Present and Future Value

Which would you rather have, fifty dollars now or one hundred dollars in the future?

If someone approached you on the street and asked you this, there would probably be a lot of questions running through your mind, like ‘Why do you want to give me money?’ and ‘Just who are you, anyway?’  If the offer turns out to be genuine, though, another question should come up, ‘When in the future would I get the hundred dollars?’  The value of one hundred dollars to you in the future will depend on how long in the future you need to wait in order to receive it.  If you would get the hundred dollars next week, that’s a good deal; there aren’t any (legal, guaranteed) ways you could invest fifty dollars in order to double its value in seven days.

On the other hand, if you would receive the $100 in ten years, it would be a much wiser course to take the fifty dollars today.  Not only could you safely invest the $50 in a way that it will be worth more than $100 in the future (barring any badly timed downturns), but the one hundred future dollars will not have the same spending power due to the effects of inflation.

This is known as the time value of money; inflation and potential investment gains make the same nominal amount of money more valuable in when received or spent in the present compared to the future.  As a result, many financial advisers and planners will talk about future value, the worth of an investment after a set period of time has passed, and present value, the needed amount of money to invest currently to achieve that future value.  The equation relating these values is:

FV = PV*(1+i)^n

Where FV is future value, PV is present value, i is the interest rate (which could be the rate of return, the inflation rate, or debt interest, depending on the calculation), and n is the number of years between the present and future values.

Confused?  That’s understandable; let’s go through an example to see how all these numbers relate and show how we could calculate each of these values.  In our example, you’re about to inherit a lump sum of money from the estate of your great-uncle Mort.  After attending his funeral (and avoiding any completely unfunny comments about his name), you want to crunch some numbers to see if your inheritance alone will be enough to allow you to retire early in a few decades.  You want to be able to set the inheritance money aside, allowing it to grow, and not have to invest anything more to fund your retirement.

You’re currently twenty-five, and would like to retire when you turn fifty, with twenty-five times your current income in savings (adjusted for inflation).  That way, you think, you will be able to withdraw 4% each year, spend an (inflation-adjusted) amount equal to your current salary, and still have little chance of running low on money.  If you earned $40k last year, you’re shooting for a total retirement fund of $1,000k (an even one million dollars) in present value to completely replace your salary at a 4% withdrawal rate.  If you think there’s going to be a 4% inflation rate over the next few decades (a bit higher than the historic rate), the future value calculation will be:

FV = PV*(1+i)^n = $1,000k*(1+0.04)^25 = $1,000k*(2.772) = $2,772k

You will need $2,772k in order to replace your current income.  Now, let’s assume that in order to achieve this investment goal, you plan to invest in bonds and bond funds, which you expect to return at least 6% throughout the next two and a half decades.  We can rearrange our future value equation in order to determine what present investment value we need to get to this sum:

PV = FV/(1+i)^n = $2,772k/(1+0.06)^25 = $2,772k/4.292 = $646k

If good old Mort left you an inheritance of at least six hundred fifty thousand dollars, you’re sitting pretty; you can simply invest that money, and when you reach fifty, you can live off the proceeds without putting in another dollar towards your investments.  But, to continue our explanation of future value (as well as remind you not to depend on old relatives dying to fund your retirement), let’s assume you didn’t get quite that much; if your inheritance was only $400k, it would take longer to reach your goal.  To figure out how much longer, let’s solve the equation for n, using the same interest rate (be warned, we’re going to have to use natural logarithms to calculate n):

n = [ln(FV/PV)]/[ln(1+i)] = [ln($2,772k/$400k)]/[ln(1+0.06)] = [ln(6.93)]/[ln(1.06)] = 1.94/0.0583 = 33.2 years

If you’re willing to delay your retirement from fifty to fifty-eight, you can use your desired, very conservative investment plan, and still retire early by most people’s standards.  If you want to keep your retirement at age fifty, though, you could add some stocks to your portfolio and increase your expected returns.  To figure out just how much, we’ll solve for i in the final variation on our equation:

i = [(FV/PV)^(1/n)]-1 = [($2,772k/$400k)^(1/25)]-1 = [(6.93)^(0.04)]-1 = [1.081]-1 = 0.081 or 8.1%

Thus, to still build your money up fast enough to meet your retirement plans without adding more funds, you will need to average a return of 8.1% over twenty-five years.  You can probably achieve this level of return if you’re willing to add some stocks to your asset mix; it will be tough to get enough of a return with just bonds (without taking on significant risk with junk bonds or similar riskier bonds).  A good balanced fund will likely provide the level of return that you need.

Now of course, this is a simplified example, using only a single, large investment with compounding only once a year.  If you put in small amounts into your investments on a regular basis, the calculations get trickier.  Still, having an idea of what your investments will be worth when you cash them out or knowing what you need to invest in order to meet your goals is always a plus.


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