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What is this Time Value of Money Business?

Time value of money is a probably the most important concept in personal finance.  It is one of the first concepts taught in introductory finance classes in college.  And it is probably the most underestimated and probably the most misunderstood concept in personal finance.

The typical text book explanation of time of value of money is: since we can earn interest on money, money received today is worth more than money received in the future.  True, but if you look at the formulas below this concept can easily be lost.  The thing to remember is that investments/savings grow over time because of interest earned or compound interest so its better to get the money now sock it away and earn interest on that money than to just get some dollars in the future that haven’t earned any interest.  This growth in investment/savings is an the most important example of time value of money.

Definition time:  Compound interest is interest paid on interest and is probably the most powerful tool at our disposal in accumulating wealth.  Compound interest happens when interest paid on an investment is reinvested and added to the principal and then process is started again earning more interest on an even higher principal amount - continuously.  The most important thing about investing/saving is getting compound interest to work in our favor - primarily through saving more.

But there is more to time value of money then just compound interest and understanding how investments grow over time.  Time value of money allows us to strip away the effects of inflation (and other costs/risk) and see what future cash flows are worth in today’s dollars.

FUTURE VALUE

What is the value of our investment at some future point in time?  This is where compound interest comes into play.  Let’s start with a basic formula to see how Future Value of money and compound interest works:

Future Value =  Beginning amount (or Present Value) + Interest Earned
                                                                    FV =  PV + PV(i)
                                                                    FV =  PV(1 + i)

i =  the interest earned on the balance at the beginning of the year and is paid at the end of the year

Notice how i or interest earned is very important to equation and its effect on Future Value.  Now, this the very basic formula for Future Value of just ONE period.  Here is the formula for Future Value that considers compounding:

FV =  PV(1 + i)ⁿ
n = the number of years during which the compounding occurs

Let’s take a look at an example:

Harvey just won $10,000 in the Pick 5 state lottery.  Harvey doesn’t need the money right now and decides to save it for a cruise in 4 years.  He deposits the winnings into savings account earning 2% annual interest compounded annually.  How much will this deposit be worth in 4 years?

FV =  PV(1 + i)ⁿ

= $10,000 (1 + .02)⁴
= $10,000 (1.08243)
= $10,824.32

So, at the end of 4 years Harvey will have $10,824.32 in his savings account.  Obviously, this assumes he don’t make any other deposits or withdrawals.

Compounding periods are not always annual - as in the case above.  The compounding period could be quarterly, daily or even continuous basis.  The important thing to remember is that money grows faster as the compounding period becomes shorter (ex. annual compounding to daily compounding).  That is because interest is earned on interest more frequently as length of compounding period decreases.

PRESENT VALUE

What is the value today of a sum of money to be received in the future?  In Present Value the issue is discount rate instead of compound interest rate.  “Discount rate” is the interest rate used to bring future money back to present.  Let’s look at the formula for Present Value:

We can derive the Present Value formula with a little bit of algebra and the Future Value formula from above to get:

PV =  FV ( 1/(1 + i)ⁿ)

n = number of years until the payment is received
i = annual discount (or interest) rate

The best way to understand Present Value is by using an example:

Say a long lost uncle promises to leave you $500,000 in 30 years.  Assuming the appropriate discount rate (the interest rate used to bring future money back to present) is 3%.  What is the current or Present Value of this $500,000 in 30 years?

PV =  FV ( 1/(1 + i)ⁿ)
  =$500,000 (1/(1+.03)³⁰)
  =$500,000 (.41199)
  =$205,993.38

So that $500,000 30 years in the future is worth $205,993.38 today.

This is a good place to stop.  There is much more to time of value of money.  Present and Future Value are just a start.  The important thing to remember is that the value of money changes as we move through time.

Good luck.