The IntMath Newsletter - 12 Oct 2008

In this Newsletter:

1. Math tip - Money math, probability and the current crisis
2. Encryption - Solution from Last Time
3. IntMath Poll results - Math Anxiety
4. From the math blog
5. Final thought - the thrill of the chase

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1. Math tip - Money math, probability and the current crisis

We are witnessing the worst financial crisis since the Great Depression. How did such a mess happen? Poor misunderstanding of math — from consumers to banks to governments — is behind most of the problems.

Let's look at one aspect of the crisis - bad understanding of probability.

If you are an insurance company, people pay you for assuming risk. So if you sell house insurance, you look at the probability that a house will burn down (or be destroyed in a cyclone or earthquake) and you charge the customer based on that probability. If the house does not burn down (high probability) then you make a profit. If it does burn down, you make a loss, but it is OK because lots of other houses that you have insured will not burn down. Overall, you make a profit.

Likewise, if you sell life insurance, you examine the likelihood that someone will die during the next year. It turns out that this probability is very low. For example, the probability for a 10 year-old Australian boy to die in the next 12 months is only 0.000107.

An 18 year-old boy is 6 times more likely to die (0.000685) than a 10 year-old. (Source) Of course, smokers, unfit people, fighter pilots and F1 drivers are much more likely to die than people who choose safer life styles.

The probabilities for life insurance are generally independent. If someone has a car accident, it doesn't mean everyone else in the country will also have a car accident at the same time.

The life insurance company needs to charge an amount that covers the likelihood of paying out on the death of the person, plus an amount for profit.

Back to the economic crisis.

Let's look at mortgage bonds, one of the many financial products that have turned sour over the last 12 months. Bonds are offered by governments and companies when they need to raise money (for roads, bridges, schools, expansion, whatever). An investor would buy the bond and then the government (or company) would give them back their money plus interest, say 5 years later.

Mortgage bonds are similar but are connected to the housing industry. They used to work like this (before the whole thing collapsed).

Say I wanted to buy a $200,000 house but I didn't have very much money and I only worked part time. I went to Dodgy Mortgage Company down the road, and the nice man there assured me that there was money available and I could buy the house. The monthly payment looked good (about the same as I was paying for rent) and he mumbled something about that payment going up after 2 years.

Now, Dodgy Mortgage Company had arranged 100 similar loans that month and needed to obtain money to lend to me and the other customers. They would go to Big Investment Bank Company and would offer the big institution a mortgage bond for the all the money that they needed (100 customers x $200,000 = $20,000,000).

Everyone is happy. I'm happy because I can buy a house. Dodgy Mortgage Company is happy because they can make money out of me (a percentage of my monthly payments). Big Investment Bank Company is happy because they will get (most of) my monthly interest payments.

Mortgage bonds are secured by real estate (house, land, office or equipment). This means that if the mortgage holder defaults (stops paying), then the investors (in our case, Big Investment Bank Company) can sell off the real estate and get their money back.

This is fine if the value of that real estate goes up all the time, but as most of you know by now, house prices in the US have been dropping for almost a year and there have been many mortgage defaults. This sets up a cycle where investors want their money back and try to sell the houses. There are many more sellers than buyers and the houses are sold at whatever price can be achieved.

Now the risk on these mortgage bonds (and hence the price investors pay for them and the cost of insuring them) is determined by probability, pretty much like the house insurance and life insurance that we talked about before.

Trouble is, the risk assessors (and the companies selling the mortgages and mortgage bonds) got the probability wrong. They assumed that the probabilities involved were independent (like our life insurance case above), however, as it has become painfully clear, this was not the case.

These low quality mortgages had a "honeymoon period" where the interest rate was low to start with, but would ratchet up 1 or 2 years into the loan. Many people got themselves into difficulty because their budget did not stretch to pay the higher interest amount. Such probabilities are very much dependent since it is quite possible for hundreds of defaults to occur together because one event is causing the problem (the increase in interest payments).

So with a great deal of "toxic debt" out there, the banks and other lenders got caught out. There was no income from house buyers and the lenders could not get reasonable prices when they tried to sell the houses. The mortgage bonds became worthless.

What has been happening in the last few weeks is a spillover to the larger worldwide economy. Confidence in the system is at an all-time low and people are getting out of whatever investment market they are in, causing prices of just about everything to crash.

Who caused the mess? It's a combination of inadequate government oversight (they left everything 'to the market'), greedy investors (who assumed house prices would always go up), and clueless customers (who didn't read the fine print about what was going to happen to them when the interest rate went up and if they lost their job).

It is vital that students should learn financial mathematics. Look at the mess we get when people don't know this stuff.

See also:

• Financial literacy for students
• Money Math
• Independent and Dependent Events in Probability

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2. Encryption - Solution from Last Time

In the last Newsletter, I challenged you to crack the following code. Did you manage to solve it? (It was actually quite straightforward.)

18 C 9 16 9 17 5 B 9 12 D 19 17 D 12 5 10 10 13 A 19 17

There was some helpful information given (it uses hexadecimal numbers and a shift cipher) and a few clues (e.g. the most commonly used letters of the English alphabet in order are E, T, A, O, I, N).

One way to solve it: As we learned last time, using hexadecimal numbers (without shifting anything) we would code the alphabet as follows A=1, B=2, C=3, etc. Here's the 26 letters of the alphabet coded using hexadecimal numbers:

A

B

C

D

E

F

G

H

I

J

K

L

M

1

2

3

4

5

6

7

8

9

A

B

C

D

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

E

F

10

11

12

13

14

15

16

17

18

19

1A

Next, looking at the frequency of the letters used in the coded message, we get:

Used 3 times: 9, 17
Used 2 times: 5, 10, 12, 19, D
Used 1 time: 13, 16, 18, A, B, C

There is a good probability that 9 and 17 in my code represent one of the letters E, T, A in the original message (since 9 and 17 are the most common in the code and E, T, A are the most common letters in English).

Let's try 9 = E and see what we get from it. If 9 = E, and if it is correct, then we have used a 4-letter shift cipher (we would have shifted all the coded letters to the left by 4). So the coding system would be as follows:

A

B

C

D

E

F

G

H

I

J

K

L

M

5

6

7

8

9

A

B

C

D

E

F

10

11

N	O	P	Q	R	S	T	U	V	W	X	Y	Z
12	13	14	15	16	17	18	19	1A	1	2	3	4

Note that V=1A and W=1. What's going on?

"1A" in hexadecimal means "one sixteen plus ten = 26".

The next hexadecimal number is "1B" which means "one sixteen plus eleven = 27". However, there are only 26 letters in the alphabet so we use modulo arithmetic, start again at the beginning and write "1".

Now, given the above de-coding system, we can see if the other letters actually give us a message.

18	= T
C	= H
9	= E
16	= R
9	= E

This is looking promising! We have a word already and it looks like we cracked it on the first attempt! I'll leave you to decrypt the rest of the coded message.

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3. IntMath Poll results - Math Anxiety

The most recent IntMath poll asked readers, "How is your Math Anxiety?". It seems like many of you are suffering from quite high math anxiety. Here are the votes of over 2,000 readers:

52%  High math anxiety
28%  Moderate math anxiety
20%  No math anxiety

I'll be writing some articles about math anxiety in the near future.

The current IntMath poll asks readers whether they like (or hate) being asked to come to the board and answer a question in a math class. Many people report this event as one which causes a lot of math anxiety. How about you?

You can respond to the poll on any page in Interactive Mathematics.

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From the math blog

1) What part of ... didn't you understand?
A mathematical take on that rhetorical question favored by those in authority.

2) Financial literacy for students
Why is the world descending into a financial meltdown? Lack of financial math knowledge on the part of consumers and financial institutions has a lot to do with it. Here's a resource that aims to improve the situation.

3) Friday Math Movie - Say that Funky Number, Math Guy
Here's a fun mathematical parody of the 70's song Play That Funky Music.

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5. Final thought - the thrill of the chase

Here's an interesting observation about chemicals in the brain and achieving goals.

It turns out that it is actually the pursuit of a goal that releases dopamine into the brain -- a kind of "feel-good" chemical that brings pleasure. When the goal is achieved, the dopamine release stops. Therefore, pleasure stops. So it turns out that there seems to be less satisfaction and personal pleasure in achieving a goal than there is in relentlessly pursuing the goal. [Source]

The lesson here is that we should continue to challenge ourselves and to not be satisfied with second best.

Till next time.

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