Transcript Savings Power Point Lecture

Financial Mathematics
Savings
Adapted from “Compound Interest”
Powerpoint by Patrick Callahan, Ph.d
First, a review
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$1.00 in 2007 does NOT equal $1.00 in 2008
Why?
B/c $1.00 in 2007 buys MORE than $1.00 in ’08
CPI – Method for converting $ from year to year,
to make comparisons
For example…
• You buy a bike in 2007 and spend $200.
• Your sister bought a similar bike in 1998
for $175.
• Who’s bike cost more?
– In nominal dollars, yours did: $200 > $175
– But in real (or constant) dollars, hers did
– B/c $175 (in 1998) = $223 (in 2007)
Another example
• Wages are also affected by inflation
– What you can buy with your earnings in 2008
is less than what you could buy with the same
wages in 1998.
• Need your wages to increase at a rate at
least equal to inflation.
What is central is the idea that
CPI helps to give us CONTEXT
for understanding the value of
money across time
Inflation
• The change in the CPI from year to year is
the inflation rate
• So, one goal is to have your income keep
pace with inflation
• Another goal would be to have your
income outpace inflation
– which would give you some leftover…to
spend or to save
Why save money?
• To get enough to purchase a big ticket
item (car, house, pay for graduate school)
• To have money to live on after you retire
and no longer have a steady income from
your work
Why not stuff it in your mattress?
• You’ll dig it out and spend it
• Someone will break into your home and
steal it
• Your house will burn down and it will go up
in smoke
• Inflation cuts into the value
– $20,000 stuffed away in 1970
= $3743 in 2007
(Carmela Soprano knew this!)
Savings Accounts
• When you put money into a savings
account, it earns interest
• This means the amount GROWS over
time
• (The bank pays you money to lend them
your money, which they then lend out to
others at a slightly higher rate. – More on
this later!)
Two Kinds of Savings Accounts
• Basic Savings Account
– Usually no minimum balance required
– Pays a very low interest
– Can withdrawal money whenever you want
• Money Market Account
– Usually has a minimum balance
– Pays a higher interest rate
– Often limits the number of withdrawals/month
Two types of interest
• Simple interest: Fixed percentage of
original amount invested or deposited.
• Compound interest: Fixed percentage of
original amount plus accumulated
interest.
– You earn interest on your interest.
Example: $1000 invested at 10%
Simple
Compound
Original Amount
$1000
$1000
Year 1
$1100
$1100
Year 2
$1200
$1210
Year 3
$1300
$1331
Simple v. compound
• Simple interest = linear growth
• Compound interest = exponential growth
• Which is better?
Formula for compound growth
Balance=Principle(1+r/n)yn
Balance = How much in your account
Principle = What you started with (originally)
r = annual interest rate
n = compounding frequency
y = number of years
Money can compound at
different time periods
Balance=Principle (1+r/n)yn
This changes the value of n:
Annually: n=1
Quarterly (every 3 months): n=4
Monthly: n=12
Different Compounding
• Basic Formula: Balance=Principle(1+r/n)yn
• Various Versions:
Yearly: Balance = Principle (1+r)y
Quarterly: Balance=Principle (1+r/4)4y
Monthly: Balance=Principle (1+r/12)12y
An Example:
• 5% APR,
– compounded quarterly, for 7 years
Balance=Principle(1+r/n)yn
Balance=Principle (1+.05/4)7*4
=1,000 (1.0125)28
= $1415.99
Another example:
• 5% APR
– Compounded monthly, for 7 years
Balance=Principle(1+r/n)yn
Balance=Principle (1+.05/12)7*12
= 1,000 (1.0041667)84
= $1418.04
Excel Example
1
A
B
C
D
Year
Annually
Quarterly
Monthly
2
0
1000
3
1 =B2*(1+.055)^1
4
2
5
3
6
4
7
5
8
6
9
7
10
8
11
9
12
10
1000
=C2*(1+.055/4)^4
1000
=D2*(1+.055/12)^12
Annual percentage yield [APY]
• In formulas, r was annual percentage rate
or APR
• When interest compounded more often
than once per year, actual interest earned
in a year is greater than APR
Example: $10,000 invested for 10
years at 8% APR
Annually:
$21,589.25
Quarterly:
$22,080.40
Monthly:
$22,196.40
Computing APY
1. Compute the balance for one period.
2. Calculate percentage change from two
consecutive periods
(new balance-old balance)/old balance
Computing APY
• Another version
• APY = (1 + r/n )n – 1 where r is the stated
annual interest rate and n is the number of
times you’ll compound per year.
• Example: 8% rate, compounded monthly
• APY = (1+.08/12)12 – 1
• APY = 8.29