Question

PP2:

You've bought an inflation-adjusted annuity to give you a constant real income during your retirement years.

• The annuity will make 20 annual payments.
• The first payment of $80,000 will occur one year from now
• Annual payments will then grow at the rate of inflation, 3%, for 19 more years and then stop. (There are 20 total payments and the last payment is made at the end of year 20.)
• The interest rate is 8%.

What is the present value of this annuity if you do not recieve the first cash flow at the end of year 1?

(Everything else about the annuity remains the same. In other words, the first cash flow you recieve will be $80,000×(1+0.03)$80,000×(1+0.03) at the end of year 2, and the last cash flow you recieve will be $80,000×(1+0.03)19$80,000×(1+0.03)19 at the end of year 20.)

Answer 1

If All Payments received, Present Value of Growing Annuity

Here Annuity = 80,000

r = Interest Rate = 8% = 0.08

g = Growth factor = Inflation Rate = 3% = 0.03 [ Annuity is Growing 3% each Year ]

n = 20 Years

PV = 980,003.3

Now If First Installment was missed

Value of First Installment = 80,000* ( 1+ g) =  80,000* ( 1+ 3%) = 82,400

PV of First Installment = Value of First Installment / ( 1 + Interest Rate) ^ 01

= 82,400 / 1.08 = 70,644.7

So Present Value = Present Value of Growing Annuity -  PV of First Installment

=  980,003.3 -   70,644.7 = 905,929.3

Ans :  present value of this annuity if you do not receive the first cash flow at the end of year 1 = 905,929.3 (Ans)