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PRESENT VALUE  MEASURING THE TIME VALUE OF MONEY

Imagine that someone offered to give you \$100 today or \$100 in 10 years, Which would you choose? This is an easy question. Getting \$100 today is better because you can always ‘deposit the money in a bank, still have it in 10 years, and earn interest on the \$100 along the way. The lesson tony today is more valuable than the same amount of money in the future Now consider a harder question: Imagine that someone offered you \$100 today or \$200 in 10 years Which would you choose? To answer this question, you need some way to compare sums of money from different points in time. Economists do this with a concept called present value. The present value of any future sum of money is the amount today that would be needed, at current interest rates, to produce that future sum. To learn how to use the concept of present value, let’s work through a couple of simple examples:

Question: If you put \$100 in a bank account today, how much will it be worth in N years? That is, what will be the future value of this \$loo Answer: Let’s use r to denote the interest rate expressed in decimal form (so an interest rate of 5 percent means r = 0.05). Suppose that interest is paid annually and that the interest paid remains in the bank account to earn more interest-a process called compounding. Then the \$100 will become

(1 + r) X \$100 after 1 year,
(1 + r) X (1 + r) X \$100 = (1 + r”f X \$100  after 2 years,
(1 + r) X (1 + r)X (1 + r) X \$100 = (1 + r)~ X \$100  after 3 years, …
(1 + r}'” X \$100 after  N years..

For example, if we are investing at an interest rate of 5 percent for 10 years, then the future value of the \$100 will be (1.05)10 X \$100, which is \$163.

Question: Now suppose you are going to be paid \$200 in N years. What is the present value of this future payment? That is, how much would you have to deposit in a bank right now to yield \$200 in N years?
Answer: To answer this question, just turn the previous answer on its head. In the last question, we computed a future value from a present value by multiplying by the factor (1 + rYV.To compute a present value from a future value, we divide by the factor (1 + rYV.Thus, the present value of \$200 in N years is \$2001 (1 + rYV. If that amount is deposited in a bank today, after N years it would become (1 + rYV X [\$200/(1 + ryv), which is \$200. For instance, if the interest rate is 5 percent, the present value of \$200 in
10 years is \$200/(1.05)10, which is \$123. This means that \$123 deposited today in a bank account that earned 5 percent would produce \$200 after 10 years.

This illustrates the general formula:

If r is the interest rate, then an amount X to be received in N years has a present value of XI(1 + rYV.

Because the possibility of earning interest reduces the present value below the amount X. the process of finding a present value of a future sum of money is called discounting. This formula shows precisely how much future sums should be discounted Let’s now return to our earlier question: Should you choose \$100 today or \$200 in 10 years? We can infer from our calculation of present value that if the interest rate is 5 percent, you should prefer the \$200 in 10 years. The future \$200 has a present value of \$123, which is greater than \$100. You are better off waiting for the future sum Notice that the answer to our question depends on the interest rate. If the interest rate were 8 percent  then the \$200 in 10 years would have a present value of \$2001(1.08)10, which is only \$93. In this case, you should take the \$100 today. Why should the interest rate matter for your choice? The answer is that the higher the interest rate. the more you can earn by depositing your money at the bank, so the more attractive getting SI()()today becomes.

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