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April 21, 2014



Future Value of Regular Contributions

If you regularly add money to a 401(k), IRA, or bank account with the goal of withdrawing that money in the future, you need to able to calculate how much your money will be worth at the time of withdraw.  If you made a single, one time contribution, it’s easy enough to calculate the future value by using the appropriate future value calculations.  But what if you are adding money  to the account; how can you determine how much your money will be worth when it is time to withdraw?

If you wanted, you could take each contribution and use the future value calculation to determine how much it will be worth when it is your time to use the money.  However, that would be incredibly tricky to do, as well as requiring more time to do then we really want to expend (even setting up a spreadsheet could take hours).  Luckily, there are ways to simplify the math; one (relatively) simple equation that allows us to combine all the future value calculations into one:

AV = C * [({1+i}^n - 1)/i]

Where AV is accumulated value, C is the contributions, i is the interest rate, and n is the number of periods (usually years, if i is expressed as annual interest).  Crunch the numbers and you can determine how your contributions will add up over time.  Of course, even with this formula, there’s still a lot of number crunching that you might need to do in order to calculate the future value.  If you don’t want to go to all that trouble, I made up a shortcut table to make things easier on you:

accumulation-annuity

This table shows the growth of annual contributions at fixed interest rates.  Take the number of years you want to stay invested, go down to the expected interest rate, and you’ll have the multiplier you need.  For a retirement in forty years and a 10% growth rate, for example, you have a growth factor of 442.6; if you invest $5000 a year, you’ll end up with $2.2 million when you retire.

Putting it all together

How can we use this equation to help plan our retirement goals?  Well, let’s consider someone who is thirty years old, makes $50,000 a year, and has no debt or retirement savings yet.  She decides that she wants to retire with $50,000 a year during retirement (adjusted for inflation) and hopes to retire in thirty years.  She wants to invest fairly agressively for twenty years (at about 10% return) and then dial back her risk for the ten years before retirement (so she’ll only earn 7% return).  She figures she can easily invest $5000 a year without decreasing her standard of living.  How can she determine if she’ll meet her goals?

First, if she uses a safe withdraw rate of 4% of her final portfolio, she should have twenty-five times her desired annual income in her nest egg when she retires, a total of $1.25 million.  Of course, given inflation, the actual amount needed will be much higher; we can use the future value calculation to determine how much money we’ll need in thirty years to be the equivalent of 1.25 million.  We can’t know for certain how much inflation we’ll experience during this time period, so we’ll choose a reasonable but high value (like 5%) to stay conservative:

Future Value = Present Value * (1 + inflation rate)^number of years = $1.25 million*(1+0.05)^30 = $5.4 million

So, we figure on a total of $5.4 million for our example woman to retire.  Quite an obstacle, but we have the power of compound interest on our side.  If look at our table, we see that the intersection of twenty years and 10% interest gives us a factor of 57.3.  With our given contributions of $5000, at the end of the first stage of her investment plan, she’ll have:

Accumulated Value = Contribution * Our Regular Contribution Factor = $5000 * 57.3 = $286,500

Not a bad start, but she needs more than that if she hopes to meet her goals.  For the second part of her retirement plan, she’ll need to calculate her added contributions separately from the money she’s already accumulated.  For the $286,500 she already has saved, the future value calculation at seven percent interest for ten years yields:

FV= PV*(1+i)^n = $286,500*(1+0.07)^10 = $563,600

And her new contributions, $5000 a year for ten years at 7% yield (for a factor of 13.8):

AV = C*Accumulation Factor (AF) = $5000*13.8 = $69,000

For a total of $632,600, well below the $5.4 million she needs to meet her goals.  Now, she could try to run the calculations again, changing some of the assumptions that she made the first time around, with the hopes that by shifting things around a bit, she can increase her nest egg and decrease how much she’ll need.  If she feels she can get by on only 80% of her salary, she can cut the amount she needs to $1 million in present value.  Further, if she assumes that inflation is only at the average rate (about 3%)between now and when she retires, the amount of her desired nest egg will decrease (although, if she is incorrect in her estimates, she can find herself short when retiring).  By delaying her retirement for ten years, she can give herself more time to build up her funds; she can also give her money more time to compound by investing aggressively for thirty years and conservatively for ten.  Finally, if she stretches her budget and invests $10,000 per year, she can increase the power of compounding that will work for her.

Her new desired nest egg:

FV = PV*(1+i)^n = $1 million * (1 + 0.035)^40 = $3.26 million

How her contributions grow over the first thirty years (on our table, thirty years and 10% returns lead to a factor of 164.5):

AV = C*AF = $10,000 * 164.5 = $1.65 million

For the last ten years, this amount of money will grow as follows:

FV = PV*(1+i)^n = $1.65 million*(1+0.07)^10 = $3.25 million

Along with her continued contributions (ten years at seven percent interest gives a factor of 13.8):

AV = C*factor = $10,000*13.8 = $138,000

For a grand total of $3.39 million, more than she estimates she needs (which somewhat makes up for the less strict numbers she used in her initial calculations).  This example shows not only how to run these calculations, but also how a few changes can shift the amount of money you expect to have at retirement significantly.

So, now that you know how to calculate the future value of your regular contributions, how many of you still feel your retirement plans are on track to meet your needs?

Comments

  1. Hello,

    I am trying to write a program in MatLab to give an amount of money saved after a number of months with an initial value, a monthly contibution an some interest rate. I started with a simple for loop to iterate

    new amount = amount(1 + interest) + contribution

    I am trying to optimize my program by calculating the whole vector at once. For some reason it only works when I use.

    FV = PV + C*[({1+i}^n - 1)*i]

    Am I missing something?

    Thank you

  2. @Carlos: I’m not sure why you are having problems; not being familiar with MatLab, I don’t know if it’s something particular to that program. The calculations I have listed are correct as best I can tell, but it’s possible that something is wrong. Sorry I can’t pinpoint any errors.

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