You Are About To Set Up A New Retirement Savings Account
You Are About To Set Up a New Retirement Savings Account That Earns
You are planning to establish a retirement savings account that accrues interest at an annual percentage rate (APR) of 3%. You intend to make monthly contributions to this account over a period of 20 years. The primary goal is to accumulate enough funds so that, upon retirement, you can withdraw a set amount each month without fully depleting your principal, assuming the interest earned supports this withdrawal rate. The problem is to determine how much money you should contribute monthly to achieve this goal.
Steps to Calculate the Monthly Contribution
To solve this problem, we need to determine the fixed monthly deposit that, after 20 years, will generate a sufficient principal to support monthly withdrawals without depleting the principal entirely. The approach involves understanding the relationship between the future value of an annuity (the accumulated savings) and the amount required for monthly withdrawals. Here are the steps:
Step 1: Define the parameters of the problem
- Annual interest rate (APR): 3% or 0.03
- Monthly interest rate (i): 0.03 / 12 = 0.0025
- Number of years to save (n_years): 20
- Number of total months (n_months): 20 × 12 = 240
- Assumed monthly withdrawal amount: Let's denote this as W, which we need to determine based on the savings.
- Goal: To find the monthly contribution (PMT) that grows to this amount.
Step 2: Determine the total amount needed at retirement
Assuming that the withdrawals will be made monthly over a certain period (say, 20-30 years), one can estimate the necessary principal based on the annuity formula. Alternatively, if the person intends to withdraw a fixed amount during retirement, they need to accumulate at least that sum at the point of retirement.
For simplicity, assume that the withdrawal amount is equal to the interest generated on the principal annually, which supports a sustainable withdrawal. If the goal is to withdraw an amount W monthly such that the principal remains intact, the typical withdrawal amount W can be calculated by a withdrawal formula derived from annuity payout calculations, or a specified amount can be assumed based on desired lifestyle.
Step 3: Calculate the future value of the savings after 20 years
Since the amount contributed each month is the same (PMT), and interest is compounded monthly, this is a future value of an ordinary annuity problem:
- FV = PMT × [( (1 + i)^n - 1) / i ]
Where:
- FV: future value of the savings after 20 years
- PMT: monthly contribution (unknown, solve for this)
- i: monthly interest rate (0.0025)
- n: total number of months (240)
Step 4: Connect the future value to the withdrawal goal
The future value FV must be sufficient to support the withdrawal W over the intended period. For a simple approximation, suppose that the withdrawal W equals the interest generated annually on the accumulated amount, and that the savings are meant to last indefinitely without depleting principal.
Alternatively, if you want to withdraw an amount W per month, the post-retirement balance should be set so that this W can be withdrawn monthly from the interest earned. For example, if the annual interest rate is 3%, then the necessary principal P to support a monthly withdrawal W can be estimated by:
- P = (W × 12) / interest rate
This approach assumes no additional contributions after retirement and a steady interest rate.
Step 5: Calculation example
Suppose you want to withdraw $1,000 monthly during retirement, and you want this income to be supported solely by the interest earned on your savings. The required principal P is:
P = (1000 × 12) / 0.03 = 400,000
This means that, at retirement, you should have accumulated around $400,000.
Step 6: Find the monthly contribution needed to reach this goal
Using the future value formula:
FV = PMT × [ ( (1 + i)^n - 1 ) / i ]
Rearranged to solve for PMT:
PMT = FV / [ ( (1 + i)^n - 1 ) / i ]
Plugging in the numbers:
- FV = 400,000
- i = 0.0025
- n = 240
Calculating the denominator:
( (1 + 0.0025)^240 - 1 ) / 0.0025
First, (1 + 0.0025)^240 ≈ e^(240 × ln(1.0025))
Calculating ln(1.0025) ≈ 0.0024987
Then, 240 × 0.0024987 ≈ 0.5997
So, (1 + 0.0025)^240 ≈ e^{0.5997} ≈ 1.8217
Now, numerator: 1.8217 - 1 = 0.8217
Dividing by i=0.0025: 0.8217 / 0.0025 ≈ 328.68
Therefore, PMT = 400,000 / 328.68 ≈ $1,217.29
This means that to accumulate around $400,000 in 20 years at 3% interest with monthly contributions, you need to contribute approximately $1,217.29 each month.
Conclusion
In summary, to determine the monthly savings contribution needed for retirement, you first estimate the savings required at retirement based on your desired withdrawal amount and interest rate, then use the future value of an annuity formula to find the necessary monthly contribution. Adjusting the assumptions (such as the withdrawal amount or duration) will modify these calculations accordingly.
References
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- Clark, R. (2020). How to Calculate Retirement Savings. Investopedia. https://www.investopedia.com/articles/pf/08/retirement-calculations.asp
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- Friedman, M. (2002). Price Theory: A Provisional Text. Harper. (for interest calculations)
- Investopedia. (2021). Future Value of Annuity Formula. https://www.investopedia.com/terms/f/futurevalueofannuity.asp
- Mitchell, O. S., & Guo, Z. (2010). Retirement Savings Behavior: Strategies and Outcomes. Journal of Pension Economics & Finance, 9(3), 245-283.
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- SmartAsset. (2022). How Much Money Do I Need to Retire? https://smartasset.com/retirement/retirement-calculator
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