The Time Value of Money: Present and Future Values
The story so far: In the last class, we discussed why NPV was so important. Briefly, if a manager focused on maximizing NPV he would be able to reconcile all the differing demands from his different classes of shareholders. The capital market would take care of the immediate consumption and investment requirements of these shareholders. However, to compute NPV, we needed two things - the cash flows arising from the project or company being valued and the discount rate used to value this project. Before we actually figured out how to compute these two things, we needed to be sure we could use them properly once we had them. So this and the next lesson are devoted to finding out how to compute present and future values ... Now read on ... Required reading: RWJ Chapter 3 or my notes below. Let’s start with a simple question. In 1626, the Algonquin Indians sold Manhattan - 57 km2 of land to Peter Minuit of the Dutch Indian Company for 60 guilders ($24). (The site where the deal was concluded is now the U.S. Custom House on Bowling Green). Was this a bad deal for the Indians? To answer this question, we need to figure out the present value of $24.
Compounding a single lump sum
Suppose we invest 100 dollars in the bank for one year at 10% interest rate. At the end of one year, we have $10 interest plus $100 original principal to get $110.
MGMT 610W: Financial Management Krannert School, Purdue University Raghavendra Rau Finance Department
$110
=
$100 × (1 + .10)
If we now invest the $110 for another year at 10%, we get $11 dollars interest and $110 back as principal to get a total of $121. $121 = $110 × (1 + .10) = $100 × 1.1 × 1.1 = $100 × 1.12
Similarly, for three years, we get: $133.10 = =
$121 × (1 + .10) = $100 × 1.1 × 1.1 × 1.1 $100 × ________
In general, the future value, FVt, of $1 invested today at r% for t periods is given by: FVt = $1 × (1+r)t
The expression (1 + r)t is the future value interest factor. The term r is called the interest rate or discount rate. Example: Suppose you deposit $5,000 today in an account paying 12%. How much will you have in 6 years? How much is simple interest? How much is compound interest? A. Multiply the $5000 by the future value interest factor: $5000 (1 + r) = $5000 × ___________ = $5000 × 1.9738227 = $9869.1135
At 12%, the simple interest is .12 × $5000 = $_____ per year. After 6 years, this is 6 × $600 = $_____ ; the difference between compound and simple interest is thus $_____ - $3600 = $_____ Example 2: In 1934, the first edition of a book described by many as the “bible” of financial statement analysis was published. Security Analysis has proven so popular among financial analysts that it has never been out of print. According to an item in The Wall Street Journal, a copy of the first edition was sold by a rare book dealer in 1996 for $7,500. The original price of the first edition was $3.37. What is the annually compounded rate of increase in the value of the book?
MGMT 610W: Financial Management Krannert School, Purdue University
Raghavendra Rau Finance Department
A: Set this up as a future value (FV) problem. Future value = $7,500 Present value = $3.37 t = 1996 - 1934 = 62 years FV = PV × (1 + r)t so, $7,500 = $3.37 × (1 + r)62 (1 + r)62 = $7,500/3.37 = 2,225.52 Solve for r: r = (2,225.52)1/62 - 1 = .1324 = 13.24% Example 3: You have just won a $1 million jackpot in the state lottery. You can buy a ten year certificate of deposit which pays 6% compounded annually. Alternatively, you can give the $1 million to your brother, who promises to pay you 6% simple interest annually over the ten year period. Which alternative will provide you with more money at the end of ten years? A: The future value of the CD is $1 million × (1.06)10 = $1,790,847.70. The future value of the investment with your brother-in-law, on the other hand, is $1 million + $1 million (.06)(10) = $1,600,000. Thus, compounding (or interest on interest), results in incremental wealth of nearly $191,000. (Of course we haven’t even begun to address the risk of handing your brother $1 million!) What would the advantage to the CD be (if any) if your brother had offered 7.5% simple interest annually? Example 4: Want to be a millionaire? No problem! Suppose you are currently 31 years old, and can earn 10 percent on your money (about what the typical common stock has averaged over the last six decades - but more on that later). How much must you invest today in order to accumulate $1 million by the time you reach age 65? A: Once again, we first define the variables: FV = $1 million r = 10 percent t = 65 - 31 = 34 years PV = ? Set this up as a future value equation and solve for the present value:
MGMT 610W: Financial Management Krannert School, Purdue University Raghavendra Rau Finance Department
$1 million = PV × (1.10)34 PV = $1 million/(1.10)34 = $39,142.51. Of course, we’ve ignored taxes and other complications, but stay tuned - right now you need to figure out where to get $39,000! As we can see, in this example, we know the future value but are using the future value equation to give us the present value. This leads to the second important equation of today - which tells us how to get the present value of a fixed lump sum which will be given to us some time in the future.
Discounting a single lump sum
Example 5: Suppose you need $20,000 in two years to pay your college tuition. If you can earn 8% on your money, how much do you need today? A: Here we know the future value is $20,000, the rate (8%), and the number of periods (2). What is the unknown present amount (called the present value)? From before: FVt = PV × (1 + r)t $20,000 = PV × __________ Rearranging: PV = $20,000/(1.08)2 = $_____________
In general, the present value, PV, of a $1 to be received in t periods when the rate is r is PV = $1/(1+r)t The expression 1/(1 + r)t is called the present value interest factor.
MGMT 610W: Financial Management Krannert School, Purdue University
Raghavendra Rau Finance Department
The following graph shows what is the present value of $1 for different periods and rates.
We can use these two formulae to derive any of the missing numbers in a problem. For example: Example 6: Suppose you deposit $5000 today in an account paying r percent per year. If you will get $10,000 in 10 years, what rate of return are you being offered? A: Set this up as a present value equation: FV = $10,000 PV = $ 5,000 t = 10 years PV = FVt/(1 + r)t $5000 = $10,000/(1 + r)10 Now solve for r: (1 + r)10 = $10,000/$5,000 = 2.00
MGMT 610W: Financial Management Krannert School, Purdue University Raghavendra Rau Finance Department
r = (2.00)1/10 - 1 = .0718 = 7.18 percent
Example 7: Benjamin Franklin died on April 17, 1790. In his will, he gave 1,000 pounds sterling to Massachusetts and the city of Boston. He gave a like amount to Pennsylvania and the city of Philadelphia. The money was paid to Franklin when he held political office, but he believed that politicians should not be paid for their service(!). (The will is interesting in itself since it provides an example of compounding.) Franklin originally specified that the money should be paid out 100 years after his death and used to train young people. Later, however, after some legal wrangling, it was agreed that the money would be paid out 200 years after Franklin’s death in 1990. By that time, the Pennsylvania bequest had grown to about $2 million; the Massachusetts bequest had grown to $4.5 million. The money was used to fund the Franklin Institutes in Boston and Philadelphia. Assuming that 1,000 pounds sterling was equivalent to 1,000 dollars, what rate did the two states earn (the dollar didn’t become the official U.S. currency until 1792)? A. For Pennsylvania, the future value is $________ and the present value is $________ . There are 200 years involved, so we need to solve for r in the following: ________ = _____________/(1 + r )200 (1 + r )200 = ________ Solving for r, the Pennsylvania money grew at about 3.87% per year. The Massachusetts money did better; check that the rate of return in this case was 4.3%. Small differences can add up!
And what happened to the Indians? Did they get a good deal or not?
MGMT 610W: Financial Management Krannert School, Purdue University
Raghavendra Rau Finance Department