A Young Engineer Decides To Save $240 Per Year Toward Retire
A Young Engineer Decides To Save 240 Per Year Toward Retirement In 40
A young engineer commits to saving $240 annually toward retirement over a period of 40 years. The question involves calculating the total accumulated amount at retirement under different investment scenarios, considering variations in interest rates and compounding frequencies. Specifically, we are asked to:
1. Determine the future value of a series of annual $240 payments invested at an annual interest rate of 9%.
2. Calculate the future value if the interest rate is increased to 12%.
3. Assess the amount accumulated if the engineer invests one-quarter of the annual $240, i.e., $60, quarterly in an account earning 12% nominal interest compounded quarterly.
4. Determine the equivalent annual effective interest rate corresponding to this quarterly compounding scenario.
This analysis provides a comprehensive understanding of how different interest rates, compounding frequencies, and investment strategies impact long-term retirement savings.
Paper For Above instruction
The goal of this paper is to evaluate the future value of a series of consistent annual savings investments, emphasizing the effect of different interest rates and compounding periods. These calculations are critical for understanding how savings grow over time and for making informed investment decisions, especially for retirement planning.
Part 1: Future Value of Annual Savings at 9% Interest
The first scenario considers an ordinary annuity where the engineer saves $240 at the end of each year for 40 years at an annual interest rate of 9%. The future value (FV) of an ordinary annuity can be calculated using the formula:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
where:
- \( P \) = annual payment = $240
- \( r \) = annual interest rate = 0.09
- \( n \) = number of years = 40
Plugging in the values:
\[ FV = 240 \times \frac{(1 + 0.09)^{40} - 1}{0.09} \]
Calculating \( (1 + 0.09)^{40} \):
\[ (1.09)^{40} \approx 50.218 \]
Thus,
\[ FV = 240 \times \frac{50.218 - 1}{0.09} = 240 \times \frac{49.218}{0.09} \]
\[ FV = 240 \times 546.867 \approx \$131,288.08 \]
Therefore, if the engineer invests $240 annually at 9%, they will accumulate approximately $131,288.08 by retirement.
Part 2: Future Value at 12% Interest Rate
If the interest rate increases to 12%, the future value becomes:
\[ FV = 240 \times \frac{(1 + 0.12)^{40} - 1}{0.12} \]
Calculating \( (1.12)^{40} \):
\[ (1.12)^{40} \approx 93.051 \]
Then,
\[ FV = 240 \times \frac{93.051 - 1}{0.12} = 240 \times \frac{92.051}{0.12} \]
\[ FV = 240 \times 767.092 \approx \$184,098.99 \]
Consequently, by achieving a 12% interest rate through investment, the engineer could potentially amass approximately $184,099.
Part 3: Quarterly Deposits of $60 in a 12% Nominal Interest Account
The third scenario involves depositing $60 quarterly, totaling $240 annually, into an interest-bearing account earning a nominal annual interest rate of 12%, compounded quarterly. The effective quarterly interest rate is:
\[ i_{quarter} = \frac{0.12}{4} = 0.03 \quad \text{or} \ 3\% \]
The future value of an ordinary annuity with quarterly contributions is:
\[ FV = P_q \times \frac{(1 + i_{quarter})^{m} - 1}{i_{quarter}} \]
where:
- \( P_q \) = quarterly deposit = $60
- \( i_{quarter} \) = 0.03
- \( m \) = total number of quarters = \( 40 \times 4 = 160 \)
Calculating:
\[ FV = 60 \times \frac{(1.03)^{160} - 1}{0.03} \]
Calculate \( (1.03)^{160} \):
\[ (1.03)^{160} \approx e^{160 \times \ln(1.03)} \]
\[ \ln(1.03) \approx 0.0295588 \]
\[ e^{160 \times 0.0295588} = e^{4.729} \approx 113.44 \]
Thus,
\[ FV = 60 \times \frac{113.44 - 1}{0.03} = 60 \times \frac{112.44}{0.03} \]
\[ FV = 60 \times 3748.00 \approx \$224,880 \]
The engineer could accumulate approximately $224,880 through quarterly deposits of $60 in a 12% nominal interest account compounded quarterly over 40 years.
Part 4: Equivalent Annual Effective Interest Rate
To find the annual effective interest rate \( i_{eff} \) corresponding to the quarterly compounding scenario, we use the relationship:
\[ (1 + i_{eff}) = (1 + i_{quarter})^{4} \]
\[ (1 + i_{eff}) = (1.03)^4 \]
Calculating:
\[ (1.03)^4 = 1.1255 \]
Therefore,
\[ i_{eff} = 1.1255 - 1 = 0.1255 \text{ or } 12.55\% \]
The effective annual interest rate equivalent to quarterly compounding at 12% nominal interest is approximately 12.55%.
Conclusion
This analysis highlights the significant impact of interest rates and compounding frequency on long-term savings growth. Increasing the interest rate from 9% to 12% yields substantial additional wealth ($53,000+). Quarterly contributions, with compounding, surpass annual contributions in total accumulated value due to the benefits of more frequent compounding intervals. Moreover, understanding the effective annual interest rate allows investors to compare different investment vehicles accurately, emphasizing the importance of compounding frequency in financial planning.
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