Engineering Finance
Lessons


Project Evaluation
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
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  • Be able to compute the growth in money over time with simple and compound interest.
  • Be able to indicate the effect of changes in interest and time on the time value of money.
  • Use compounding to move money from the present to a future time.
  • Use discounting to move money from a future time toward the present.
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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.

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3.1 Engineering Economic Analysis
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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.

Time Value of Money

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?

Growth with 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 year-old 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.

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The Answer to
the Original Question

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.

Division

Summary

 

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Time Value of Money Summary
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Engineering Finance
by Paul A. Jensen
Copyright 2005 - All rights reserved

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