Present and Future Value
Suppose you could choose between receiving a new flat screen TV today, and receiving the same TV one month from now. Unless you have unusual preferences, you would choose receiving the TV today. In general, we prefer present consumption over future consumption because we discount the future. How much we should discount the future is determined by the interest rate and the present value formula. The relationship between present and future value is often referred to as “the time value of money,” but it is important to keep in mind that it is what we do with the money (e.g. consumption or investment) that really matters. One of the easiest ways to see this relationship is to imagine you are putting money into a savings account at your local bank. If you put $100 into the account now, you would like to know how much money you will have in one, two and three or more years. This initial $100 is referred to as the principal, and the extra income earned on the principal are interest payments. We normally assume there is compound interest, meaning that the interest earned each period is applied to the principal and the interest income from all previous periods. If the bank calculates and pays compound interest on an annual basis, we can use the following formula:
FVt PV (1 r ) t ,
(1)
where FVt is the future value in t years, PV is the present value and r is the interest rate (or yield) per annum. In this case the present value is $100. If the annual interest rate is 5% then r is expressed as a decimal so that the value of the deposit after one, two and three years will be (rounding to the nearest cent), Year One: Year Two: Year Three:
FV1 PV (1 r ) $100 (1.05) $105
FV2 PV (1 r ) 2 $100 (1.05 ) 2 $110 .25
FV3 PV (1 r ) 3 $100(1.05) 3 $115.76 .
In general, to find the future value in t periods, multiply the present value by (1 + r) to the t power. One can also think of r as a growth rate for any variable which grows at a constant rate. Up to this point, compounding has occurred in discreet periods. Continuous compounding is essentially making the compounding periods infinitely small, so that interest is compounding continuously. In order to calculate future value when using continuous compounding the following equation must be used.
FVt PVe rt
(2)
We can do a simple example using the concept of retirement savings. Assume you save $2,000 for retirement your first year out of college when you are 22 years old and you plan to retire when you are 65 years old (t = 43). Let’s try this with interest rates of 5% and 10%.
5% interest rate: FVt PVe rt 2,000 e (.05*43) $17 ,169 .72 10% interest rate: FVt PVe rt 2,000 e (.10*43) $147 ,399 .59 The $2,000 you saved as a 22 year old will be worth $17,170 if it earns 5% interest and the $2,000 would be worth $147,400 if you can earn 10%. It’s no wonder Albert Einstein supposedly said, “The most powerful force in the universe is compound interest.” Most people can easily grasp the idea of a present value growing through time to become a future value. What comes less easily for some is the intuition behind discounting a future value to arrive at a present value. For any future value we can rearrange formula (1) to calculate a present value,
PV FVt (1 r ) t
.
(3)
For example, if the interest rate is 7%, what is the present value of receiving $150 in ten years? We simply plug the appropriate values into the present value formula to get,
PV $150 $76 .25 . (1.07 )10
The present value formula answers the following question: How much would I need to save now in order to have $150 in ten years? Based on an annual compound interest rate of 7%, the answer is $76.25. As an exercise, you can calculate the present value when the interest rate is 10%. Your answer should confirm that the present value is inversely related to the interest rate. Present value is a useful way to compare different dollar amounts in different time periods. Again you have two options: receiving $200 in two years (option A) and $250 in three years (option B). Which is better? To answer the question we use the market interest rate to convert both options into present value. If the interest rate is 10%, the present values are,
PV A $200 $200 $165 .29 (1.1) 2 1.21
$250 $250 $187 .83 , 3 1.331 (1.1)
PV B
so option B is better. Since the present values will change with different interest rates, we cannot say that option B is always better; only that it is better with an interest rate of 10%. [Exercise: find the interest rate in which someone is indifferent between the two options.]
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