Question

Find the present worth of the following infinite cash flows, using i = 8%/year: $0 for years 1 ~ 3, $500 for even years 4, 6, 8, ..., and $900 for odd years 5, 7, 9, ....

Answer 1

Lets fiirst calculate the value of the cashflows till year 4 (since 1 to 3 are all zero), then we will bring that value till current time to get the present worth.

Now, it is given that the cashflow is 500 for even years and 900 for odd years. Or, we can also look at it as 500 till pretuity each year, and then another cashflow of 400 at each odd year. Its pretty easy to calculate the PW of 500 till prepetuity. That will be

=500/.08 (where .08 is the interest rate)

=6250.

Now what remains is the 400 cashflow every 2 years (starting from year 5). First, lets find an equal cashflow of 2 years whose value would be 400 overall. In other words

400=x+x/1.08

x=207.7.

What we are saying here is that we can convert that 400 once every 2 years to 207.7 every year and the results would be the same. Now effectively we have an infinite cashflow of 207.7 every year, starting at year 5. So, its PW at year 5

=207.7/.08

= 2596.25

But this is at year 5. Lets bring it to year 4

Value at year 4= 2596.25/1.08

= 2403.94.

So now we have the value of all cashflows at year 4, it is

Value of 500 cashflow at year 4+value of 400 each other year cashflow at year 4

=6250+2403.94

= 8653.94

Finally, since this is in year 4, lets bring it to current year.

= 8653.94/(1+8%)⁴

=6360.90