am going to post one more, because it is a real bear. To solve it, you trust that the math will lead you to the answer.

Given a k year bond with semiannual coupons, and a yield rate of 10% convertible semi-annually, sold at a price p.

If the coupon rate had been r-0.04, the price would be P-200.

Calculate the present value of a 3k year annuity immediate paying 100 at the end of each 6 month period, at a rate of 10% semiannually.

Start at the end. We need to find:

100 \frac{1-1.05^{-6k}}{0.005}

Which means that what we really need is k (although v^k will do).

Dive in:

P=1000\frac r 2 \frac{1-1.05^{-2k}}{0.05}+1000(1.05)^{-2k} \\    P-200 = 1000\frac{r-0.04}{2}\times \frac{1-1.05^{-2k}}{0.05}+1000(1.05)^{-2k} \\    \text{We can see that the redemption values will not be significant}\\    1000 \frac r 2 a_{\overline{2k}\lvert }=1000 \frac{r-0.04}{2}a_{\overline{2k}\lvert }+200 \\    1000 \frac r 2 a_{\overline{2k}\lvert }-1000 \frac{r-0.04}{2}a_{\overline{2k}\lvert }=200 \\    500a_{\overline{2k}\lvert }(r-r+0.04)=200 \\    a_{\overline{2k}\lvert } = 10 \\    k = 7.1 \\    100 \frac{1-1.05^{-6\times7.1}}{0.05} =1749.75

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?

Only 13 days left

Only thirteen days until the exam. That really crept up on me. I have been taking a full practice exam each day, plus lots of other problems, plus trying to fill in weak areas.
I feel that I am more prepared than I was for the last exam. The practice exams that I am taking and more difficult than the actual test, and are quite humbling. This is good, but in the few days before the exam I am just going to study easy stuff, to get my confidence up.
I promise that I will post regularly until the exam.

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.

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.