Present value

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!