Category Understanding the Mathematics of Personal Finance


Spreadsheets and online calculators handle this calculation very nicely. This section, deriving the formula, can be skipped if you wish.

If R is the annual interest rate and y is the number of payments per year, which we’re assuming is the same as the number of compounding intervals per year, then

Подпись: R У is the interest per payment period. As a notational convenience, let


I — 1 + .


The balance at the time of taking the loan, that is, after 0 payment periods, is just the principal. If we let Bn be the balance after the nth payment, then if P is the principal:

Bo — P.

The balance after the first payment period is just the principal plus the interest accrued during this period minus the payment (S):

Bi — P + PR- S — P|1 + R |-S — Pi – S.

У V y)

This relationship is recursive...

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When interest rates go down, it’s time to investigate opportunities for refinancing your home. This means that you look for a new mortgage loan that will replace your old mortgage loan at a lower interest rate. In many cases, the up-front costs of the new loan are very low (sometimes even 0). As always, folding the up-front costs into the new loan gives you an effective interest rate and a resulting monthly payment number.

There are some “gotchas” in refinancing a home that you have to keep in mind. Let’s say that you are 15 years into a 30-year mortgage and the opportunity comes along to refinance at a lower rate...

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In the chapter on Life Insurance, I presented the idea of the probability of a random event (when you are going to die) and the calculation of an expected value (the average of when many thousands of people just like you are going to die). I’d like to extend these ideas to games of chance.

The expected value of my return on a game of chance is the sum of all the pos­sible things that could happen multiplied by the probability of each of them happen­ing. That sounds worse than it is. Look at the simple example of a coin flip game.

One of us flips a coin. If the coin lands heads up (heads), you give me a dollar; if it lands tails up (tails), I give you a dollar. In terms of the money in my pocket, giving me a dollar is +$1 and giving you a dollar is a dollar leaving my pocket, or -$1...

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You won’t see the term up-front costs in your mortgage contract. I’m using the term as a catchall for all the costs and fees involved in starting up a mortgage loan. This includes points. Points are a start-up fee that many lenders charge for giving you the loan. One point represents 1% of the loan amount. Then there are appraisal fees, paperwork fees, various state and county taxes, and so on. While you as the borrower certainly want to scrutinize every one of these items and make sure you’re getting the best deal available (i. e., the lowest amount of up-front money necessary to get yourself the loan), for my purposes, I’m just going to sum them all into up-front costs.

Many lenders will offer you several deals, for example, “6% plus 3.5 points, or 7% plus 2 points...

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If you want to buy a 20-year policy using the data shown in Table 10.3, the premium is $9,340. What if you don’t have this money available? The reasonable answer is to take a loan. You’ll be insured for 20 years; why not “pay as you go?”

A 20-year loan for $9,340, at 6% compounded monthly, requires payments of about $67 each month. This doesn’t seem like a lot but keep in mind that this is just an example.

If you take the loan, it looks like everything is taken care of. The insurance company has its premium, you have your 20-year insurance policy, and you have monthly payments on the loan you took that you can handle...

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If a credit card company decides that your risk as a debtor has changed, the credit card company might increase your interest rates immediately. This decision can be triggered by many factors in your financial life and need not be due to any history of late or insufficient payments to this credit card company.

One situation, so egregious that lawmakers have addressed it, is that some credit card agreements allow the credit card companies to change interest rates not only on new purchases and cash advances but also on existing balances. This is analogous to, for example, the bank that gave you an auto loan calling you sometime and telling you to “throw away your payment schedule, we’ve raised your interest rate and we’ll be sending you a new payment schedule with higher payments.”


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