Present Value
Discounting a Cashflow¶
Now, the question is how to do we do this in reverse? For example, what if we start with getting $100 in five years?
#Now the question is, what is $100 in 5 years worth to us today
timelinePlot(5,(100,5))
Let's take a simple example first. What about $105 in one year? What value would we have had to been given at time 0 if we wanted to have that one year later?
#Start with $105 in one year
timelinePlot(1,(105,1))
If we had $100 and a 5\% interest rate we know we would have ended up at the same place in a year.
timelinePlot(1,(100,0))
What about $100 in two years? What would we need at time 0 to have gotten this? We can divide by 1.05 twice to get the number.
timelinePlot(2,(100,2))
PV = 100 / 1.05 / 1.05
timelinePlot(2,(PV,0))
This brings us to the formula for present value. It is simply the reverse of compounding.
$ PV = \frac{FV}{(1+r)^t}$
where
$ PV = \text{Present Value} $
$ FV = \text{Future Value at time t} $
$ t = \text{Time period for future value} $
$ r = \text{Discount rate} $
Returning to our original question, what is $100 in 5 years from today supposed to be worth if the discount rate is 5%. We can easily solve like below.
PV = 100 / 1.05 ** 5
print(PV)
The two cashflows below are equivalent in terms of present value.
#Plot the equivalent PV cash flows
timelinePlot(5, (100, 5))
timelinePlot(5, (PV, 0))
