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Time Value of Money

Key Economic Concepts:

• Interest Rate

Description:

Suppose your brother or sister owed you $500. Would you rather have this money repaid to you right away, in one payment, or spread out over a year in four installment payments? Would it make a difference either way?

Introduction:

Suppose your brother or sister owed you $500. Would you rather have this money repaid to you right away, in one payment, or spread out over a year in four installment payments? Would it make a difference either way?

• According to a concept that economists call the time value of money, you would probably be better off getting your money right away, in one payment. You could invest this money and earn interest on it or you could use this money to pay off a all or part of a loan. There are a million things you could do with this money. The time value of money refers to the fact that a dollar in hand today is worth more than a dollar promised at some future time.
• But how can that be? A dollar is a dollar, isn't it? Yes, but a dollar in hand today can be invested in an interest-bearing account that would grow in value over time. This explains in part why the value of money is related to time.
• The time value of money is related to another concept called opportunity cost. The cost of any decision includes the cost of the best forgone opportunity.  If you pay $10.00 for a movie ticket, your cost of attending the movie is not just the ticket price, but also the time and cost of what else you might have enjoyed doing instead of the movie. Applying this concept to the $500 owed to you, you see that getting the money in installments will saddle you with opportunity cost. By taking the money over time, you lose the interest on your investment or any other use for the initial $500, such as spending it on something you would have enjoyed more.
• The trade-off between money now and money later depends on, among other things, the rate of interest you can earn by investing.

Process:

 First, consider future value. Future value (FV) refers to the amount of money to which an investment will grow over a finite period of time at a given interest rate. Put another way, future value is the cash value of an investment at a particular time in the future. Start by considering the simplest case, a single-period investment.

Investing For a Single Period:

Suppose you invest $100 in a savings account that pays 10 percent interest per year. How much will you have in one year? You will have $110. This $110 is equal to your original principal of $100 plus $10 in interest. We say that $110 is the future value of $100 invested for one year at 10 percent, meaning that $100 today is worth $110 in one year, given that the interest rate is 10 percent.

In general, if you invest for one period at an interest rate r, your investment will grow to (1 + r) per dollar invested. In our example, r is 10 percent, so your investment grows to 1 + .10 = 1.10 dollars per dollar invested. You invested $100 in this case, so you ended up with $100 x 1.10 = $110.

Investing For More Than One Period:

Consider your $100 investment that has now grown to $110. If you keep that money in the bank, what will you have after two years, assuming the interest rate remains the same? You will earn $110 x .10 = $11 in interest after the second year, making a total of $100 + $11 = $121. This $121 is the future value of $100 in two years at 10 percent. Another way of looking at it is that one year from now, you are effectively investing $110 at 10 percent for a year. This is a single-period problem, so you will end up with $1.10 for every dollar invested, or $110 x 1.1 = $121 total.

This $121 has four parts.

• The first part is the first $100 original principal.
• The second part is the $10 in interest you earned in the first year.
• The third part, is the other $10 you earn in the second year, for a total of $120.
• The fourth part is $1 which is interest you earned in the second year on the interest paid in the first year: ($10 x .10 = $1 )

The process of leaving the initial investment plus any accumulated interest in a bank for more than one period is reinvesting the interest. This process is called compounding. Compounding the interest means earning interest on interest so we call the result compound interest. With simple interest , the interest is not reinvested, so interest is earned each period is on the original principal only.

Extension Activity:

Interest on Interest

1. Suppose you locate a two-year investment that pays 14 percent per year. If you invest $325, how much will you have at the end of two years? How much of this is simple interest? How much is compound interest?

   At the end of the first year, you will have $325 x (1 + .14) = $370.50 . If you reinvested this entire amount, and thereby compound the interest, you will have $370.50 x 1.14 = $422.37 at the end of the second year. The total interest you earn is thus $422.37 -- 325 = $97.37. Your $325 original principal earns $325 x.14 = $45.50 in interest each year, for a two-year total of $91 in simple interest. The remaining $97.37 -- 91 = $6.37 results from compounding. How much will you have in the third year?

2. Complete the Columns Below

   End of Year	Amount on Interest	Interest Rate	Interest Earned	Balance
   1	$1,000	10%	$100	$1100
   2	1,100	10%	[110]	[1210]
   3	[1,210]	10%	[121]	[1331]
   4	[1,331]	10%	[133.10]	[1464.10]
   5	[1,464.10]	10%	[146.41]	[1610.51]
   6	[1,610.51]	10%	[161.06]	[1771.56]
   7	[1,771.56]	10%	[177.16]	[1948.72]
   8	[1,948.72]	10%	[194.87]	[2143.59]
   9	[2,143.59]	10%	[214.36]	[2357.95]
   10	[2,357.95]	10%	[235.80]	[2593.75]
   11	[2,593.75]	10%	[259.38]	[2853.13]
   12	[2,853.13]	10%	[285.31]	[3138.44]

   How many years did it take to double your money? [8]

Suppose you go in for an interview for a part-time job. The boss offers to pay you $50 a day for a 5-day, 10-week position OR you can earn only one cent on the first day but have your daily wage doubled every additional day you work. Which option would you take?

Additional funding for this lesson was provided by the Mortgage Bankers' Association of America.

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