You are expecting to get $3000 in five years. Assume interest rate is 8% and quarterly compounding. 1) What will be the value in year 2? 2) What will be the value in year 8?
Understand the Problem
The question is asking us to calculate the future value of an investment at two different points in time (year 2 and year 8), given that the expected future value in year 5 is $3000, the interest rate is 8% compounded quarterly. We will determine the present value first, and then the future value at year 2 and year 8.
Answer
Future value at year 2 $\approx$ $2365.76 Future value at year 8 $\approx$ $3804.96
Answer for screen readers
The future value at year 2 is approximately $2365.76. The future value at year 8 is approximately $3804.96.
Steps to Solve
- Calculate the quarterly interest rate
The annual interest rate is 8%, and since it's compounded quarterly, we need to divide the annual rate by 4. $$ \text{Quarterly interest rate} = \frac{8%}{4} = 2% = 0.02 $$
- Calculate the number of quarters until year 5
Since the future value is given at year 5, we need to calculate the number of compounding periods (quarters) in 5 years. $$ \text{Number of quarters} = 5 \times 4 = 20 $$
- Calculate the present value (PV)
We know the future value (FV) at year 5 ($3000) and the quarterly interest rate (2%). We can use the future value formula to find the present value (PV): $$ FV = PV (1 + r)^n $$ Where $FV = 3000$, $r = 0.02$, and $n = 20$. Solving for PV: $$ PV = \frac{FV}{(1 + r)^n} = \frac{3000}{(1 + 0.02)^{20}} = \frac{3000}{(1.02)^{20}} \approx \frac{3000}{1.4859} \approx 2019.08 $$ So, the present value is approximately $2019.08.
- Calculate the future value at year 2
Now we need to find the future value at year 2. This means we need to calculate the number of quarters in 2 years: $$ \text{Number of quarters} = 2 \times 4 = 8 $$ Using the present value calculated earlier ($2019.08) and the quarterly interest rate (2%), we can calculate the future value at year 2: $$ FV_2 = PV (1 + r)^n = 2019.08 (1 + 0.02)^8 = 2019.08 (1.02)^8 \approx 2019.08 \times 1.1717 \approx 2365.76 $$ So, the future value at year 2 is approximately $2365.76.
- Calculate the future value at year 8
Next, we need to find the future value at year 8. This means we need to calculate the number of quarters in 8 years: $$ \text{Number of quarters} = 8 \times 4 = 32 $$ Using the present value calculated earlier ($2019.08) and the quarterly interest rate (2%), we can calculate the future value at year 8: $$ FV_8 = PV (1 + r)^n = 2019.08 (1 + 0.02)^{32} = 2019.08 (1.02)^{32} \approx 2019.08 \times 1.8845 \approx 3804.96 $$ So, the future value at year 8 is approximately $3804.96.
The future value at year 2 is approximately $2365.76. The future value at year 8 is approximately $3804.96.
More Information
The calculations show how an initial investment grows over time with compound interest. The earlier you start investing, the more significant the impact of compounding.
Tips
A common mistake is to use the annual interest rate directly without converting it to the quarterly rate. Another mistake is miscalculating the number of compounding periods (quarters). Failing to correctly apply the future value formula is also a common error. Always double-check these values before performing the calculations.
AI-generated content may contain errors. Please verify critical information
