Suppose I want to pay off a loan 4 months and 20 days after taking the loan. The bank isn’t going to just use the results of the fourth compounding interval calculation and give me the interest they earned in the last 20 days. On the other hand, I don’t think it’ s fair for the bank to use the fifth compounding interval calculation and charge me 10 days’ worth of interest that the bank hasn’t earned.
Let’s say that there are 30 days in the current month. Take the interest rate per
month ^ R j and divide it by 30, giving us an interest rate per day, then we multiply
this by the number of days, giving us an effective interest rate for the 20 days of the month:
The interest is just the above number times the balance after 4 months... Read More
An exponent is another neat notation. Suppose I want to multiply an expression by itself (called “squaring” the expression):
(j + 7)0 + 7) = (j + 7)2.
The little “2” placed high up in the upper right means “square the expression” or more directly, “write the expression down twice, making it clear that you mean multiplication.” This is also sometimes called “raising the expression to the power of 2. ”
Similarly, I can “cube” the expression
(j + 7)0 + 7)0 + 7) = (j + 7)3
and so on.
In general, (anything)" is called raising the expression “anything” to the nth power.
The following discussion of exponents is not needed in order to understand the book; I just thought that some readers might be curious as to why raising an expression to the power of 2 is called “squa... Read More
Very often, a bank or loan company will charge some sort of loan initiation fee at the outset of the loan (I call these up-front costs). Suppose, still using Table 2.2 as our example, that the bank wants $100 up front for setting up the loan. One way to handle this would be for the bank to just give you $9,900 while recording the loan as $10,000; then Table 2.2 would still be correct.
Usually, however, this is not what happens. When you take a $10,000 loan, you probably want to walk out of the bank with $10,000 for whatever your purpose is. In this case, the bank adds the extra $100 into the loan. That is, it pretends that it really loaned you $10,100. Columns 2 and 3 of Table 2.4 shows the new balance worksheet for this loan... Read More
This topic is not necessary for understanding the rest of the book. It’s an introduction, however, to a powerful notation that allows us to work with numbers from arbitrarily large lists in a very concise manner.
Looking at Table 1.3 again, suppose that I want to add up the values of e from e5 to e8. That is, add up e5 + e6 + e7 + e8. Just for the record, I don’t know why I’d ever want to do this with a Life Table. There are cases, however, such as adding up the interest payments on a loan, where I often want to do this.
The shorthand notation involves the use of the uppercase Greek letter sigma:
How to read this: Beneath the sigma you see “i = 5.” Above the sigma you see “8... Read More
When looking at relationships between variables, the formula tells it all. Very often, however, a picture is indeed worth a thousand words in “giving us a feeling” for what the formula is telling us.
We will often be presented with a graph that we’ll study to gain some insight into the information the graph is presenting. Conversely, we will often need to be able to create a graph to show a formula that we are interested in. I’ll take this latter approach first.
Let’s start with a simple formula:
у = 27,000 – 2,000x.
This formula gives us a value for the variable у when we give it a value for the variable x. These variables might stand for the depreciation of a car’s value, the interest on a loan, the number of years that you will hold a loan, and so on.
Before I draw a graph... Read More