An Insurance Company is offering annuities for a term of 30 years with annual payments at the end of each year. The interest rate is 8% compounded annually. The company also guarantees a 10% increase in annual payments every 5 years, beginning at the end of year 6, and every 5 years thereafter to the 30th and last payment. If we invest $500,000 in this annuity, what would be the first payment? Any help would be appreciated. Thank you.
Because there is an increase of 10% in payments from year 6 and every 5 years thereafter, the payments would look like this: $1 for years 1-5. $1.10 for years 6 -10 inclusive. $1.21 for years 11 - 15 inclusive. $1.331 for years 16 - 20 inclusive. $1.4641 for years 21 - 25 inclusive. $1.61051 for years 26 - 30 inclusive. NOTE: The increase of 10% is compounded, since it is added to the previous increase.
Using cash flow analysis, will have to find the PV of all these increases employing these 2 formulas:
PV=P{[1 + R]^N - 1.[1 + R]^-N} R^-1=PV OF $1 PER PERIOD, and then this common PV formula:
PV=FV[1 + R]^-N=PV OF $1 IN THE FUTURE.
Summing them all up, we get the PV =13.0880063714. Then, we just divide:
$500,000 / 13.0880063714
=$38,202.92 - This is the payment for the first 5 years. Then we have:
=$38,202.92 x 1.1 =$42,023.21 - This is the payment for years 6 -10 inclusive.
=$42,023.21 x 1.1 =$46,225.53 - This is the payment for years 11 - 15 inclusive.
=$46,225.53 x 1.1 =$ 50,848.09 - This is the payment for years 16 - 20 inclusive.
=$50,848.09 x 1.1 =$55,932.90 - This is the payment for years 21 - 25 inclusive.
=$55,932.90 x 1.1 =$61,526.18 - This is the payment for years 26 - 30 inclusive.
And that is it!!.
P.S. I have checked these calculations using amortizations, and the balance of the annuity comes to zero after the 30th and last payment.
