In order to tie comments to specific locations in the book, I have used the available page ruler sheet to identify lines in the text. A copy of that page ruler sheet can be printed from The Index Sheet.
Chapter 5: The Mathematics of FinanceAlso see the Financial Equations page. | ||
| 5.1 Simple Interest | ||
| Page | Line # | Notes |
| 348 | 19 | The text starts by identifying the initial amount as the principal or present value. In most earlier math classes we have talked about the terms principal , rate, time, and interest. Because we are going to look at much more complex situations, we introduce the term present value to mean the "the amount of money at the start of the process." Note that present value does not mean the value at this moment. Rather, it is the value at the start of some period of time. It is only when we are referencing "now" as the starting point that present value means the value now. If we say, in 1629 the present value of the tools and trinkets was $24, we are saying that the value of the tools and trinkets was $24 in 1629. We are not making any assertion about the value of those tools and trinkets today. |
| 348 | 20 | The important point in simple interest is that the interest is "sent" to you. The book sneaks in the phrase "in the form of a check" in the second sentence of the paragraph. This payment directly to you is important. |
| 348 | 24 | The text gives the formula for "annual" interest as INT = PVr. A bit later, at the bottom of the page, the book has a "Note on Multiletter Variables." The use of such multiletter variables, (e.g., INT for annual interest and PV for present value) is quite common in financial applications. Although such multiletter variables make the equations easier to convert to an English sentence, they do put an extra burden on us to separate variables. Thus, PVr is meant to be "Present Value" times the "annual interest rate expressed as a decimal." |
| 348 | 30 | The revised formula for total simple interest over a period of years is given as
INT = PVrt where we understand that INT is now the "total interest",
PV is the value of the money at the start of the process, r
is the annual interest rate expressed as a decimal, t is the time in years, and
that there is an implied multiplication of the three values on the right side of the equation.
One additional feature of this equation, a feature that is not discussed in the book until it is snuck in at line 27 on page 350, is that the time, the value of t, does not have to be whole years. The equation works for any period of time as long as it is expressed in years or fractions of years. This, in turn, begs the question of how long is a year? Strangely enough, in the "old days", financial computatiosn assumed that a year was composed of 12 months, each 30 days long. Thus, a financial year only had 360 days in it. Given the introduction of calculators and computers, a financial year is now generally taken as having 365 days even though we know that 365.25 is a more accurate figure (one that takes into account most of the leap years). Furthermore, we consider that a year has 12 months so that a two month period is taken as 1/6 of a year. We do this even though the two months of January and February (in a non-leap year) have 59 days or 59/365ths of a year, while the months of July and August have 62 days or 62/365ths of a year. We even go so far as to assume that a year has 52 weeks, when we know that 52*7 is 364 and every year has at least one more day than that. |
| 349 | 4 | This is the introduction of the term future value to mean the value at the end
of the time period that we are considering.
This is some time after the date of the present value. Thus, if we say, in 1629
the present value of the tools and trinkets was $24, we might ask about the future value
of those tools and trinkets in 1776. In that case, the future value
is tied to a year that is centuries ago, not in our future.
The subsequent formula for future value |
| 350 | 27 | As noted above, here we have an introduction, within an example problem, of the fact that time, our value of t in the equations, can be a part of the year. |
| 351 | 27 | This formula is derived from the one given at the top of page 349
for future value. That formula was |
| 351 | 40 | It is important to note that in Example 4 the text talks about the Treasury bill earning 3.67% interest. We will compare that to the discount rate presented at line 28 of page 352. |
| 352 | 28 | The discount rate is the annualized percent of the maturity value that is the difference between the selling price and the maturity value. Thus, if we have the discount rate and the maturity value then we can find the selling price by taking the rate time the number of years times the maturity value and subtract that from the maturity value. We might think of the selling price as the present value of the bond or note, and it is. but the discount rate is not the rate that one could apply to the present value in order to get the future value, i.e., the maturity value. |
| 354 | 18 | Technology: note that there is no discussion of using the calculator to do these simple interest and bond (note) problems. For staight simple interest the formuli are pretty simple and most of the time we can just type in the problem, assuming that we can remember the formuli. In the interest of providing a technology solution for these problems, we have a program for the calculators, SIMPFIN, and a web page that illustrates the use of that program. |
| 5.2 Compound Interest | ||
| Page | Line # | Notes |
| 356 | 36 | The presentation of compund interest in the text is pretty straight forward,
especially if we have mastered the terms PRESENT VALUE and FUTURE VALUE
that were fist given in Section 5.1.
The associated Technology Guide (pages 388-391) gives a quick run through of
using the TVM (Time Value of Money) feature on the TI calculators to do most of these
computations.
I have put together a compound interest/annuities web page that goes through all of the different types of problems and shows different solutions, either using a program I provide, FINANCL, or by using the TVM feature on the calculator. |
| 5.3. Annuities, Loans, and Bonds | ||
| Page | Line # | Notes |
| 367 | 36 | |
| 348 | 24 | |
©Roger M. Palay
Saline, MI 48176
January, 2012