the nature of financial management copyright © cengage learning. all rights reserved. 11

THE NATURE OF FINANCIALMANAGEMENT

Copyright © Cengage Learning. All rights reserved.

11

Copyright © Cengage Learning. All rights reserved.

11.1 Interest

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Amount of Simple Interest

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Amount of Simple Interest

Simply stated, interest is money paid for the use of money.The amount of the deposit or loan is called the principal or present value, and the interest is stated as a percent of the principal, called the interest rate.

The time is the length of time for which the money is borrowed or lent.

The interest rate is usually an annual interest rate, and the time is stated in years unless otherwise given.

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Amount of Simple Interest

These variables are related in what is known as the simple interest formula.

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Amount of Simple Interest

Suppose you save 20¢ per day, but only for a year. At the end of a year you will have saved $73.

If you then put the money into a savings account paying 3.5% interest, how much interest will the bank pay you after one year?

The present value (P) is $73, the rate (r) is 3.5% = 0.035, and the time (t, in years) is 1.

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Amount of Simple Interest

Therefore,

I = Prt

= 73(0.035)(1)

= 2.555

Round money answers to the nearest cent: After one year, the interest is $2.56.

You can do this computation on acalculator:

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Example 1 – Find the amount of interest

How much interest will you earn in three years with an initial deposit of $73?

Solution:I = Prt = 73(0.035)(3) = 7.665.

After three years, the interest is $7.67.

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Future Value

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Future Value

The future value is the amount you will have after the interest is added to the principal, or present value.

Let A = FUTURE VALUE.

Then

A = P + I

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Example 2 – Find an amount of interest

Suppose you see a car with a price of $12,436 that is advertised at $290 per month for 5 years. What is the amount of interest paid?

Solution:The present value is $12,436. The future value is the total amount of all the payments:

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Example 2 – Solution

Therefore, the amount of interest is

I = A – P

= 17,400 – 12,436

= 4,964

The amount of interest is $4,964.

cont’d

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Interest for Part of a Year

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Interest for Part of a Year

There are two ways to convert a number of days into a year:

Exact interest: 365 days per year

Ordinary interest: 360 days per year

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Example 4 – Find the amount to repay a loan

Suppose that you borrow $1,200 on March 25 at 21% simple interest. How much interest accrues by September 15 (174 days later)?

What is the total amount that must be repaid?

Solution:We are given P = 1,200,

r = 0.21, and

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Example 4 – Solution

I = Prt

= 121.8

The amount of interest is $121.80.

To find the amount that must be repaid, find the future value:

A = P + I = 1,200 + 121.80 = 1,321.80

The amount that must be repaid is $1,321.80.

Simple interest formula

Substitute known values.

Use a calculator to do the arithmetic.

cont’d

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Interest for Part of a Year

We derive a formula for future value because sometimes we will not calculate the interest separately as we did inExample 4.

FUTURE VALUE = PRESENT VALUE + INTEREST

A = P + I

= P + Prt

= P(1 + rt)

Substitute I = Prt.

Distributive property

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Interest for Part of a Year

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Example 5 – Find a future value

If $10,000 is deposited in an account earning simple interest, what is the future value in 5 years?

Solution:We identify P = 10,000,

r = 0.0575, and

t = 5.

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Example 5 – Solution

A = P(1 + rt)

= 10,000(1 + 0.0575 5)

= 10,000(1 + 0.2875)

= 10,000(1.2875)

= 12,875

The future value in 5 years is $12,875.

cont’d

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Example 6 – Find the amount necessary to retire

Suppose you have decided that you will need $4,000 per month on which to live in retirement. If the rate of interest is 8%, how much must you have in the bank when you retire so that you can live on interest only?

Solution:We are given I = 4,000,

r = 0.08, and

t = (one month = year):

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Example 6 – Solution

I = Prt

48,000 = 0.08P

600,000 = P

You must have $600,000 on deposit to earn $4,000 per month at 8%.

cont’d

Divide both sides by 0.08.

Multiply both sides by 12.

Simple interest formula

Substitute known values.

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Compounding Interest

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Compounding Interest

Most banks do not pay interest according to the simple interest formula; instead, after some period of time, they add the interest to the principal and then pay interest on this new, larger amount.

When this is done, it is called compound interest.

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Example 7 – Compare simple and compound interest

Compare simple and compound interest for a $1,000 deposit at 8% interest for 3 years.

Solution:

Identify the known values:

p = 1000,

r = 0.08, and

t = 3.

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Example 7 – Solution

Next, calculate the future value using simple interest:

A = P(1 + rt)

= 1000(1 + 0.08 3)

= 1,000(1.24)

= 1,240

With simple interest, the future value in 3 years is $1,240.

cont’d

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Example 7 – Solution

Next, assume that the interest is compounded annually. This means that the interest is added to the principal after 1 year has passed.

This new amount then becomes the principal for the following year.

Since the time period for each calculation is 1 year, we let t = 1 for each calculation.

cont’d

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Example 7 – Solution

First year (t = 1): A = P(1 + r)

= 1,000(1 + 0.08)

= 1,080

Second year (t = 1): A = P(1 + r)

= 1,080(1 + 0.08)

cont’d

One year’s principal is previous year’s total.

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Example 7 – Solution

= 1,166.40

Third year (t = 1): A = P(1 + r)

= 1,166.40(1 + 0.08)

= 1,259.71

With interest compounded annually, the future value in 3 years is $1,259.71. The earnings from compounding are $19.71 more than from simple interest.

cont’d

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Compounding Interest

Most banks compound interest more frequently than once a year. For instance, a bank may pay interest as follows:

Semiannually: twice a year or every 180 days

Quarterly: 4 times a year or every 90 days

Monthly: 12 times a year or every 30 days

Daily: 360 times a year

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Compounding Interest

If we repeat the same steps for more frequent intervals than annual compounding, we again begin with the simple interest formula A = P(1 + rt).

Semiannually, then t = : A = P(1 + r )

Quarterly, then t = : A = P(1 + r )

Monthly, then t = : A = P(1 + r )

Daily, then t = : A = P(1 + r )

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Compounding Interest

We now compound for t years and introduce a new variable, n, as follows:

Annual compounding, n = 1: A = P(1 + r)t

Semiannual compounding, n = 2: A = P(1 + r )2t

Quarterly compounding, n = 4: A = P(1 + r )4t

Monthly compounding, n = 12: A = P(1 + r )12t

Daily compounding, n = 360: A = P(1 + r )360t

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Compounding Interest

We are now ready to state the future value formula for compound interest, which is sometimes called the compound interest formula.

For these calculations you will need access to a calculator with an exponent key.

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Compounding Interest

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Example 8 – Compare compounding methods

Find the future value of $1,000 invested for 10 years at 8% interest

a. compounded annually.

b. compounded semiannually.

c. compounded quarterly.

d. compounded daily.

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Example 8 – Solution

Identify the variables: P = 1,000, r = 0.08, t = 10.

a. n = 1: A = $1,000(1 + 0.08)10 = $2,158.92

b. n = 2: A = $1,000(1 + )2 · 10 = $2,191.12

c. n = 4: A = $1,000(1 + )4 · 10 = $2,208.04

d. n = 360: A = $1,000(1 + )360 · 10 = $2,225.34

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Continuous Compounding

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Continuous Compounding

Suppose $1 is invested at 100% interest for 1 year compounded at different intervals.

The compound interest formula for this example is

where n is the number of times of compounding in 1 year.

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Continuous Compounding

The calculations of this formula for different values of n are shown in the following table.

Looking only at this table, you might (incorrectly) conclude that as the number of times the investment is compounded increases, the amount of the investment increases withoutbound.

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Continuous Compounding

Let us continue these calculations for even larger n:

The spreadsheet we are using for these calculations can no longer distinguish the values of (1 + 1/n)n for larger n. These values are approaching a particular number.

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Continuous Compounding

This number, it turns out, is an irrational number, and it does not have a convenient decimal representation. (That is, its decimal representation does not terminate and does not repeat.)

Mathematicians, therefore, have agreed to denote this number by using the symbol e. This number is called the natural base or Euler’s number.

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Continuous Compounding

The number e is irrational and consequently does not have a terminating or repeating decimal representation and its value is approximately 2.7183.

It is easy to see how this formula follows from the compound interest formula if we let so that n = mr.

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Continuous Compounding

As m gets large, approaches e, so we have

A = Pert

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Example 9 – Find future value using continuous compounding

Find the future value of $890 invested at 21.3% for 3 years, 240 days, compounded continuously.

Solution:We use the formula A = Pert where P = 890, r = 0.213.

For continuous compounding, use a 365-day year, so

3 years, 240 days = 3.657534247 years

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Example 9 – Solution

Remember that t is in years, and also remember to use this calculator value and not a rounded value.

A = 890e0.213t 1,939.676057

The future value is $1,939.68.

cont’d

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Inflation

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Inflation

Any discussion of compound interest is incomplete without a discussion of inflation.

The same procedure we used to calculate compound interest can be used to calculate the effects of inflation. The government releases reports of monthly and annual inflation rates.

In 1980 the inflation rate was over 14%, but in 2010 it was less than 1.25%.

Keep in mind that inflation rates can vary tremendously.

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Example 11 – Find future value due to inflation

If your salary today is $55,000 per year, what would you expect your salary to be in 20 years (rounded to the nearest thousand dollars) if you assume that inflation will continue at a constant rate of 6% over that time period?

Solution:Inflation is an example of continuous compounding. The problem with estimating inflation is that you must “guess” the value of future inflation, which in reality does not remain constant.

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Example 11 – Solution

However, if you look back 20 years and use an average inflation rate for the past 20 years—say, 6%—you may use this as a reasonable estimate for the next 20 years.

Thus, you may use P = 55,000,

r = 0.06, and

t = 20 to find

A = Pert = 55,000e0.06(20)

cont’d

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Example 11 – Solution

Note:

Be sure to use parentheses for the exponent:

Display: 182606.430751

The answer means that, if inflation continues at a constant 6% rate, an annual salary of $183,000 (rounded to the nearest thousand dollars) will have about the same purchasing power in 20 years as a salary of $55,000 today.

cont’d

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Present Value

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Present Value

Sometimes we know the future value of an investment and we wish to know its present value. Such a problem is called a present value problem.

The formula follows directly from the future value formula (by division).

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Example 12 – Find a present value for a Tahiti trip

Suppose that you want to take a trip to Tahiti in 5 years and you decide that you will need $5,000. To have that much money set aside in 5 years, how much money should you deposit now into a bank account paying 6% compounded quarterly?

Solution:In this problem, P is unknown and A is given: A = 5,000. We also have r = 0.06,

t = 5, and

n = 4.

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Example 12 – Solution

Calculate:

= $3,712.35

cont’d