Jack is 45 years old and a small business owner. He intends to retire at 65. To that end, he purchased a stripped coupon for $311,804.73 to mature on his retirement date. On his retirement, he plans to withdraw annual annuity payments from his retirement fund for 25 years. All we know is that his annuity will pay him 6% compounded annually. In addition, his annuity payments will be indexed for inflation, which is forecast to be 2.50% annually. His 25th and last payment will be $111,446.60.
Now, the questions:
1)-How much was Jack’s fund worth on his retirement date?
2)-What was the amount of Jack’s 1st annuity payment?
3)-What was the discount rate of the stripped coupon that he purchased?
4)-How much interest did Jack earn on his stripped coupon?
5)-How much interest did Jack earn on his annuity over its lifetime of 25 years?
P.S. Only standard TVM formulae are used to answer the above 5 questions
Hmm.........Here is a new twist to deferred annuity:
First thing to notice is that the annuity payments are indexed to inflation rate. This will complicate things a bit but not by much. Notice that they give the 25th and the last payment only. But, it should be relatatively easy to get the 1st. payment, since they increase by a constant inflation rate of 2.50%.Since the 25th and last payment is $111,446.60, then it follows that the first payment would be:$111,446.6(1.025)^-24=$ 61,616.08.
This 1st payment of his annuity is already indexed by 2.50%, so that net payment would be:
$61,616.08/1.025=$60,113.25. This is the payment that he would receive each year for 25 years without being indexed for inflation. Now, we simply have to find the PV of these 25 payments, and that should give us the amount of his annuity when he retired. One slight complication is that we have to find the net interest rate to use for this purpose.
Since we know he earns 6% annual compound and the inflation rate is 2.50%, the net investment rate would be: 1.06/1.025=3.415%. Now, we can find the PV of his 25 even annuity payments of $60,113.25 using this common formula:PV=P{[1 + R]^N - 1.[1 + R]^-N} R^-1=PV OF $1 PER PERIOD. When we plug all the numbers into this formula, we get PV=$1,000,000, which is the balance of his fund at his retirement.
And, of course he earned of:$1,000,000 - $311,804.73=$688,195.27 being the interest on his stripped coupon. Since we have both the PV and FV value of his intitial purchase, we can easily find the rate at which he purchased his stripped coupon:1,000,000=311,804.73(1 + i)^20. When we solve for i, we find that it is 6% compounded annually.
The only thing left unanswered is the total interest he earned on his 25-year annuity. But, since we calculated the 1st payment as being $61,616.08, we simply find the FV of these 25 payments at the inflation rate of 2.5%, using this common formula:FV=P{[1 + R]^N - 1/ R}=FV OF $1 PER PERIOD. When we plug all the numbers into this formula we get FV=$2,104,667.52.
Finally, the total interest he earned on his annuity is=$2,104,667.52 - $1,000,000=$1,104,667.52.
This answers all the above 5 questions.
