Compound Interest
Motivation—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.
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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
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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
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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
…
After 12 years
Formula for
wealth
50,000
=50,000(1.07)
=50,000(1.07)2
=50,000(1.07)3
…
=50,000(1.07)12
Wealth
50,000.00
$53,500.00
$57,245.00
$61,252.15
…
$112,609.58
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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.
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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
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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
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Present value
𝐹𝑉 = 𝑃𝑉 1 + 𝑟 𝑇
𝐹𝑉
= 𝑃𝑉
𝑇
1+𝑟
𝐹𝑉 1 + 𝑟 −𝑇 = 𝑃𝑉
𝑃𝑉 = 𝐹𝑉 1 + 𝑟
FV = future value
PV = present value
r = interest rate
T = # years
−𝑇
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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
𝑃𝑉 = 𝐹𝑉 1 + 𝑟 −𝑇
𝑃𝑉 = 265.33 1.05 −20
𝑃𝑉 = $100
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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
𝑃𝑉 = 𝐹𝑉 1 + 𝑟 −𝑇
𝑃𝑉 = 40,000 1.03 −5
𝑃𝑉 = $34,504.35
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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. It’s a bad investment.
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