Nineteen Compound Interest and Present Value Copyright 2014 by The McGrawHill Companies Inc All rights reserved McGrawHillIrwin Compare simple interest with compound interest Calculate the compound amount and interest manually using algebraic formulas and with a financial calculator ID: 131980
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Slide1
Chapter Nineteen
Compound Interest and Present Value
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
McGraw-Hill/IrwinSlide2
Compare simple interest with compound interest.
Calculate the compound amount and interest manually, using algebraic formulas and with a financial calculator.Explain and compute the effective rate (APY).LU 19-1: Compound Interest (Future Value) – The Big PictureLearning unit objectivesLU 19-2: Present Value -- The Big Picture
Compare present value (PV) with compound interest (FV).
Compute present value using algebraic formulas and with a financial calculator.
Check the present value answer by compounding.
19-2Slide3
Compound Interest (Future Value)
Compound Interest – The interest on the principal plus the interest of prior periodsCompounding – 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 period
19-3Slide4
Compounding Terms
Compounding Periods Interest CalculatedCompounding Annually Compounding Semiannually Compounding Quarterly Compounding Monthly Compounding Daily
Once a year
Every 6 months
Every 3 months
Every month
Every day
19-4Slide5
Future Value of $1 at 8% for Four Periods (Figure 19.1)
Number of periodsCompounding 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
19-5Slide6
Tools for Calculating Compound Interest
Number of periods (N) Number of years multiplied by the number of times the interest is compounded per yearRate 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%Slide7
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 TI = $80 x .08 x 4I = $25.60MV = $80 + $25.60MV = $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?
Interest: $108.83 -- $80.00 =
$28.83
19-
7
Simple
CompoundedSlide8
Calculating Compound Amount 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 PV amount, n, and i into the following formula: FV = PV(1 + i)
n
Step 4
. Solve. This gives the compound amount.
19-
8Slide9
Calculating Compound Amount using the compound interest formula
19-9Bill 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)
y
x
4
x
80
=
$108.84Slide10
Calculating compound amount using YOUR TI BA II PLUS calculator
Remember to clear the TVM each time you work with new data: 2ND CLR TVMTo 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-
10Slide11
Compounding (fv)figure 19.2
19-11Slide12
CALCULATING THE COMPOUND INTEREST
FV – PV = Compound interest19-12$108.84 - $80.00 = $28.84Slide13
Nominal versus Effective Rates annual Percentage yield (APY)
Truth in Savings LawAnnualPercentageYield
Effective rate (APY)
4
=
Interest for 1 year
Principal
Nominal Rate (stated rate) –
The rate on which the bank calculates interest
19-
13Slide14
Calculating Effective Rate (APY)
19-14Slide15
Nominal and Effective Rates (APY) of Interest Compared (Figure 19.3)
19-15Slide16
Present Value of $1 at
8% for Four Periods (Figure 19.4)19-16Slide17
Relationship of compounding (fv) to Present value (pv) – bill smith example
19-17Slide18
Calculating Present Value
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 FV amount,
n
, and
i
into the following formula:
Step 4
. Solve. This gives the present value.
19-
18
PV = FV
(1 + i)
nSlide19
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 = $80.00Calculator keystrokes for this problem are:(1
+ .08) yx 4
= STO
1 108.84
÷ RCL
1
=
$80.00
19-
19Slide20
Calculating Present Value using A FINANCIAL CALCULATOR
19-20Step 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:Slide21
Comparing Compound Interest (FV) with Present Value (PV)
19-21Slide22
Comparing compound interest (fv) with Present value (pv)
19-22Slide23
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.Slide24
Problem 19-13
Solution:Mystic4 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)Slide25
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
$ 59,533.90Slide26
Problem 19-23
Solution:8 years x 2 = 16 periods6% 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