Transcript Chapter 10 Section 1

Chapter 10 Section 1
Interest
Terms
• Interest : Fee that is paid for the use of
money
• Principal : Amount of initial deposit or
initial/current balance
• Compound Amount : Amount to which the
principal grows (after the addition on
interest). Alternate term: Balance
• Compounded : Computed
Compound Periods
• The number of times interest is
compounded in a single year
• Denoted by : m
• Table 2 (page 470)
Compound Periods Table
Interest
Compounded …
Number of Interest
Periods Per Year
Annually
m=1
Semiannually
m=2
Quarterly
m=4
Monthly
m = 12
Weekly
m = 52
Daily
m = 365
Annual Interest Rate
• Denoted by : r
• Also known as Nominal Rate or Stated
Rate.
• Number which is stated / advertised and
used to calculate the interest rate per period.
• Use decimal form when calculating by
hand.
Interest Rate Per Compound Period
• Denoted by : i
• Number which is used to calculate interest
for each compounding period.
• Use decimal form when calculating.
• Formula on next slide (page 470 – Bluegray box).
Interest Rate Per Period Formula
• Formula:
i=r/m
where
r = annual interest rate ( in decimal form)
m = number of compound periods in a year
Example of Interest Rate Per Period
• Find the interest rate per period of an account
that earns 6.25% interest compounded weekly.
• Solution:
Given:
r = 0.0625 and
m = 52
i = r/m
= 0.0625 / 52 ~ 0.00120
Interest rate per period is approximately 0.12 %
Compound Interest Problems
•
Basic idea for compound interest accounts
1. Deposit an initial amount of money into an
account.
2. Step back and watch it grow.
3. You do not deposit or withdraw any
additional money while interest is
accumulating.
Diagram for Compound Interest
Balances:
B0
Interest:
B1
B2
B3
B4
i ·B0
i ·B1
i ·B2
i ·B3
…
Deposits
or
Withdraws
P
B = Balance
P = Principal = Initial Deposit
Each tick mark represents a compound period
Balance for Compound Interest
New balance based on the old balance
Bnew = Bprevious + i·Bprevious
which simplifies to
Bnew = (1+ i)Bprevious
(Note that this is in the form
of a difference equation)
Note that i·Bprevious represents the amount of interest that one earns for
the compound period
Balance after n
interest/compounding periods
F = ( 1 + i )n ·P
Where: F = compounded amount after n
compounding periods.
P = Principal (in the form of an initial
deposit or current balance).
Notational Differences
Book
Calculator
Term
n
N
Number of compounding
periods
F
FV
Future Value
P
PV
Principal Value
m
P/Y
&
C/Y
Number of compounding
periods in a year
R
PMT
Rent / Payment per period
Accessing the TVM Solver
1. Hit APPS key
2. Select 1:Finance function (Hit ENTER key)
3. Select 1:TVM Solver …function (Hit ENTER key)
TVM Solver Variables
•
•
•
•
•
•
•
•
N = Number of compound periods
I% = Annual Interest Rate (in percent form ( r% ))
PV = Principal Value (or) “Previous/Current” Balance
PMT = Rent / Payment Per Compound Period
FV = Future Value
P/Y = Payments Per Year = m
C/Y = Compounding Periods Per Year = m
PMT:END = Payments(/Interest) made(/calculated)
at the end of the compounding period
Using the TVM Solver
• Enter the numbers for each variable of
interest.
• Move the cursor to the variable that you
want to solve for.
• Hit the ALPHA (green) key and then the
ENTER (/solve) key.
• The answer will appear next to the variable
that you are solving for.
When using the TVM Solver on the
calculator
• Think:
1. Outflow = NEGATIVE cash flow
(i.e. You DO NOT have the instantaneous use of your money )
2. Inflow = POSITIVE cash flow
(i.e. You do have the instantaneous use of your money )
Exercise 5 (page 477)
Formula Solution
• Calculate the compound amount of $1,000 after 2 years
if deposited at 6% interest compounded monthly.
• Solution:
n = 2 ·12 = 24
i = r/m = 0.06/12 = 0.005
F = ( 1 + i )n ·P
F = ( 1 + 0.005 )24 ·1000
F = ( 1.12715977…) ·1000
F = 1127.15977
Answer :$1,127.16
Exercise 5 (page 477)
TVM Solver Solution
• Calculate the compound amount of $1,000 after 2 years
if deposited at 6% interest compounded monthly.
• Solution:
N = 2 ·12 = 24
I% = 6
PV = – 1000
PMT = 0
FV = 1127.159776
P/Y = C/Y = 12
Note the negative sign!!!
Answer :$1,127.16
Set cursor on FV and Solve ( ALPHA key and then ENTER
key)
Effective Rate of Interest
•
•
Page 474
Used to
1. Compares two annual interest rates that have
two different yearly compounding periods.
2. When money from the interest is reinvested in
the account, will tell you the ‘true’ interest
rate that you are earning.
Effective Rate of Interest Formula
Formula:
reff = ( 1 + i )m – 1
where:
reff = Effective Rate of Interest
i = Interest Rate Per Period = r / m
m = Number of compounding periods in a
single year
Effective Rate of Interest on the
Calculator
• Access
– Hit APPS key
– Select 1:Finance function
– Use down (or up) arrow key to select C:
function
• Syntax
Eff( r% , m )
( r% = Annual Interest Rate in % form)
Eff(