Category Understanding the Mathematics of Personal Finance

ONLINE CALCULATORS

These calculators are so numerous that I’m sure you can find many more in addition to the ones listed below. These look fairly good and are easy to use without first having to register or “get a quote.” You’ll also see some unusual numbers such as “average monthly interest.” Please don’t take these too seriously—read about Present Value in Chapter 7 . Most of these sites will present a full amortization table if requested. This can be very convenient:

1. http://www. bankrate. com/brm/mortgage-calculator. asp;

2. http://www. homefair. com/tools/mortgage-payment-calculator/index. asp? cc=1;

3. http://realestate. yahoo. com/calculators/payment. html.

4.1 FIXED RATE MORTGAGES

Mortgage loan interest rates can be arranged in any of several ways...

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INTRODUCTION TO THE LIFE TABLES

Table 10.1 is the 2004 Life Table for all males in the United States. The first column to the left is the age of the man at his last birthday. In the rest of the table entry defi­nitions, this single number is just referred to by the letter x.

The second column is called q(x) . This notation is referred to as functional notation. It’s saying that the variable q “is a function of x.” Don’t worry about this. Column q is the probability of dying sometime between your xth and (x + 1)st birthday. For example, the probability of dying between your twentieth and twenty – first birthday, that is, when you’re 20 years old, is 0.001266. This definition, as stated, is correct—but should be elaborated on. This probability that you ’ tl die during your twentieth year is 0...

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DAILY BALANCE

The daily balances of your purchases and/or cash advances are the amounts on which finance charges are calculated. To understand the details of the finance charge calculation, you first need to understand the calculation of the daily balances for each of the days in the billing period (usually a month). Note that there are two separate balances: one for purchases and one for cash advances.

To calculate a day’s daily balances,

1. start with the previous day’s daily balances;

2. add transactions and fees charged that day (to the appropriate balance);

3. add finance charges accrued on previous day’s daily balance when appropriate;

4. subtract any credits and payments applied that day against the balances...

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LIFE ANNUITIES

The immediate in immediate annuity means that payments start immediately upon paying the premium. Often, the premium is accumulated by the purchaser over time. With a deferred annuity, the premium can be built up with a number of payments that grow with the tax on the interest deferred. This increases the growth of your payments as compared with putting the money into a savings bank. When payments begin, the exclusion ratio is lower than in the case of the immediate annuity, so that more of the payment gets taxed. The IRS is reclaiming some of the tax it deferred during the accumulation period.

Returning to the example above, I need a balance of $413,930 in my account on the day that I want the annuity to start making payments...

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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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