## Summary of Chapter 5 inFinite Mathematics/Finite Mathematics & Applied CalculusTopic: Mathematics of Finance

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Tool: Time Value of Money Calculator

Simple Interest | Compound Interest | Effective Interest Rate | Future Value of a SInking Fund | Present Value of an Annuity

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

 INT = PVrt INT = Interest PV = Present Value

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

FV = PV + INT = PV(1 + rt)

We can also solve for the present value PV to obtain

PV = FV/(1 + rt).

Examples

1. For a 8.5% simple interest 4-year \$20,000 loan, the total interest is

INT = PVrt = (20,000)(0.085)(4) = \$6,800,
While the future value is

FV = PV + INT = 20,000 + 6,800 = \$26,800.

2. A US Treasury bill paying \$5,000 after 2 years yields 3.5% simple annual interest. Its present value is

PV = FV/(1 + rt) = 5,000/(1 + 0.0352) = \$4,672.90

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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

FV = PV(1 + r/m)mt
or
FV = PV(1 + i)n
where i = r/m is the interest per compounding period and n = mt is the number of compounding periods.

Again, we can solve for the present value PV to obtain

PV = FV/(1 + r/m)mt
Example

For 4-year investment of \$20,000 earning 8.5% per year, with interest re-invested each month, the future value is

FV = PV(1 + r/m)mt
= 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").

 FV = PV = r = m = t =

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Effective Interest Rate

If money is invested at an annual rate r, compounded m times per year, the effective interest rate is

reff = (1 + r/m)m - 1.

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 rnom in the textbook.

Example

A CD paying 9.8% compounded monthly has a nominal rate of rnom = 0.098, and an effective rate of

 reff = (1 + rnom /m)m = (1 + 0.098/12)12 - 1 = 0.1025.
Thus, we get an effective interest rate of 10.25%, since the compounding makes the CD paying 9.8% compounded monthly really pay 10.25% interest over the course of the year.

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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

 FV   =   PV(1 + i)n + PMT (1 + i)n - 1i ,

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

FV=
 PV(1 + i)n + PMT (1 + i)n - 1i
=
 5,000(1+0.05/12)120 + 100 (1+0.05/12)120 - 10.05/12
=
 \$23,763.28.

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 TI-83, and Excel:

If a quantity is paid out, it is entered as negative.

Thus, to redo the above example, you can enter:

PV = -5000
PMT = -100,
r = 5%       (or 0.05)
m = 12
t = 10
and compute FV.

 FV = PV = PMT = r = m = t =

Press here to bring the above utility up in its own window for your convenience.

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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

 PV   =   FV(1 + i)-n + PMT 1 - (1 + i)-ni ,
where i = r/m is the interest paid each period and n = mt is the total number of periods.

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
This is an (ordinary) annuity with FV = 10,000, PMT = 2,000, r = 0.07, m = 2 and t = 15, so i = 0.07/2 = 0.035 and n = 2.15 = 30. Substituting gives

 PV = 10,000(1+0.035)-30 + 2,000 1 - (1+0.035)-300.035 PV = \$40,346.87

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
A mortgage is essentially an annuity (the bank has deposited money with you at interest and withdrawing it through monthly payments) with a future value of 0. The present value is PV = \$90,000, r = 0.08, m = 12, and t = 30. Substituting all these into the formula on the left and solving for PMT gives PMT = \$660.39.

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).

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Last Updated: March, 2006
Copyright © 2000-2006 Stefan Waner & Steven. R. Costenoble

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