Final answer:
To acquire an annuity paying $90,000 annually for 4 years starting 9 years from now at an interest rate of 9%, the client should invest $146,215.86 today. This is found by calculating the present value of the annuity payments and then discounting that value back 8 years to account for the first payment starting in the 9th year.
Step-by-step explanation:
To calculate the present value of an annuity that starts payments in the future (also known as a deferred annuity), we need to use the present value formula for annuities and adjust for the deferral period. Given that the annuity pays $90,000 each year for 4 years, with the first payment occurring 9 years from now and an interest rate of 9%, we can perform the following calculations:
- Calculate the present value of the annuity as if it started today: This is done using the present value formula for an ordinary annuity,
which in this case is:
PV = P * [(1 - (1 + r)^(-n)) / r]
where:
- PV is the present value of the annuity
- P is the payment per period, which is $90,000
- r is the interest rate per period, which is 0.09
- n is the number of periods, which is 4
Calculating the present value of just the annuity portion gives us:
PV = $90,000 * [(1 - (1 + 0.09)^(-4)) / 0.09] = $90,000 * 3.23972 = $291,575.36
- Adjust for the deferred period: Next, we must discount the present value back to today, 8 years before the first payment since we're at the end of the current year, considering the first payment is at the end of the 9th year.
The formula for the present value of a single sum is:
PV = FV / (1 + r)^n
Applying this to adjust the annuity present value, we use $291,575.36 as the future value (FV), r as 0.09, and n as 8:
PV = $291,575.36 / (1 + 0.09)^8 = $291,575.36 / 1.99384 = $146,215.86
Therefore, the client would need to invest $146,215.86 today to receive the annuity described.