Financial Mathematics

Savings

Adapted from “Compound Interest” Powerpoint by Patrick Callahan, Ph.d

First, a review

• $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

  A B C D

1 Year Annually Quarterly Monthly

2 0 1000 1000 1000

3 1 =B2*(1+.055)^1 =C2*(1+.055/4)^4 =D2*(1+.055/12)^12

4 2      

5 3      

6 4      

7 5      

8 6      

9 7      

10 8      

11 9      

12 10      

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