Question

Serena receives a fifty-year annuity-due that has payments that start at $2,000 and increase by 2.6% per year through the twenty-fourth payment, then stay level at $4,000. Find the accumulated value of this annuity at the end of fifty years if the annual effective rate of interest remains 6.3% throughout the time of the annuity.

Answer 1

First payment=R=$2000

Growth rate=g=2.6%

Rate of interest=i=6.3%

First period of annuity=n1=24

Second period of uniform annuity=n2=26

Uniform payment=C=$4000

It is a case of annuity due i.e. payments are made at the beginning of period.

We first calculate the FV in case of ordinary annuity then multiply it a factor of (1+i) to get the accumulated value of annuity due.

FV of growing ordinary annuity for 24 years is given by

Let us calculate the FV of uniform annuity for 26 years

(F/P,6.3%,26)=(1+6.3%)^26=4.896265

Now let us calculate the FV of annuity due

FV=[(F/P,0.063,26)*FVg+FVc]*(1+0.063)

FV=[4.896265*134137.36+247381.88]*(1+0.063)=961115.64