Question

Annette has just inherited $180,000. She invests this money at a rate of return of 6.19% per year compounded quarterly. After some period of time, she purchases an annuity that pays $3080 at the beginning of each month for 17 years.

Investing the $180,000 today, how many years until Annette can purchase her annuity? No units are required.

Answer 1

Given that Annette has purchased an annuity that pays $3080 at beginning of each month for 17 years.

Here the annuity price is not given

Generally the annuity price will be the present value of the future cash flows

Therefore Annuity price= PV of all cash flows received every month till 17 years

PV of all cash flows can be calculated using following formula

PV= P+P[1-(1+r)-(n-1)]/r

Where P = periodic payment

r= rate per period

n= number of periods

Periodic payment in this case =$3080

r=6.19%/12 =0.0051583

{ as no rate is specified, normal return of 6.19% is considered and it is divided by 12 to convert it to monthly rate}

n=17 years * 12 months =204 periods

By substituting the above values in the formula

PV= 3080+3080[1-(1+0.0051583)-(204-1)]/0.0051583

=3080+3080[1-(1.00516)-203]/0.0051583

=3080+3080[1-(0.351767)]/0.00516

=3080+3080(0.6483)/0.00516

=3080+387097

=390177

Therefore annuity price is $390177

If Annette has $180000 now and invests at a return of 6.19% compounded quarterly then how many years it would take for this amount to become $390177?

It can be calculated as follows

compound interest formula is A= P(1+r/4)4n

Where A is the amount after compounding=390177

P is the principal =180000

r is the annual rate of return 6.19%

4n indicates number of years to be compounded

By substituting these values we get

390177=180000(1+6.19%/4)^4n

390177/180000=(1+0.015)^4n

2.16765=(1.015)^4n

This above equation can be expressed in terms of log as follows

log 2.167651.015=4n (1.015 is the base)

51.9621=4n

n=12.99

Therefore n=13 years i.e., it takes 13 years for Annette to purchase the annuity she wishes if she invests $180000 today