1. Present value
An important idea in Finance is the _Time Value of Money_. Basically, this means that 100 dollars today is worth more than 100 dollars a month from now. We have already done examples that demonstrate this fact well.
2. Time value of money
If I gave you
3. Time value of money
100 dollars worth of apple stock today, and told you that
4. Time value of money
it would gain 10% this month, you could easily tell me that
5. Time value of money
at the end of the month you would have 110 dollars. Because of this, the 100 dollars of apple stock I handed you today
6. Time value of money
is worth more than the 100 dollars I promise to give you in a month.
7. Future value and present value
A related idea to this, which will be used in the next few exercises,
8. Future value and present value
is _Present Value_, and _Future Value_.
9. Future value and present value
By moving the 100 dollars I handed you today forward one month
10. Future value and present value
to become 110 dollars, you are saying that
11. Future value and present value
the future value of that 100 dollars is 110 dollars. Instead of moving forward, you can also move backwards by _discounting_ money. For example, I promised to pay you
12. Future value and present value
100 dollars 1 month from now, but how much is that worth, right now, in _today's_ money?? The calculation for this is simple. Instead of multiplying today's 100 dollars by 1-point-10,
13. Future value and present value
we divide the 100 dollars I promise to give you in 1 month by 1-point-10. This corresponds to
14. Future value and present value
a present value of 90-point-91,
15. Future value and present value
which is less money compared to if I would give you 100 dollars today. Notice that we can write this as
16. Future value and present value
100 dollars times 1-point-10 to the negative 1, and the equation is exactly the same.
17. Present value - multiple periods
To go one step further,
18. Present value - multiple periods
if I promise to give you
19. Present value - multiple periods
100 dollars 2 months from now, with a 10% increase each month, how much is that 100 dollars worth in today's money? In other words, what is the present value of 100 dollars to be received 2 months from now? Just like when we chained together return multipliers for multiple months, we can chain together the division
20. Present value - multiple periods
to discount back by multiple months. The result is 100 dollars,
21. Present value - multiple periods
divided by 1-point-10 to move back one month,
22. Present value - multiple periods
divided again by 1-point-10
23. Present value - multiple periods
to move back a second month. Written a different way,
24. Present value - multiple periods
this is 100 dollars times 1-point-10 to the negative 2.
25. Present value - general formula
In this form, we can generalize the calculation of present value to the following formula: cash_flow times the multiplier raised to the negative of the number of periods you are moving the money backwards. In R, creating the variables and calculating the present value would look like this. Looks like we got the same answer as our slightly rounded one. Great!
26. Let's practice!
That was a lot. Present value is a tricky topic the first time around, but the exercises will definitely help you to grasp the concept. Good luck!