Question

20. An annuity due that lasts for 30 years has annual payments of 500 at the beginning of each year for 15 years, and 1000 at the beginning of each year for the followiing 15 years. A perpetuity due has payments of P each year for 20 years and then annual payments of 2P thereafter. The present values of the annuity and the perpetuity are the same if d = 9%. Find P.

Answer 1

Present value of Annuity = 500 * PVAF(9%,0^th year) + PVAF(9%,1 to 14^th yr) *500 + PVAF(9%,15 to 29^th yr) * 1000

= 500*1 + 7.786*500 + 2.412*1000

= 6805.21

Present value of Perpetuity (Assuming payments are made at the beginning of the year)

= P * PVAF(9%,0^th year) + PVAF(9%,1 to 19^th yr) *500 + PVAF(9%,20 to 29^th yr) * 2P

= P*1 + 8.95*P + 1.248 * 2P

= 12.446P

Since PV of Annuity = PV of Perpetuity

which implies that 12.446P = 6805.21

Now, P = 6805.21/12.446

and we have P = 546.76

Note

PVAF(9%, 1 to 9 yr) means summation of present values of year 1 to 9 on a discount rate of 9%.