Category Understanding the Mathematics of Personal Finance

IN THE LIMIT-CONTINUOUS COMPOUNDING

This section just shows a point that is probably interesting for those comfortable with the math. It’s not a necessary section. However, I recommend looking at the graph and reading through the description of the axes.

The balance of a loan that’s compounded once a year at, say, 10% annual interest grows by 10% a year. As I have shown above, if the same loan is compounded monthly (12 times a year), the balance grows by 10.47% a year. What if the loan is compounded more often—weekly, daily, or even hourly? Does the effective interest rate just keep growing?

Подпись: 0.6 10. 0

1 10 100 1,000 10,000

Number of compounding intervals per year Figure 2.1 Effective interest rate versus the number of compounding intervals per year.

The answer to this question is shown in Figure 2.1...

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PRESENT VALUE OF PREPAYMENT PENALTIES

Analogous to the present value is the future value. Present value is the value today of some amounts of money that are known on different dates. Future value is the value at some future date, which must be specified, of some amounts of money that
are known on different dates. The point here is that in various situations, you will be interested in the value of some amounts of money on some date and you must calculate, using known or at least estimated interest rates, what this value is. The only real difference is that present value means the value today (a unique date), whereas future value means the value at some date in the future that must be speci­fied; “in the future” is not a unique date...

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Gambling

You may have been told that any place you put your money that’s not absolutely safe with insured protection is, to some extent, gambling. Accordingly, any invest­ing, whatever the risk level, is gambling.

I’m going to be more restrictive. I’ll only consider gambling to be betting your money on games of chance. Mixed systems, such as poker, combine skill and chance. You hope that your skill level will be higher than that of any of the other players so that in the long run, you will prevail.

Let me state my conclusion even at the start of the discussion. The only people who make money on games of chance are the casino owners. For games of chance, there are no systems; there are no lucky numbers; and there are no incantations— nothing can help you other than a little luck...

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ADJUSTABLE RATE MORTGAGES (ARMs)

Not all mortgage loans commit to a certain rate of interest for the life of the loan. The ones that don’t commit to this are called, reasonably enough, ARMs.

An ARM will typically offer an attractive low rate for a certain period of time (e. g., 5 or 10 years) after which time the rate can change. Just how much it can change, how often it can change, and what factors will be used to calculate a change will be specified in the contract. Also, there are laws that limit just how aggressively the lender can change the interest rate. Typically, the new mortgage rates will be tied to various federal government interest rates.

An ARM can be a very attractive way to get into a new home with modest monthly payments...

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EXPECTED VALUES

Assume that on January 1, 1900, four men whose lives we’d like to follow were born. They of course didn’t know it at the time, but their lives would end on:

First man: In 1922, when he was 22 years old.

Second man: In 1968, when he was 68 years old.

Third man: In 1974, when he was 74 years old.

Fourth man: In 1988, when he was 88 years old.

The average life span, or equivalently, the expected life span, of these four men is the average of the number of years that each man lived:

22 + 68 + 74 + 88 ^

———————- = 63.0.

4

Suppose we started studying these men in 1950. In 1950, there are only three of these men left alive. If we kept track of them until they were all dead, we would find that their expected life span is

Подпись: 76.7.68 + 74 + 88
3

Many people interpret results such as these as telling them...

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SOME CALCULATION EXAMPLES

For the following examples, I’m going to start each month (which will have 30 days and will also be the billing period) with the same initial situation: [17]

EXAMPLE 6.1 Table 6.2

I pay my bill in full ($821.08) and vow to never use my credit card again. As you can see from the table, my balances just come forward day after day until day 12. Since I have sent in the full payment, everything gets zeroed out and my daily balances are 0 from the twelfth day until the end of the month.

At the end of the month, the bank again calculates the ADB. This is a simple calculation—in each category, add up all the daily balances and then divide this number by the number of days in the month. Shown in the table are the purchase ADB of $123.77 and the cash advance ADB of $177.29.

These ADBs in turn ...

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