Final answer:
To determine the better option for Mariam, the lump sum of $625,000 is compared to the future value of a $4,500 monthly annuity over 15 years with a 3.3% annual compound growth rate. The future value of the annuity is calculated, and whichever amount is higher indicates the better deal by showing the difference between the two.
Step-by-step explanation:
To determine which option is better for Mariam, we will compare the lump sum payment of $625,000 to the future value of the annuity of $4,500 per month for the next 15 years. We will need to take into account the growth rate of the economy of 3.3% compounded annually.
Firstly, the annuity needs to be converted to an equivalent annual amount:
- $4,500 × 12 months = $54,000 per year.
Next, we calculate the future value of the annuity using the formula for the future value of an annuity:
Future Value = P × {[(1 + r)^n - 1] / r}
Where:
- P is the periodic payment,
- r is the interest rate per period, and
- n is the number of periods.
For Mariam's case:
- P = $54,000
- r = 0.033 (3.3% expressed as a decimal)
- n = 15 years
Plugging these values into the formula:
Future Value = $54,000 × {[(1 + 0.033)^15 - 1] / 0.033}
After calculating, we compare this future value to the $625,000 lump sum to see which is higher. If the future value of the annuity is greater than $625,000, then the annuity is the better deal for Mariam; otherwise, the lump sum is preferable. We also calculate the difference between the two amounts to see how much better the chosen option is.