Time Value of Money
Everyone realizes that an amount of money at
one point of time has a different value than the same amount
of money at some other point of time. I might ask: "Are
you willing to give me $100 today if I give you $100 in one year?" If
you respond "Yes,"
you really need to learn this material. If you say "No,"
I might then ask: "How much are you willing to give me today
if I promise to give you $100 in one year?" You might be
wondering: "Can I trust this guy? I just met him. Can I
be sure that he will give me the money?" Then you are considering
the issue of risk. For now we will neglect risk and
assume all cash flows are deterministic or without risk. Just
for purposes of this lesson, assume that the deal is without
risk.
Now you might think: "I can't turn down the deal. This professor will probably give me a bad grade in this course." Well, forget it. There's no politics involved here. Just concentrate on the money involved. There will be plenty of reasons you might not do well in this course, and I promise never to really ask you for money.
Well you say: "What about inflation? Because of the economic situation, money usually loses purchasing value with time. A dollar today can't buy what it bought ten years ago. Certainly I can't give you $100 today because when I get it back in a year it won't be worth $100." Well that is pretty observant, but forget inflation for now. Assume that purchasing value does not change with time. We also neglect the effects of government taxes on any profit you might make.
You also might be thinking: "Whatever the
deal, I don't have any money to give you. I'm just a poor student." This
is the issue of the availability of capital. Again,
for purposes of discussion we neglect this matter. For now
assume that you have the capital and the question is whether
you will forego the other things you could do with the money
for the promise of receiving $100 in one year.
You already know that there are many, many issues concerned
with getting and spending money. Although we hear that money
isn't everything, we do spend a lot of time and effort considering
it. It has been said that the very purpose of a publicly owned
corporation is to make money, now and in the future. I do not
really think that money should be the entire purpose of living
or running a business, but for this lesson and much of this
course we isolate money from other issues and concentrate on
the financial aspects of decision making.
The question in the first paragraph involves
the time
value of money. We need some method to move money around
so we can find the equivalent value today of $100
received in the future. The methods introduced in this lesson
are used throughout the remainder of this course, first for
the evaluation of engineering projects and then for the comparison
between alternative project solutions. The methods are also
used for the evaluation of the cash flows associated with
project management. Time value of money also plays a role
for personal finance and every citizen of a modern economy
will benefit from this knowledge.
We answer this question at the bottom of this page. First, some important background.


Goals 



Text 

Chapter 3 in the texbook is the
primary reference for this part of the course. For this lesson
read the Introduction in Section 3.1.


3.1 Engineering
Economic Analysis 




Interest and Compounding 


Use compounding with a specified interest rate to answer questions involving moving money from one time to another. 


Click the Quicktime symbol below to
see the presentation.

One characteristic of biological populations is that they reproduce. People make more people, birds make more birds and viruses make more viruses. The capitalistic view is that money is similar to biology. Money makes money. One reason a business is willing to pay investors to obtain capital is that it thinks that it can use the money to buy knowledge, raw materials and productive capacity to create, produce and sell products. The plan (and hope) is that the receipts from sales will be sufficient so that the business can repay the capital with interest and still retain a profit for the owners of the business.
The term interest refers to a charge for a financial
loan. It is essentially a way to pay the lender for forgoing
the use of the money during the period of the loan. It is also
a payment for the risk involved in permitting the borrower
the use of the lender's money. Interest is stated as a percentage
per time interval. When otherwise unstated, the time interval
is one year, although we will have occasion to use other time
intervals.
The flash movie below shows the growth of an amount invested at time 0 with different interest rates. One possible rule for the growth of value in time is through simple interest when the interest added is proportional to the initial investment. This is called simple interest. Another rule is that interest is proportional to the current principal balance. This is called compound interest. Experiment with the movie to see the effects of different interest rates for the two methods. As an investor should you prefer simple or compounded interest? 


For purposes of this section, we define the notation at the left. We use the period for the interval of time between one interest calculation and the next. Often, as in the example, the period is one year. We consider later when the period is a fraction of a year.


The future value of an invested amount grows linearly with the number of periods invested. For the case shown:


Although simple interest provides a return for the investor, most people would not be satisfied with this arrangement. After the first year the principal (or the value of the investment) would be $110, so the earnings in the second year should be (110)(0.1) = $11. The difference between $10 and $11 doesn't seem like much, but it makes a big difference as time goes on. When the interest is computed on the basis of the total value available each year, the result is called compounded interest. The chart with compounded interest after 10 years is much greater than with simple interest, $259 instead of $200.

The future value of an invested amount grows exponentially with the number of periods invested. For the case shown:

The formulas describe discrete compounding in
that that interest is computed at the end of each period, the
end of each year for the example. An alternative is to use continuous
compounding when interest is accumulated continuously.
Because of numerical simplicity we use discrete compounding
in this class, although continuous compounding is useful in
many practical situations and is convenient for some mathematical
analyses. The compounding period may differ for different situations,
but the interest rate used for computations must be consistent
with the period. Thus if the period is one month, the interest
rate is the rate per month.
Because of the earning power of money, an amount of money invested at one time will provide a different amount at another time. The value depends on time so we use the term time value of money. In the expression above P is the value at time 0, or the present worth. F is the value at time 10, or the future worth. We use this concept of time value of money computed through compounding throughout this class. 

The time value of money depends on the interest rate 

Of course the time value of money depends a great
deal on the interest rate. For those conservative folk who prefer
to protect their money in their mattress or strong box, the interest
rate is 0%. The amount of money available after 10 years is the
same as the amount invested. The money is safe, but there is
no growth. As shown earlier a 10% return yields $259. A 20% return
yields $619, and a 30% return yields a remarkable $1379 in ten
years. A 10% return is not unreasonable for an investor because
10% is near the historic return rate of the U.S. stock market.


The time value of money depends on time 

Since compounding formulas describe exponential growth, the amount of money grows significantly if you wait long enough. If a 20 yearold person puts away $100 until he or she is 70 years old at 10% annual interest, the fund will grow to almost $12,000. Careful saving and exponential growth are the keys to a secure retirement.


Discounting 

Our examples have shown the growth in an investment
due to interest earned. Compounding is a way to move money from
the present to the future. What about going the other way in
time, that is, what is the value of the promise of some future
amount worth today? The answer is provided by the inverse of
compounding, discounting.

The present value of an
invested amount declines inversely to the compound amount
factor. For the case shown:
At 10% interest, the value of $100 promised
in ten years is less than $36 today. The operation
of finding the present value of a future amount
is called discounting.



This discussion should have provided a method to answer my original question.
"How much are you willing to give me today if I promise to give you $100 in one year?"

The discounting formula can be used to answer
this question. You know F and N and are you
are asking for the value of P. But what is the value
of i?
As an investor, you must specify what rate of return that you
require. We call the interest rate used in the computation
the minimum
acceptable rate of return, or MARR. The answer will be
different for different people and it will be different for
different companies.
We will not provide a complete answer to the question. In part it depends on how much capital is available for investment. For an individual willing to invest some money, it is at least the return that could be earned if the $100 were invested elsewhere. For persons with debt, it should be at least the interest charged by the lender. For persons without debt, it should be at least the interest rate that could be earned in a safe investment. Let's say that you are a student with credit card debt charging 1.5% per month on the outstanding balance. This is roughly equivalent to 18% a year. You compute:
You should answer:
"Since my MARR is 18%, I'll give you $84.75."

If your MARR is greater than 18%, the amount
you should be willing to give me should be less than $84.75.
If your MARR is less that 18%, you should give me more. Whatever
you offer, it is then up to me to decide whether I should accept
the deal. That decision depends on my MARR.
For most of this class, the MARR is given in any specific instance. Most organizations set an MARR value, just for questions like this. Time value of money is a rational approach for making decisions about money when the amounts involved occur at different times. We will see how it is used in the next few lessons.


Summary 



Time
Value of Money
Summary 



Return to Top of Page 