chapter nineteen compound interest and present value copyright © 2014 by the mcgraw-hill companies,...
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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
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 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:
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