Summary of Chapter 5 in

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Simple Interest
The simple interest INT on an investment (or loan) of PV (present value) dollars at an annual interest rate of r for a period of t years is
The future FV (or maturity value) of a simple interest investment of P dollars at an annual interest rate of r for a period of t years is
We can also solve for the present value PV to obtain

Examples
1. For a 8.5% simple interest 4year $20,000 loan, the total interest is
2. A US Treasury bill paying $5,000 after 2 years yields 3.5% simple annual interest. Its present value is


Compound Interest
The future value of an investment of PV dollars earning interest at an annual rate of r compounded m times per year for a period of t years is
Again, we can solve for the present value PV to obtain

Example
For 4year investment of $20,000 earning 8.5% per year, with interest reinvested each month, the future value is
= 20,000(1 + 0.085/12)^{(12)(4)} = $28,065.30, Notice that the interest earned is $28,065.30  $20,000 = $8,065.30  considerably more than the corresponding simple interest (see above). Compound Interest Utility Here is a little Javascript utility that computes any one of the five quantitites FV, PV, r, m, t given the other four. To use it, fill in any four of the five fields, and press "Compute" to obtain the missing quantity. You can enter r either as a percentage (example: "5.3%") or as a decimal (example: "0.053").


Effective Interest Rate
If money is invested at an annual rate r, compounded m times per year, the effective interest rate is
This is the interest rate that would give the same yield if compounded only once per year. In this context r is also called the nominal rate, and is denoted as r_{nom} in the textbook.  Example
A CD paying 9.8% compounded monthly has a nominal rate of r_{nom} = 0.098, and an effective rate of


Future Value of a Sinking Fund
A sinking fund is an investment that is earning interest, and into which regular payments of a fixed amount are made. If you make a payment of PMT at the end of each compounding period into an investment with a present value of PV, paying interest at an annual rate of r compounded m times per year, then the future value after t years will be
where i = r/m is the interest paid each period and n = mt is the total number of periods. 
Example
You deposit $100 per month into an account that now contains $5,000 and earns 5% interest per year compounded monthly. After 10 years, the amount of money in the account is
Annuity Utility To use the little utility below, fill in any five of the six fields (see the note below), and press "Compute" to obtain the missing quantity. Note: We use the following convention used in standard financial calculators, the TI83, and Excel: Thus, to redo the above example, you can enter:
Press here to bring the above utility up in its own window for your convenience. 

Present Value of an Annuity
An annuity is an investment that is earning interest, and from which regular withdrawals of a fixed amount are made. If you invest PV dollars at an annual rate of r compounded m times per year, you receive a payment of PMT at the end of each compounding period, and the investment has value FV after t years, then
If you make your withdrawals at the end of each compounding period as above,, you have an ordinary annuity. If, instead, you make withdrawals at the beginning of each compounding period, you have an annuity due. In this book we concentrate on ordinary annuities. 
Examples
1. You wish to establish a trust fund from which you can withdraw $2,000 every six months for 15 years, whereafter you want to be left with $10,000 in the fund. The trust will be invested at 7% per year compounded every six months. How large should the trust be?
Solution
Thus the trust should start with $40,346.87. To use the utility to obtain this answer, enter all the quantities except PV as positive (since they are all amounts paid to you) and press "Compute" to obtain PV (it will be negative). 2. You are buying a house, and have taken out a 30 year, $90,000 mortgage at 8% per year. What will your monthly payments be? Solution To use the utility to obtain this answer, enter all the quantities as positive (they are amounts coming in to you) and press "Compute" to calculate PMT (it will be negative). 