Wednesday

 Mostly summarized from Gregory Mankiw’s Principles of Economics, 5th Ed.

PART 9 The Real Economy in the Long Run
Chapter 27 of 36 Basic Tools of Finance
Section 3 of 16
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If someone offers you $100 today or $100 in 10 years, which would you choose?
Getting $100 today is better.
You can deposit the $100 in a bank savings account and in 10 years still have it plus the interest you earn on it over the 10 years.
Money today is more valuable than the same amount of money in the future because of interest.
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If someone offers you $100 today or $200 in 10 years, which would you choose?
To decide, you must compare money value at different points in time.
This is done with a present value calculation.
The present value of any future sum of money
· is the amount of money today that would be needed
· at current interest rates
· to produce that future sum
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Question#1:
If you put $100 in your bank account today, how much will it be worth in 10 years?
In other words, what will be the future value, 10 years later, of the $100 you have today?
For the following calculations
· use the letter r for the interest rate expressed in decimals, with r of 0.05 meaning an interest rate of 5 percent
· assume no inflation
Assume interest is paid annually.
The interest paid remains in your bank account to earn more interest, a process called compounding.
The $100 will become
· after one year: (1 + r) x $100
· after 2 years: (1 + r) x (1 + r) x $100 = (1 + r)² x $100
· after 3 years: (1 + r) x (1 + r) x (1 + r) X $100 = (1 + r)³ x $100
· after N years: (1 + r)ᴺ x $100
Answer to Question #1 above is $163.
If you put $100 in your bank,
-at an interest rate of 5 percent
-and keep it there for 10 years
-the future value of the $100 will be (1.05)¹⁰ x $100 = $163
Receiving $100 today or $163 in ten years would have equal values.
Note, this and the following are easily calculated at ChatGPT or any AI.
“If you put $100 in your bank account today, with 5% compounded interest how much will it be worth in 10 years?”
ChatGPT: In 10 years, your $100 deposit at 5% compounded interest would grow to $162.89.
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Question #2:
If you are going to be paid $200 in ten years, what is the present value of this future payment?
To answer, we calculate using present value, which is the inverse of future value calculation used in question #1.
In the previous future value question we computed the future value from a present value by multiplying by the factor (1 + r)ᴺ
Here, to compute a present value from a future value we instead divide by the factor (1 + r)ᴺ
So, the present value of $200 in N years is $200 / (1 + r)ᴺ
Answer to Question #2 is $123:
· if the interest rate is 5 percent
· the present value of $200 in 10 years is $200 / (1.05)¹⁰ = $123
Otherwise stated, if the interest rate is 5% you would have to deposit $123 in a bank now to receive $200 in 10 years.
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present value and future value
genzai kachi to shōrai kachi
現在価値 と 将来価値