# 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.

#### From Givens:

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

#### 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:

$F(i-r)v^{n-k+1}$

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. $S_0 =800, r=0.02, \delta = 0\quad F_{0, 1yr}?$

Solutions:

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.

# 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.

I can’t wait to finally post my Exam P cheat sheet.  LaTeX is great 🙂

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.