# How much money do we need to invest today at an interest rate of 3

```Compound Interest
Motivationβa choice between two investments
(1) Pay \$50,000 now for an expected profit of
\$100,000 after twelve years
(2) Invest \$50,000 at an interest rate of 7% for
twelve years.
Which option provides you with the most wealth in twelve years?
Need to understand compound interest for an answer.
1
Option 2: Invest \$50,000 at an interest rate of 7% for
twelve years.
Time
now
after 1 year
Formula for
wealth
50,000
50,000 +
50,000(0.07)
=50,000(1.07)
Wealth
50,000
\$53,500
2
Option 2: Invest \$50,000 at an interest rate of 7% for
twelve years.
Time
now
Formula for
wealth
50,000
Wealth
50,000
after 1 year
=50,000(1.07)
\$53,500
After 2 years
={50,000(1.07)}(1.0 \$57,245
7)
=50,000(1.07)2
3
Option 2: Invest \$50,000 at an interest rate of 7% for
twelve years.
Time
now
after 1 year
after 2 years
after 3 years
β¦
After 12 years
Formula for
wealth
50,000
=50,000(1.07)
=50,000(1.07)2
=50,000(1.07)3
β¦
=50,000(1.07)12
Wealth
50,000.00
\$53,500.00
\$57,245.00
\$61,252.15
β¦
\$112,609.58
4
Which option makes us wealthier?
(1) Pay \$50,000 now for an expected profit of
\$100,000 after twelve years
(2) Invest \$50,000 at an interest rate of 7% for
twelve years. Provides us with \$112,609.58 in
twelve years. Clearly this is the better option.
5
Terminology
What is the future value of
\$50,000 invested at a 7%
interest rate for twelve years?
FV = (PV)(1+r)T
FV = (50,000)(1.07)12
FV = \$112,609.58
FV = future value
PV = present value
r = interest rate
T = # years
6
More practice
What is the future value of
\$100 invested at a 5%
interest rate for 20 years?
FV = (PV)(1+r)T
FV = (100)(1.05)20
FV = \$265.33
FV = future value
PV = present value
r = interest rate
T = # years
7
Present value
πΉπ = ππ 1 + π π
πΉπ
= ππ
π
1+π
πΉπ 1 + π βπ = ππ
ππ = πΉπ 1 + π
FV = future value
PV = present value
r = interest rate
T = # years
βπ
8
Present value
How much money do we need
to invest today at an interest
rate of 5% to get \$265.33
in 20 years?
FV = future value
PV = present value
r = interest rate
T = # years
ππ = πΉπ 1 + π βπ
ππ = 265.33 1.05 β20
ππ = \$100
9
Present value
How much money do we need
to invest today at an interest
rate of 3% to get \$40,000
in 5 years?
FV = future value
PV = present value
r = interest rate
T = # years
ππ = πΉπ 1 + π βπ
ππ = 40,000 1.03 β5
ππ = \$34,504.35
10
Back to the original question
(1) Pay \$50,000 now for an expected profit of
\$100,000 after twelve years. How much money do
we need to invest today at 7% interest to receive
\$100,000 after 12 years?
PV = (100,000)(1.07)-12 = \$44,401.20
Clearly we prefer \$50,000 dollars today over
\$44,401.20 dollars today. Itβs a bad investment.
11
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