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MODULE 24: THE TIME VALUE OF MONEY

The purpose of this module is to introduce students to the time value of money. This is important because investments usually require money to be paid in the present, but the firm will not receive earnings from the investment until the future. We need a way to equate dollars in the present with dollars in the future.

Student learning objectives:

• Why a dollar today is worth more than a dollar a year from now.

• How the concept of present value can help you make decisions when costs or benefits come in the future.

Key Economic Concepts For This Module:

• The key economic concept is simply that money in the present is worth more than money in the future. Through the use of several examples of borrowing and lending, we can see that the interest rate (r) is the key to equating future dollars with present dollars.

• The present value of $1 received t years from now is $1/(1+r)t.

• The future value, after t years of time, of $1 invested today is $1*(1+r)t.

Common Student Difficulties:

• The most prevalent student difficulty is simply the math. Several simple examples can overcome this difficulty.

• Another difficulty is that students often have no direct experience as borrowers, lenders, or investors. However the students are investing in something, their human capital. Try to get the students to think about why it is so important to complete school and go on to college. It is so expensive to make this investment, yet so many do. The instructor can talk about how the future returns from the education are similar to the future returns of a firm’s investment in physical capital.

Module 24: The Time Value of Money 219

In-Class Presentation of Module and Sample Lecture

Suggested time: This module can be covered in one hour-long class session, with additional time spent on more examples.

I. The Concept of Present Value

A. Borrowing, Lending, and Interest

B. Defining Present Value

C. Using Present Value

I. The Concept of Present Value

Ask the students: “Suppose you could have $1000 today or $1000 next year? Which would you choose?” $1000 today! Of course, but why?

It would allow me the satisfaction of buying or saving today, rather than waiting.

For example, if I need to buy food or pay my rent, I can’t wait a year to get my hands on that money. The other reason is that if you had the money today, you could put it in the bank and in a year you would have more than $1000.

So for both reasons, $1000 today is worth more than waiting a year to get $1000.

A. Borrowing, Lending, and Interest

Note: it can be useful to get the students to think about lending money to someone for a year. They have a self-interest to start thinking about the benefits of receiving interest rather than paying interest in a borrowing example.

Example:

You are going to lend your friend $100, and he is going to pay you back in one year.

• Assume no inflation, you agree to a 10% interest rate, the going rate you could receive if you had simply saved the money. Why do you need to receive interest on this loan?

• The opportunity cost of lending your friend $100 is the interest you could have earned, $10, after a year had passed.

• So the interest rate measures the cost to you of forgoing the use of that $100.

• Rather than saving it, you could have spent $100 on clothing right now that would have provided immediate benefit to you.

Repayment received on lending $100 for one year = $100 + $100*.10 = $100*(1+.10)

What if you were going to lend your friend the money for two years?

Repayment in two years = $100(1.10)*(1.10) = $121

Generalization:

• Your friend, as a borrower, must pay you $21 to compensate you for the fact that he has your $100 for a period of two years.

• You, as a saver, could put the $100 in the bank today, two years from now you would have $121 to spend on goods and services.

• This implies that you would be completely indifferent between having $100 in your pocket today or $121 two years from today.

• They are equivalent measures of purchasing power, just measured at two different points in time, and it is the interest rate that equates the two.

220 Section 5: Financial Sector

B. Defining Present Value

As the above examples demonstrate, there is a difference between dollars received today and dollars received in the future. We will provide some more specifics to this relationship.

Generalization:

To see the relationship between dollars today (present value PV) and dollars 1 year from now (future value FV), a simple equation is applied:

Future Payment, or FV = PV*(1+r) or, using our example,

FV = $100*(1.10) = $110

In other words, one year into the future, $100 in the present will be worth $110. This is true whether you saved it or lent it to your friend.

We can also rearrange our equation and solve for the present value PV:

PV = FV/(1+r)

Using our example again, PV = $110/(1.10) = $100

This tells us that $110 received a year from now is worth $100 in today’s dollars.

Now let’s look again at the decision to lend the money for a period of t=2 years: Repayment in two years = $100(1.10)*(1.10) = $121

Generalization:

FV = PV(1+r)(1+r) = PV(1+r)t

Or

PV = FV/(1+r)t

• Money today is more valuable than the same amount of money in the future.

• The present value of $1 received one year from now is $1/(1+r).

• The future value of $1 invested today is $1*(1+r).

• Interest paid on savings and interest charged on borrowing is designed to equate the value of dollars today with the value of future dollars.

B. Using Present Value

Decisions often involve dollars spent, or received, at different points in time. We can use the concept of FV to evaluate whether we should commit to a project (or choose between projects) today when benefits may not be enjoyed for several years.

Example:

What if you could invest $10,000 now and receive a guaranteed (after inflation) $20,000 later? Good deal? Maybe.

What if you had to wait 10 years to receive your $20,000?

If I put my $10,000 in an alternative investment earning 8%:

FV = 10,000*(1.08)10 = $21,589.25

What is the $20,000 in 10 years worth today?

PV = 20,000/(1.08)10 = $9263.87

Module 24: The Time Value of Money 221

So you would only have to invest $9263.87 to get $20,000 in 10 years, rather than the aforementioned $10,000.

Either way, you’re wise to pass on this investment opportunity.

Example: Many environmental programs are costly now, but pay off later. Examples: an energy efficient furnace, insulation, or a hybrid car.

A hotel manager needs to replace all of the light bulbs and has heard about these new compact fluorescents. Each CFL costs $15 and each regular incandescent bulb costs $1. The hotel has 1000 fixtures.

The energy bill will be $800 if you use the CFL’s and $4800 if you use the incandescent. They will both last 4 years

Cash Outlays for Investing in Lighting

| | | | |Year | | |

|Option |0 (today) |1 |2 | |3 |4 |Total |

|A. CFL’s |$15,000 |$800 |$800 | |$800 |$800 |$18,200 |

|B. |1000 |4800 |4800 | |4800 |4800 |20,200 |

|Incandescents | | | | | | | |

|B – A: |-$14,000 |4000 |4000 | |4000 |4000 |$2000 |

|Savings from | | | | | | | |

|CFL’s | | | | | | | |

Should you do it? At first it seems like a no-brainer. You should switch the bulbs.

But you could have invested the $14,000 in something else, or just put it in the bank.

Let’s look at the PV of the savings from the initial expenditure to year four.

PV = -14,000/(1+r)0 + 4000/(1+r)1 + 4000/(1+r)2 + 4000/(1+r)3 + 4000/(1+r)4

If PV>0, it makes sense to switch the bulbs.

PV of Savings from CFL’s at 2 Discount Rates

| | | | |Year | | |

|Interest Rate |0 |1 |2 | |3 |4 |Total |

|0% |-$14,000 |4000 |4000 | |4000 |4000 |$2000 |

|5% |-$14,000 |3810 |3628 | |3455 |3290 |+183 |

|10% |-$14,000 |3636 |3305 | |3005 |2732 |-1320 |

The higher the interest rate, the less value is placed upon future dollars (they’re more heavily discounted) and more emphasis is placed upon current dollars.

A higher interest rate makes an alternative (like a simple savings account) more attractive.

222 Section 5: Financial Sector

In-Class Activities and Demonstrations

Students will benefit from additional practice in present value problems, especially those that allow them to see how the interest rate matters and can change decisions.

Grandma’s Generous Graduation Gift

If your grandma said you could have $5,000 when you graduate from college in four years, what is that money worth today?

PV = $5,000/(1+r)4

It depends on the interest rate.

If r=3%, PV = $4,442.43

What do you make of this figure?

You are indifferent between $4,442.43 now and waiting 4 years to receive $5,000.

If r=5%, PV = $4,113.51

If r=10%, PV = $3,415.07

Should you go to college?

Suppose that a person only lives 2 years and has 2 choices. She can go to school in next year, or go straight into the work force. If a person immediately starts working, she will earn $20,000 in both years 1 and 2. If a person goes to school in year 1, she must pay $5,000, but she would earn $47,500 in year 2. If the interest rate is 5%, calculate the present value of a person who goes to school and a person who does not. Does the investment in school make sense? Does it make sense if the interest rate is 6% or 4%?

School:

PV = -5,000/(1.05) + 47,500/(1.05)2 = $38,322.00

Workforce:

PV = 20,000/(1.05) + 20,000/(1.05)2 = $37,188.21

Module 24: The Time Value of Money 223

224 Section 5: Financial Sector

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