Calculating the Present Value of a Future Car Purchase

Introduction

Planning for a major expense in the future, such as buying a car, requires careful financial preparation. In this scenario, we will explore how much money one should deposit today into a bank account earning a 10% annual interest rate in order to have $15,000 available for the car purchase in 5 years. We will consider three different scenarios for interest calculations: annually, semi-annually (twice a year), and quarterly (four times a year).

Part A: Annually When interest is compounded annually, the formula for calculating the present value (PV) of a future sum is given by:

��=��(1+�)�PV=(1+r)nFV​

Where:

PV is the present value (the amount you need to deposit today).

FV is the future value ($15,000, in this case).

r is the annual interest rate (10% or 0.10).

n is the number of years (5 years).

Using this formula, we can calculate the present value for this scenario:

PV = \frac{15,000}{(1 + 0.10)^5} = \frac{15,000}{1.61051} \approx $9,311.45

So, if you deposit approximately $9,311.45 into a bank account earning 10% interest annually, you will have $15,000 in 5 years to purchase the car.

Part B: Semi-Annually When interest is compounded semi-annually, we need to adjust the formula slightly. The interest rate and the number of compounding periods (n) will change. The formula becomes:

��=��(1+��)�⋅�PV=(1+mr​)n⋅mFV​

Where:

m is the number of times interest is compounded per year (2 for semi-annual).

Using this formula, we can calculate the present value for this scenario:

��=15,000(1+0.102)5⋅2=15,000(1+0.05)10PV=(1+20.10​)5⋅215,000​=(1+0.05)1015,000​

Now, let’s calculate:

PV = \frac{15,000}{(1.05)^{10}} \approx $8,537.68

So, if you deposit approximately $8,537.68 into a bank account earning 10% interest semi-annually, you will have $15,000 in 5 years for your car purchase.

Part C: Quarterly When interest is compounded quarterly, the formula remains similar to the semi-annual formula, with a change in the number of compounding periods (n) to 4 for quarterly compounding:

��=15,000(1+0.104)5⋅4=15,000(1+0.025)20PV=(1+40.10​)5⋅415,000​=(1+0.025)2015,000​

Now, let’s calculate:

PV = \frac{15,000}{(1.025)^{20}} \approx $8,220.86

In this scenario, you would need to deposit approximately $8,220.86 into a bank account earning 10% interest quarterly to have $15,000 available for your car purchase in 5 years.

Conclusion

To prepare for a car purchase costing $15,000 in 5 years, the amount you need to deposit today varies depending on how often the interest is compounded. For annual compounding, you should deposit approximately $9,311.45; for semi-annual compounding, the required deposit is approximately $8,537.68, and for quarterly compounding, you should deposit approximately $8,220.86. Choosing the right compounding frequency can make a significant difference in how much you need to save today to reach your financial goal in the future.