Julie received a large gift from her grandparents and wants to invest enough money today so she can have a dream wedding in 12 years. If her dream wedding will cost $35,000 and Julie can earn 9% compounded daily, how much will she need to invest today?
The correct answer and explanation is:
To find how much Julie needs to invest today, use the present value formula for compound interest: PV=FV(1+rn)ntPV = \frac{FV}{\left(1 + \frac{r}{n}\right)^{nt}}
Where:
- PVPV is the present value (amount to invest today)
- FV=35,000FV = 35,000 is the future value (cost of the wedding)
- r=0.09r = 0.09 is the annual interest rate (9%)
- n=365n = 365 is the number of times interest is compounded per year (daily)
- t=12t = 12 is the number of years
Step-by-step Calculation:
PV=35,000(1+0.09365)365×12PV = \frac{35,000}{\left(1 + \frac{0.09}{365}\right)^{365 \times 12}} PV=35,000(1+0.00024657534)4380PV = \frac{35,000}{\left(1 + 0.00024657534\right)^{4380}} PV=35,000(1.00024657534)4380≈35,0002.853287165PV = \frac{35,000}{\left(1.00024657534\right)^{4380}} \approx \frac{35,000}{2.853287165} PV≈12,269.46PV \approx 12,269.46
Final Answer:
Julie needs to invest approximately $12,269.46 today.
Explanation (300 words):
To determine how much Julie should invest today for her wedding in 12 years, the concept of present value in finance is applied. Present value allows calculation of how much money needs to be invested now in order to reach a specific future amount, taking into account the effect of compound interest over time.
Julie wants to have $35,000 in 12 years. Because she can earn 9% interest per year, compounded daily, the money she invests today will grow each day. Compounding means that each day, interest is added not only on the original amount but also on the accumulated interest. This process increases the growth rate of her investment.
The formula used takes into account the number of compounding periods in a year (365 for daily compounding), the interest rate, and the number of years the investment will grow. By plugging these values into the formula, the future value of $35,000 is discounted back to its equivalent present value.
This calculation shows that Julie doesn’t need to save the full $35,000 today. Instead, she only needs to invest about $12,269.46. Over the next 12 years, with daily compounding at a 9% annual rate, this investment will grow into the desired $35,000. This approach demonstrates how time and interest can work together to increase savings and help achieve financial goals more efficiently.