Chapter Nineteen

COMPOUND INTEREST AND PRESENT VALUE

Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

1. Compare simple interest with compound interest.2. Calculate the compound amount and interest

manually, using algebraic formulas and with a financial calculator.

3. Explain and compute the effective rate (APY).

LU 19-1: Compound Interest (Future Value) – The Big Picture

LEARNING UNIT OBJECTIVES

LU 19-2: Present Value -- The Big Picture1. Compare present value (PV) with compound interest

(FV).2. Compute present value using algebraic formulas and

with a financial calculator.3. Check the present value answer by compounding.

19-2

COMPOUND INTEREST (FUTURE VALUE)

Compound Interest –

The interest on the principal plus the interest of prior

periods

Compounding –

Involves the calculation of interest periodically over the life

of the loan or investment

Present Value –

The value of a loan or investment today

Future Value (compound amount) –

The final amount of the loan or investment at the end of the

last period19-3

COMPOUNDING TERMS

Compounding Periods Interest Calculated

Compounding Annually

Compounding Semiannually

Compounding Quarterly

Compounding Monthly

Compounding Daily

Once a year

Every 6 months

Every 3 months

Every month

Every day

19-4

FUTURE VALUE OF $1 AT 8% FOR FOUR PERIODS (FIGURE 19.1)

$0.00$0.50$1.00$1.50

$2.00$2.50$3.00$3.50$4.00$4.50$5.00

0 1 2 3 4Number of periods

Compounding goes from present value to future value

Present

value

After 1

period, $1 is worth $1.08

After 2 periods,

$1 is worth $1.17

After 3 periods,

$1 is worth $1.26

Future

Value

After 4 periods, $1 is worth $1.36

$1.00

$1.08 $1.1664 $1.2597 $1.3605

19-5

TOOLS FOR CALCULATING COMPOUND INTEREST

Number of periods (N) Number of years

multiplied by the number of times the interest is compounded per year

Rate for each period (I) Annual interest rate

divided by the number of times the interest is

compounded per year

If you compounded $1 for 4 years at 8% annually,

semiannually, or quarterly, what is N and I?

Annually:

Semiannually:

Quarterly:

Periods Rate

19-6

4 x 1 = 4

4 x 4 = 16

4 x 2 = 8

8% ÷ 1 = 8%Annually:

Semiannually:

Quarterly: 8% ÷ 4 = 2%

8% ÷ 2 = 4%

SIMPLE VERSUS COMPOUND INTEREST

Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s simple interest and maturity value?

I = P x R x T

I = $80 x .08 x 4

I = $25.60

MV = $80 + $25.60

MV = $105.60

I = P x R x T

I = $80 x .08 x 4

I = $25.60

MV = $80 + $25.60

MV = $105.60

Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s interest and compounded amount?

Year 1 Year 2 Year 3 Year 480.00$ 86.40$ 93.31$ 100.77$ x .08 x .08 x .08 x .08

Interest 6.40$ 6.91$ 7.46$ 8.06$ Beg. bal 80.00 86.40 93.31 100.77 End of year 86.40$ 93.31$ 100.77$ 108.83$

Interest: $108.83 -- $80.00 = $28.8319-7

Simple Compounded

CALCULATING COMPOUND AMOUNT

USING FORMULAStep 1. Find the periods n: Years multiplied by number

of times interest is compounded in 1 year.

Step 2. Find the rate i: Annual rate divided by number of times interest is compounded in 1 year.

Step 3. Plug the PV amount, n, and i into the following formula: FV = PV(1 + i)n

Step 4. Solve. This gives the compound amount.

19-8

CALCULATING COMPOUND AMOUNT USING THE COMPOUND INTEREST

FORMULA

19-9

Bill wants to know the value of $80 in 4 years at 8%. He begins by identifying the PV, n, and i:

PV = $80n = 4 (4 years x 1 compounding period per year)i = 8% (8% divided by 1 compounding period)Calculator keystrokes for this problem are:

(1 + .08) yx 4 x 80 = $108.84

CALCULATING COMPOUND AMOUNT USING YOUR TI BA II PLUS

CALCULATORRemember to clear the TVM each time you work with new data: 2ND CLR TVM

To solve the future value of $80 at 8% compounded annually for 4 years, using your calculator, follow these steps:

Step 1: Input 4 and then press N.Step 2: Input 8 and then press I/Y.Step 3: Input 80, press +/- and then press PV.Step 4: Input 0, and then press PMT.Step 5: Press CPT FV = 108.84

19-10

COMPOUNDING (FV)FIGURE 19.2

19-11

CALCULATING THE COMPOUND INTEREST

FV – PV = Compound interest

19-12

$108.84 - $80.00 = $28.84

NOMINAL VERSUS EFFECTIVE RATES ANNUAL PERCENTAGE YIELD

(APY)

Truth in

Savings Law

Annual

PercentageYield

Effective rate (APY)4 = Interest for 1 year

Principal

Nominal Rate (stated rate) –

The rate on which the bank calculates interest

19-13

CALCULATING EFFECTIVE RATE (APY)

19-14

NOMINAL AND EFFECTIVE RATES (APY) OF INTEREST COMPARED

(FIGURE 19.3)

19-15

PRESENT VALUE OF $1 AT 8% FOR FOUR PERIODS (FIGURE

19.4)

19-16

RELATIONSHIP OF COMPOUNDING (FV) TO PRESENT VALUE (PV) – BILL SMITH

EXAMPLE

19-17

CALCULATING PRESENT VALUE USING FORMULA

Step 1. Find the periods n: Years multiplied by number of times interest is compounded in 1 year.

Step 2. Find the rate i: Annual rate divided by number of times interest is compounded in 1 year.

Step 3. Plug the FV amount, n, and i into the following formula:

Step 4. Solve. This gives the present value.

19-18

PV = FV (1 + i)n

CALCULATING PRESENT VALUE USING FORMULA

Since Bill knows the bike will cost $108.84 in the future, he completes the following calculation:

PV = $108.84/(1 + .08)4 = $80.00

Calculator keystrokes for this problem are:

(1 + .08) yx 4 = STO 1 108.84 ÷ RCL 1 = $80.00

19-19

CALCULATING PRESENT VALUE USING A FINANCIAL

CALCULATOR

19-20

Step 1: Input 4 and then press N.Step 2: Input 8 and then press I/Y.Step 3: Input 108.84 and then press FV.Step 4: Input 0 and then press PMT.Step 5: Press CPT PV = -80.00

Remember to clear the TVM each time you work with new data: 2ND CLR TVM

Since Bill knows the bike will cost $108.84 in the future, he completes the following calculation:

COMPARING COMPOUND INTEREST (FV)

WITH PRESENT VALUE (PV)

19-21

COMPARING COMPOUND INTEREST (FV) WITH PRESENT

VALUE (PV)

19-22

PROBLEM 19-11

Solution:

8 years x 2 = 16 periods 6% 2 = 3% $40,000 x (1 = .03)16 =

$64,188.26

19-23

Lynn Ally, owner of a local Subway shop, loaned $40,000 to Pete Hall to help him open a Subway franchise. Pete plans to repay Lynn at the end of 8 years with 6% interest compounded semiannually. How much will Lynn receive at the end of 8 years? LU 19-1(2)

Step 1: Input 16 and then press N.Step 2: Input 6/2 = and then press I/Y.Step 3: Input 40,000 +/- and then press PV.Step 4: Input 0 and then press PMT.Step 5: Press CPT FV = 64,188.26.

PROBLEM 19-13

Solution:

Mystic

4 years x 2 = 8 periods

10% 2

= 5%

FV = $10,000(1 + .05)8 = $14,774.55 $14,774.55 - $10,000 = $4,774.55

Four Rivers

4 years x 4 = 16 periods

8% 4

= 2%

FV = $10,000(1 +.02)16 = $13,727.86 $13,727.86 - $10,000 = $3727.86

19-24

Melvin Indecision has difficulty deciding whether to put his savings in Mystic Bank or Four Rivers Bank. Mystic offers 10% interest compounded semiannually. Four Rivers offers 8% interest compounded quarterly. Melvin has $10,000 to invest. He expects to withdraw the money at the end of 4 years. Which bank gives Melvin the better deal? Check your answer. LU 19-1(2)

PROBLEM 19-14

Solution:

3 years x 2 = 6 periods9% 2 = 4.5%

$59,533.90(1 + .045)6 = $77,528.62

19-25

Lee Holmes deposited $15,000 in a new savings account at 9% interest compounded semiannually. At the beginning of year 4, Lee deposits an additional $40,000 at 9% interest compounded semiannually. At the end of 6 years, what is the balance in Lee’s account? LU 19-1(2)

$15,000(1 + .045)6 = $19,533.90 + 40,000.00 $ 59,533.90

PROBLEM 19-23

Solution:

8 years x 2 = 16 periods

6% 2 = 3%

$6,000(1 + .03)16 = $3,739.00

19-26

Paul Havlik promised his grandson Jamie that he would give him $6,000 8 years from today for graduating from high school. Assume money is worth 6% interest compounded semiannually. What is the present value of this $6,000?LU 19-2(2)

Step 1: Input 16 and then press N.Step 2: Input 6/2 = and then press I/Y.Step 3: Input 6000 and then press FV.Step 4: Input 0 and then press PMT.Step 5: Press CPT PV = -3739.00