INB - Time Value of Money Lesson
Time Value of Money
Your aunt just gave you an awesome birthday gift -- $1,000! However, she wants to teach you a lesson. You can have your gift now or you can have it a year from now. If you wait, she will give you a bonus (interest)!
Which would you choose? If you are like most people, you want your money and you want it now! Waiting is not something most of us like to do and we need an incentive to do it. Earning interest is one way we are encouraged to wait on our money because our money grows as it earns more interest.
In the last module, we learned the difference between simple and compound interest. With simple interest, we earn only interest on the principal. With compound interest, we earn interest on the principal and any interest we had previously earned. In either case, two things affect the amount of interest we earn: time and interest rate. The rule is this: the longer an investment earns interest and the higher the interest rate, the more money earned.
Since money invested can earn interest, and it earns more money over time, money is said to have a time value. Because money has a time value, we can calculate both its value in the future (future value) and today's value of an amount needed in the future (present value). These values can be calculated using formulas, using time value tables, or using time value calculators (you can find these online). Let's investigate some of these calculations and how they might be used.
Future Value of a Single Sum
Let's go back to your aunt's offer. You decide to accept the money today and plan to put the money in an account that will pay you 5% interest compounded semi-annually. You want to figure out what your account will be worth in five years. The formula to calculate this is: FV=PV(1.00 + I)n
FV = Future Value Amount. This is the amount that the present value will grow to. It can be an account balance or the cost of an item in the future.
PV = Present Value Amount. This could be the amount of a single deposit made a the present time, a present account balance, or the present cost of an item.
I = The interest rate per time period.
N = Number of time periods that interest will be added and compounded over the life of the deposit, cost, etc.
Let's plug in numbers to find the answer to our problem:
FV= What we are trying to find
PV = 1000
I = .025 (semiannual payments, meaning twice a year. Interest rates are reported as annual, so we would pay an equal portion of the interest each time (.05/2 = .025).
So our formula will look like this:
FV = 1000(1 + .025)10
FV = 1000(1.025)10
FV = 1000(1.280)
FV = 1280
Over a five-year period, our deposit would earn us $280 in interest (PV-FV).
Present Value of a Single Sum
If we know we are going to need an amount in the future, we might want to know how much we will need to put in now to have our present value grow to that amount. For this example let's pretend you received a lot of graduation money and you want to put some aside so that you can take a great trip to celebrate your college graduation four years from now, You would really like to have $1500 at that time, and you currently know a place where you can invest it and earn 8% compounded quarterly (4 times a year). Here is the formula to calculate the present value of a single sum: PV = FV(1/(1+i)n)
Since the letters mean the same as they did in the previous formula, let's work on the calculations:
PV = What we are calculating.
FV = 1500
I = 2 (8% a year divided into 4 payments)
N = 16
That means:
PV = 1500(1/(1+.02)16
PV = 1500(1/(1.02)16
PV = 1500(1/1.373)
PV = 1500 (.72833)
PV = 1092.49
Therefore, in order to have that $1500 in four years for your trip, you need to put away $1,092.49 of that graduation money today at 8% compounded quarterly.
Future Value of an Ordinary Annuity
The term annuity refers to a stream of payments (as opposed to one deposit). For instance, you just got your first job and you are planning to put aside $50 of your paycheck each month. To encourage saving, your bank has just started a program where new depositors can receive 12% a year for the first two years their account is open if they make automatic deposits. You want to know how much you will have in the bank at the end of 2 years.
The formula for the Future Value of an Annuity is:
In this formula, C is the amount of your monthly deposit, so our numbers look like this:
FV = the amount we are looking for.
C = 50
I = .01 (12%/12months)
N = 24
So:
How much interest did you earn?
You deposited $1200 ($50 a month for 24 months) and at the end you had $1348.65. So you earned $148.65 in interest ($1348.65 - $1200.00).
Present Value of an Annuity
When we look for the present value of an annuity or stream of payments, we are trying to find out how much we will have to deposit each time to meet a specific goal if we know the length of time, the interest rate, and the compounding frequency. Say your grandfather wanted to make sure you had some help for paying for college. He decides that he would like for you to have $2000 each semester (2 per year) to spend on college expenses. He decides he will set up an annuity at the local bank that pays 5% per year compounded semiannually. He needs to know how much he needs to deposit now, for you to have your money each semester. Let's figure it out. Here is the formula for calculating the present value of an annuity.
So:
PV = the amount we need to calculate
C = 2000
I = .025 ( .05 interest as a decimal/2times a year)
N= 8 ( 2 times a year for 4 years)
Calculating...
By depositing $14,340 in the annuity, you would be able to collect $2000 a semester for your four years of college.
Rule of 72
As you have seen, these formulas can be a bit complicated and even require special calculators to get the answers. However, there is a simple calculation that can help you determine how quickly your money will double. It is called the Rule of 72. Once you know the interest rate involved, just divide the number into 72 and you will know how many years it takes for your money to double.
For instance, if the interest rate was 6%, it would take 12 years for your money to double because 72 divided by 6 is 12. Of course, our number is not precise but it is a good rule of thumb and a way to make quick calculations based on minimal information.
Self Assessment
Determine the time value formula you need for each scenario.
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