Multiple Cashflows
Present Value Outside Endpoints¶
Up until now, we have only worked through examples where we either take out or put in money in the first period and withdraw it in the last period. This is not realistic to what can happen in the real world. Let’s look at how we might value a bond that pays \$100 in period 3 given a discount rate of 5%. First, what is going to be the present value at time 0?
#Now let's say we have a bond that will give us $100 dollars in 3 years and r=.05
timelinePlot(3,[(-100/(1.05)**3,0),(100,3)])
What if the deal that we are offered is that actaully we get to buy the bond at time 2. What price would be fair in this case? The difference between year 3 and year 2 is one year, so we can discount it by only one year in this case.
#Now the cashflow at time 2 gets divided by 1.05 ^ 1 because it is only 1 year
timelinePlot(3,[(-100/(1.05)**1,2),(100,3)])
Now you are going to notice that if we discount that present value of \$95.2 to period 0 we get the same value as discounting \$100 back to period 0.
PV = 100/(1.05)**1
timelinePlot(3,[(-100/1.05**3, 0), (PV, 2)])
timelinePlot(3,[(-100/1.05**3, 0), (100, 3)])
Future Value¶
Future value is the opposite of present value. It is seeing what a dollar today would be worth at some point in the future given a rate that we can invest at. We have seen it before because it is the same as compound interest.
$ FV = PV * (1+r)^t $
where
$ PV = \text{Present Value} $
$ FV = \text{Future Value} $
$ r = \text{Annual rate of return} $
$ t = \text{Number of years between present value and future value} $
Let's apply this to having \$100 with an interest rate of 5% for 5 years.
#Start with present value
PV = 100
#Find the future value
FV = 100 * (1.05) ** 5
timelinePlot(3,[(PV, 0), (FV, 3)])
For investments in the same period, you can add them together. For example, if we had both \$200 in 3 years and \\$100 in 2 years, we may find the present value by adding the two present values together.
#Since both $172.8 and $90.7 are the present day values at time 0, we can actually add them together
#So the total PV of these two investments would then be
PV = 200/(1.05)**3+100/(1.05)**2
timelinePlot(3,[(-PV,0),(200,3),(100,2)])
If you want to calculate the future value, you can either take the present value and multiply $1.05^3$ or you can take 200 and then add 100 * 1.05 because there is one year of compounding that you get between year 2 and year 3.
#Find the future value from the present value
print(PV * 1.05 ** 3)
#Find the future value from the two future cash flows
print(100 * 1.05 + 200)