Professor JT receives a perpetuity for his retirement that pays $300 at the end of year 6, $600 at the end of year 10, $900 at the end of year 14, with payments continuing to be made every four years thereafter at an amount equal to $300 more than the immediately preceding payment. The present value of the third payment is $535.35. Calculate the present value of this perpetuity.

PV at time zero=$12621.48

Step-by-step explanation:

Looking at the given data we can see that perpetuity is increasing arithmetically with payment after ever 4 years

Starting payment is $300 at time 6

payment reaches to $900 at time 14

Present value of the 3rd payment = $535.35

Note:

Considering the 4 year interval starting time will be at 2 which is half i. e 0.5 of 4 year that is why we are going to use 3.5 years for any time and 0.5 years for time zero

Present value of time 14 at $900 is discounted back to $535.35 for 3.5 years period

PV=P / (1+i) ^n

where:

P is the given value at the time

n is the number of years

i is the interest rate

PV is the present value in given time

Solving above equation for time 4:

535.35 * (1+i) ^3.5=900

i=0.1599≅0.16

So interest i is 16%

Perpetuity immediate at present time:

P/i + Q/i^2

where:

P is $300 which is amount of first period

Q is $300 which is the increment after each interval

300/0.16 + 300/0.16^2 = 13593.75

The above value is discount for 2 years not for 4 years

So in order to find present value at time zero we must discount it back

Now:

PV=P / (1+i) ^n

where:

P is the given value at the time

n is the number of years

i is the interest rate

PV is the present value in given time

PV=$13593.75 / (1=0.16) ^0.5

PV at time zero=$12621.48