Overview

This course deals with mineral and oil project evaluation and investment decision-making. We will start by introducing the process of investment decision-making and the compound interest rate method. To make an investment decision, one needs to experience the processes of defining the problem, analyzing the problem, developing alternative solutions, deciding upon the best solution and converting the decision into effective action.

Then, in Lesson 1, the compound interest rate will be covered. Using the compounding method, we can select the appropriate factors to calculate the future value, current value, and the annual value.

One goal of this course is the application of project evaluation methods in the mining and oil industry. Besides the evaluation techniques, this lesson will cover some background and knowledge about the mining and oil industries, through readings of news and papers.

Learning Objectives

At the successful completion of this lesson, students should:

• understand the processes of investment decision-making;
• understand how to use compounding interest rates to calculate present, future, and annual values;
• understand how to apply the evaluation techniques in the mining and oil industry; and
• understand how to use Microsoft Excel for the calculations.

What is due for Lesson 1?

Lesson 1: Reading and Assignment

Reading	Chapter 1 of the textbook by Stermole.
Assignment	Go through the Syllabus, Orientation, and Lesson 1 on the website. Prepare for Homework 1.

Questions?

If you have any questions, please post them to the discussion forum, located under the Modules tab in Canvas. The TA and I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.

Investors make decisions relying on the relative profit potential of investment alternatives. The wrong choices may be made if systematic and quantitative methods are not used. In a given investment situation, it is necessary to consider several economic and technical parameters with respect to costs, profits, savings, the choice of time, tax and loyalty, project life, etc. If a reliable approach is not used to quantify the effects of these factors, it is very difficult to correctly assess each alternative and make the best choice.

The economic viewpoint assumes that capital accumulation is the primary investment objective of capitalistic individuals, companies and societies. From the late 1980s to the late 1990s, it is estimated that more capital investment dollars were spent in the US than were spent cumulatively in the past 200 years in the US. And the numbers in the 2010s were even larger. The importance of proper economic evaluation techniques in determining the most economically-effective way to spend this money seems evident for individuals, companies, and societies. This course presents the development and application of these economic evaluation techniques.

Investment decisions are analyzed over the lifetime of a project which can be decades long, and there are many input data that are related to time such as escalation and inflation of costs and revenues. Therefore, predictions, forecasting, estimations, and assumptions are required for these data which is involved with risk and uncertainty. Consequently, results of the analysis are highly dependent on accuracy and correctness of the proposed inputs. However, the techniques provided in this text can give the decision maker much better ideas about the relative risks and uncertainties between alternatives. This information, along with the numerical economic evaluation results, can help the investors to make a better choice than without using them.

In the majority of cases, making business decisions means dealing with alternative choice problems, which includes selecting the best alternative from several possible choices. The economic evaluation techniques in this course are based on the premise that profit maximization is the investment objective; that is, the alternation that maximizes the future worth of available investment dollars. In general, this involves answering the question, “Is it better to invest cash in a given investment situation, or will the cash earn more if it is invested in an alternative situation?”

Several applicable and useful techniques for evaluating various investment situations will be covered in this course and include future, present, annual value, and break-even analysis. But, the course focuses mainly on methods such as compound interest rate of return (ROR) analysis, as the primary decision-making criterion used by the majority of firms and organizations, and net present value (NPV) analysis, as the second-most used technique, properly applied on an after-tax basis.

Taxes are a cost relevant to most evaluation situations and economic analysis must be done after-tax. This course will cover the scenarios that it is proper to neglect taxes such as government project evaluations where taxes do not apply. Also, the cases with taxes incorporated will also be discussed and analyzed.

There are two main categories of projects or investments that economic evaluation decision-making can be applied to:

1. revenue producing investments
2. service producing investments

A possible third classification, “saving producing projects” will be illustrated later in the course.

Compounding

In order to compare different alternatives in an economic evaluation, they should have the same base (equivalent base). Compound interest is a method that can help applying the time value of money. For example, assume you have 100 dollars now and you put it in a bank for interest rate of 3% per year. After one year, the bank will pay you $100+100*0.03=\$103$. Then, you will put the 103 dollars in the bank again for another year. One year later, you will have $103+103*0.03=\$106.09$. If you repeat this action over and over, you will have:

$$\begin{array}{l}\text{After one year: }100+100*0.03=100*\left(1+0.03\right)=\$103\\ \text{After second year: }103+103*0.03=100*\left(1+0.03\right)+100*\left(1+0.03\right)*0.03\\ =100*\left(1+0.03\right)*\left(1+0.03\right)=100*{\left(1+0.03\right)}^2=\$106.09\\ \text{After third year: }106.09+106.09*0.03=100*{\left(1+0.03\right)}^2 +100*{\left(1+0.03\right)}^2 *0.03\\ =100*{\left(1+0.03\right)}^2 *\left(1+0.03\right)=100*{\left(1+0.03\right)}^3=\$109.27\\ \text{After fourth year: }109.27+109.27*0.03=100*{\left(1+0.03\right)}^3+100*{\left(1+0.03\right)}^3 *0.03\\ =100*{\left(1+0.03\right)}^3 *\left(1+0.03\right)=100*{\left(1+0.03\right)}^4=\$112.57\end{array}$$

Which can be written as:

$$\begin{array}{l}\text{After first year:} P+Pi=P\left(1+i\right)\\ \text{After second year:} P\left(1+i\right)+P\left(1+i\right)i=P\left(1+i\right)\left(1+i\right)=P{\left(1+i\right)}^2\\ \text{After third year:} P{\left(1+i\right)}^2+P{\left(1+i\right)}^{2}i=P{\left(1+i\right)}^2\left(1+i\right)=P{\left(1+i\right)}^3\\ \text{After forth year: }P{\left(1+i\right)}^3+P{\left(1+i\right)}^{3}i=P{\left(1+i\right)}^3\left(1+i\right)=P{\left(1+i\right)}^4\end{array}$$

In general:

The value of money after n^th period of time can be calculated as:

Which F is the future value of money, P is the money that you have at the present time, and i is the compound interest rate.

Example 1-1:

Assume you put 20,000 dollars (principal) in a bank for the interest rate of 4%. How much money will the bank give you after 10 years?

$$F=P{\left(1+i\right)}^n =20,000*{\left(1+0.04\right)}^{10}=20,000*1.48024=29604.8$$

So the bank will pay you 29604.8 after 10 years.

Discounting

In economic evaluations, “discounted” is equivalent to “present value” or “present worth” of money. As you know, the value of money is dependent on time; you prefer to have 100 dollars now rather than five years from now, because with 100 dollars you can buy more things now than five years from now, and the value of 100 dollars in the future is equivalent to a lower present value. That's why when you take loan from the bank, the summation of all your installments will be higher than the loan that you take. In an investment project, flow of money can occur in different time intervals. In order to evaluate the project, time value of money should be taken into consideration, and values should have the same base. Otherwise, different alternatives can’t be compared.

Assume you temporarily worked in a project, and in the end (which is present time), you are offered to be paid 2000 dollars now or 2600 dollars 3 years from now. Which payment method will you chose?

In order to decide, you need to know how much is the value of 2600 dollars now, to be able to compare that with 2000 dollars. To calculate the present value of a money occurred in the future, you need to discount that to the present time and to do so, you need discount rate. Discount rate, i, is the rate that money is discounted over the time, the rate that time adds/drops value to the money per time period. It is the interest rate that brings future values into the present when considering the time value of money. Discount rate represents the rate of return on similar investments with the same level of risk.

So, if the discount rate is i=10% per year, it means the value of money that you have now is 10% higher next year. So, if you have P dollars money now, next year you will have $P+iP =P\left(1+i\right)$ and if you have F dollars money next year, your money is equivalent to $F/\left(1+i\right)$ dollars at present time.

Going back to the example, considering the discount rate of 10%:

We can calculate the present value of $2600 occurred 3 years from now by discounting it year by year back to the present time:

Value of 2600 dollars in the 2^nd years from now $=2600/\left(1+0.1\right)=2363.64$
Value of 2600 dollars in the 1^st years from now $=\left(2600/\left(1+0.1\right)\right)/\left(1+0.1\right)=2600/\left[{\left(1+0.1\right)}^2\right]=2148.76$
Value of 2600 dollars at the present time $=\left(\left(2600/\left(1+0.1\right)\right)/\left(1+0.1\right)\right)/\left(1+0.1\right)=2600/\left[{\left(1+0.1\right)}^3\right]=1953.42$

So, it seems at the discount rate of i=10%, present value of 2600 dollars in 3 years equals 1953.42 dollars, and you are better off, if you accept the 2000 dollars now.

With the following fundamental equation, present value of a single sum of money in any time in the future can be calculated. It means a single sum of money in the future can be converted to an equivalent present single sum of money, knowing the interest rate and the time. This is called discounting.

P: Present single sum of money.
F: A future single sum of money at some designated future date.
n: The number of periods in the project evaluation life (can be year, quarter or month).
i: The discount rate (interest rate).

Example 1-2:

Assuming the discount rate of 10 %, present value of 100 dollars which will be received in 5 years from now can be calculated as:

$$\begin{array}{l}F=100dollars\\ n =5\\ i =0.1\\ P=F\left[1/{\left(1+i\right)}^n\right]= 100\left[1/{\left(1+0.1\right)}^5\right]=62.1\end{array}$$

You can see how time and discount rate can affect the value of money in the future. 62.1 dollars is the equivalent present sum that has the same value of 100 dollars in five years under the discount rate of 10%

Example 1-2 was about one single sum; what if you want to add some savings to your bank account each year? So, we need to learn some more techniques to be prepared for real-world economic evaluations. First, take a look at Figure 1-2. It can help us to better understand the investment evaluation problems.

Figure 1-2: Time diagram

The horizontal line represents the time. The left-hand end shows the present time and the right-hand end shows the future. The numbers below the line (1, 2, 3, …, n) are time periods. Above each time period, there is a sum A, which shows the money that occurs in that time period; here, we assume all of them are equal payments, so:

A is a uniform series of equal payments at each compounding period;
P is a present single sum of money at the time zero;
F is a future sum of money at the end of period n. And i is the compound interest rate.

In order to understand an economic evaluation problem we have to determine:

• How much money is given?
• When is the money given (where on the timeline)?
• What is the time period (year, quarter, or month)?
• What is the interest rate?
• What needs to be calculated?

Following these steps, we just need to use the proper equation to solve the problem. Based on the unknown (asked) variable, there are six basic categories of problems here:

1. F (future value) needs to be calculated from given P
2. F (future value) needs to be calculated from given A
3. P (present value) needs to be calculated from given F
4. P (present value) needs to be calculated from given A
5. A (uniform and equal period values) needs to be calculated from given F
6. A (uniform and equal period values) needs to be calculated from given P

Table 1-1 displays a method of notation that can help summarize the given information and avoid confusion.

Table 1-1: Variable relationship between P, F, and A and the Appropriate Factor

	To be Calculated Quantity	Given Quantity	Appropriate Factor (symbol)	Relationship
1	F	P	$F/P_{i,n}$	$\begin{array}{r}\hfill F=P*F/P_{i,n}\end{array}$
2	P	F	$P/F_{i,n}$	$P=F* P/F_{i,n}$
3	F	A	$F/A_{i,n}$	$F=A*F/A_{i,n}$
4	A	F	$A/F_{i,n}$	$A=F*A/F_{i,n}$
5	P	A	$P/A_{i,n}$	$P=A*P/A{ }_{i,n}$
6	A	P	$A/P_{i,n}$	$A=P*A/P_{i,n}$

Note: “/” in the Appropriate Factor (symbol) column is not a division operator, the entire $F/P_{i,n}$or $F/A_{i,n}$, … is a factor (symbol). The first letter shows the variable that needs to be calculated and the second letter shows the given variable. The two subscripts on each factor are the given period interest rate, i, followed by the number of interest compounding periods, n.

The new notation helps us summarize the problem. The factor actually give a gives us a coefficient that when multiplied by given parameter, gives the unknown parameter.

All time value of money calculations involves writing an equation or equations to calculate F, P, or A. Each of terms in the column “Appropriate Factor (symbol)” has a name that you will learn later in this course.

Please watch the following (4:32) video:

1. Single Payment Compound-Amount Factor

The first category of six categories that were introduced explains the situation that the present value of money is given and asks you to calculate the future value according to the given interest rate of i per period and n period from now. This problem can be summarized with the factor (symbol) of $F/P_{i,n}$and can be shown as:

Figure 1-3: Single Payment Compound-Amount Factor, F/P_i,n

As explained earlier, the future value of money after n period with an interest rate of i can be calculated using the Equation 1-1: $F=P{\left(1+i\right)}^n$ which can also be written regarding Table 1-1 notation as: $F=P*F/P_{i,n}$. The mathematical expression ${\left(1+i\right)}^n$ is called the “single payment compound-amount factor."

3. Uniform Series Compound-Amount Factor

The third category of problems in Table 1-5 demonstrates the situation that equal amounts of money, A, are invested at each time period for n number of time periods at interest rate of i (given information are A, n, and i) and the future worth (value) of those amounts needs to be calculated. This set of problems can be noted as $F/A_{i,n}$ . The following graph shows the amount occurred. Think of it as this example: you are able to deposit A dollars every year (at the end of the year, starting from year 1) in an imaginary bank account that gives you i percent interest and you can repeat this for n years (depositing A dollars at the end of the year). You want to know how much you will have at the end of year n^th.

0		A	A		A	A	F=?
	0	1	2	...	n-1	n

Figure 1-4: Uniform Series Compound-Amount Factor, $F/A_{i,n}$

In this case, utilizing Equation 1-2 can help us calculate the future value of each single investment and then the cumulative future worth of these equal investments.

Future value of first investment occurred at time period 1 equals $A{\left(1+i\right)}^{n-1}$
Note that first investment occurred in time period 1 (one period after present time) so it is n-1 periods before the n^th period and then the power is n-1.

And similarly:
Future value of second investment occurred at time period 2: $A{\left(1+i\right)}^{n-2}$
Future value of third investment occurred at time period 3: $A{\left(1+i\right)}^{n-3}$
Future value of last investment occurred at time period n: $A{\left(1+i\right)}^{n-n}=A$
Note that the last payment occurs at the same time as F.

So, the summation of all future values is
$$F=A{\left(1+i\right)}^{n-1}+A{\left(1+i\right)}^{n-2}+A{\left(1+i\right)}^{n-3}+\dots +A$$

By multiplying both sides by (1+i), we will have
$$F\left(1+i\right)=A{\left(1+i\right)}^n+A{\left(1+i\right)}^{n-1}+A{\left(1+i\right)}^{n-2}+\dots +A\left(1+i\right)$$

By subtracting first equation from second one, we will have
$$\begin{array}{l}F\left(1+i\right)–F=A{\left(1+i\right)}^n+A{\left(1+i\right)}^{n-1}+A{\left(1+i\right)}^{n-2}+\dots \\ +A\left(1+i\right)–\left[A{\left(1+i\right)}^{n-1}+A{\left(1+i\right)}^{n-2}+A{\left(1+i\right)}^{n-3}+\dots +A\right]\\ F+Fi–F=A{\left(1+i\right)}^n+A{\left(1+i\right)}^{n-1}+A{\left(1+i\right)}^{n-2}+\dots \\ +A\left(1+i\right)–A{\left(1+i\right)}^{n-1}-A{\left(1+i\right)}^{n-2}-A{\left(1+i\right)}^{n-3}-\dots -A\end{array}$$

which becomes:
$$Fi=A{\left(1+i\right)}^n–A$$

then

Therefore, Equation 1-3 can determine the future value of uniform series of equal investments as $F=A\left[{\left(1+i\right)}^n-1\right]/i$ . Which can also be written regarding Table 1-5 notation as: $F=A*F/A_{i,n}$. Then $F/A_{i,n }=\left[{\left(1+i\right)}^n-1\right]/i$.
The factor $\left[\left(1+i\right)n-1\right]/i$ is called “Uniform Series Compound-Amount Factor” and is designated by F/A_i,n. This factor is used to calculate a future single sum, “F”, that is equivalent to a uniform series of equal end of period payments, “A”.

Note that n is the number of time periods that equal series of payments occur.

Please review the following video, Uniform Series Compound-Amount Factor (3:42).

Example 1-3:

Assume you save 4000 dollars per year and deposit it at the end of the year in an imaginary saving account (or some other investment) that gives you 6% interest rate (per year compounded annually), for 20 years. How much money will you have at the end of the 20th year?

0		$4000	$4000		$4000	$4000	F=?
	0	1	2	...	19	20

So
A =$4000
n =20
i =6%
F=?

Please note that n is the number of equal payments.

Using Equation 1-3, we will have
$$\begin{array}{l}F=A*F/A_{i,n}=A\left[{\left(1+i\right)}^n-1\right]/i\\ F=A*F/A_{6\%,20}=4000*\left[{\left(1+0.06\right)}^{20}-1\right]/0.06\\ F=4000*36.78559=147142.4\end{array}$$

So, you will have 147,142.4 dollars at 20th year.

Table 1-8: Uniform Series Compound-Amount Factor

Factor	Name	Formula	Requested variable	Given variables
F/Ai,n	Uniform Series Compound-Amount Factor	$\left[{\left(1+i\right)}^n-1\right]/i$	F: Future value of uniform series of equal investments	A: uniform series of equal investments n: number of time periods i: interest rate

4. Sinking-Fund Deposit Factor

The fourth group in Table 1-5 is similar to the third group but instead of A as given and F as unknown parameters, F is given and A needs to be calculated. This group illustrates the set of problems that ask you to calculate uniform series of equal payments (or investment), A, to be invested for n number of time periods at interest rate of i and accumulated future value of all payments equal to F. Such problems can be noted as $A/F_{i,n}$ and are displayed in the following graph. Think of it as this example: you are planning to have F dollars in n years and there is a saving account that can give you i percent interest. You want to know how much you have to deposit every year (at the end of the year, starting from year 1) to be able to have F dollars after n years.

0		A=?	A=?		A=?	A=?	F
	0	1	2	...	n-1	n

Figure 1-5: Sinking-Fund Deposit Factor, $A/F_{i,n}$

Equation 1-3 can be rewritten for A (as unknown) to solve these problems:

Equation 1-4 can determine uniform series of equal investments, A, given the cumulated future value, F, the number of the investment period, n, and interest rate i. Table 1-5 notes these problems as: $A=F*A/F_{i,n}$. Then $A/F_{i,n}=i/[{\left(1+i\right)}^n-1]$. The factor $i/[{\left(1+i\right)}^n-1]$ is called the “sinking-fund deposit factor”, and is designated by $A/F_{i,n}$ . The factor is used to calculate a uniform series of equal end-of-period payments, A, that are equivalent to a future sum F.

Note that n is the number of time periods that equal series of payments occur.

Please watch the following video, Sinking Fund Deposit Factor (4:42).

Example 1-4:

Referring to Example 1-3, assume you plan to have 200,000 dollars after 20 years, and you are offered an investment (imaginary saving account) that gives you 6% per year compound interest rate. How much money (equal payments) do you need to save each year and invest (deposit it to your account) in the end of each year?

0		A=?	A=?		A=?	A=?	F=200,000
	0	1	2	...	19	20

So
F=$200,000
n=20
i=6%
A=?

Using Equation 1-4, we will have
$$\begin{array}{l}A=F*A/F_{i,n }=F\left\{i/\left[{\left(1+i\right)}^n-1\right]\right\}\\ A=F*A/F_{6\%,20}=200,000*0.06/[{\left(1+0.06\right)}^{20}-1]\\ A=200,000*0.027185=5436.912\end{array}$$

So, in order to have 200,000 dollars at 20th year, you have to invest 5,436.9 dollars in the end of each year for 20 years at annual compound interest rate of 6%.

Table 1-9: Sinking-Fund Deposit Factor

Factor	Name	Formula	Requested variable	Given variables
$A/F_{i,n}$	Sinking-Fund Deposit Factor	$i/\left[{\left(1+i\right)}^n-1\right]$	A: Uniform series of equal end-of-period payments	F: cumulated future value of investments n: number of time periods i: interest rate

Note that $i/\left[{\left(1+i\right)}^n-1\right]$

5. Uniform Series Present-Worth Factor

The fifth group in Table 1-5 covers a set of problems that uniform series of equal investments, A, occurred at the end of each time period for n number of periods at the compound interest rate of i. In this case, the cumulated present value of all investments, P, needs to be calculated. In summary, P is unknown and A, i, and n are given parameters. And the problem can be noted as $P/A_{i,n}$ and displayed as:

P=?		A	A		A	A	0
	0	1	2	...	n-1	n

Figure 1-6: Uniform Series Present-Worth Factor, $P/A_{i,n}$

If we replace substitute F in Equation 1-3 from Equation 1-2, we will have the present value as:

$$\begin{array}{l}\text{Equation 1-3:} F=A\left[{\left(1+i\right)}^n-1\right]/i\\ \text{Equation 1-2:} F=P{\left(1+i\right)}^n\\ P{\left(1+i\right)}^n =A\left[{\left(1+i\right)}^n-1\right]/i\end{array}$$

then,

Equation 1-5 gives the cumulated present value, P, of all uniform series of equal investments, A, as $P=A\left[{\left(1+i\right)}^n-1\right]/[i{\left(1+i\right)}^n]$. And also can be noted as: $P=A*P/A_{i,n}$ .The factor $\left[{\left(1+i\right)}^n-1\right]/\left[i{\left(1+i\right)}^n\right]$is called the “uniform series present-worth factor” and is designated by $P/A_{i,n}$ . This factor is used to calculate the present sum, P that is equivalent to a uniform of equal end of period payments, A. Then $P/A_{i,n}=A\left[{\left(1+i\right)}^n-1\right]/\left[i{\left(1+i\right)}^n\right]$

Note that n is the number of time periods that equal series of payments occur.

Please review the following video, Uniform Series Present Worth Factor (Time 3:35).

Example 1-5:

Calculate the present value of 10 uniform investments of 2000 dollars to be invested at the end of each year for interest rate 12% per year compound annually.

P=?		A=$2000	A=$2000		A=$2000	A=$2000	0
	0	1	2	...	9	10

So,
A =$2000
n =10
i =12%
P=?

Using Equation 1-5, we will have:
$$\begin{array}{l}P=A*P/A_{i,n}=A\left[{\left(1+i\right)}^n-1\right]/\left[i{\left(1+i\right)}^n\right]\\ P=A*P/A_{12\%,10}=2000*\left[{\left(1+0.12\right)}^{10}-1\right]/\left[0.12{\left(1+0.12\right)}^{10}\right]\\ P=2000*5.650223=\$11,300.45\end{array}$$

Note that we use the factor $P/A_{i,n}$ when we have equal series of payments. i is the interest rate and n is the number of equal payments. There is an important assumption here, the first payment has to start from year 1. In that case $P/A_{i,n}$ will return the equivalent present value of the equal payments.

Now let's consider the case that we have equal series of payments and the first payment doesn't start from year 1. In that case the factor $P/A_{i,n}$ will give us the equivalent single value of equal series of payments in the year before the first payment. However, we want the present value of them (at year 0). So, we need to multiply that with the factor $P/F_{i,n}$ and discount it to the present time (year 0).
 

Example:

P=?			A=$2000		A=$2000	A=$2000	0
	0	1	2	...	10	11

Note that there are 10 equal series of $2,000 payments. But the first payment is not in year 1. The factor $P/A_{12\%,10}$ returns the equivalent value of these 10 payments to the year before the first payment, which is year 1.

P=?		$2000(P/A12%,10)					0
	0	1	2	...	10	11

However, we want the present value. So, we need to discount the value by one year to have the present value of 10 equal payments.

P=?	$2000(P/A12%,10)(P/F12%,1)						0
	0	1	2	...	10	11

$$\text{Present value}=2,000\left(P/A_{12\%,10}\right)\left(P/F_{12\%,1}\right)$$

Example: Now consider the the following case that the first payment starts at year 3:

P=?				A=$2000		A=$2000	A=$2000	0
	0	1	2	3	...	10	12

$$\text{Present value}=2,000\left(P/A_{12\%,10}\right)\left(P/F_{12\%,2}\right)$$

Table 1-10: Uniform Series Present-Worth Factor, P/A_i,n

Factor	Name	Formula	Requested variable	Given variables
$P/A_{i,n}$	Uniform Series Present-Worth Factor	$\left[{\left(1+i\right)}^n-1\right]/\left[i{\left(1+i\right)}^n\right]$	P: Present value of uniform series of equal investments	A: uniform series of equal investments n: number of time periods i: interest rate

6.Capital-Recovery Factor

The sixth group in Table 1-5 belongs to set of problems that A is unknown and P, i, and n are given parameters. In this category, uniform series of an equal sum, A, is invested at the end of each time period for n periods at the compound interest rate of i. In this case, the cumulated present value of all investments, P, is given and A needs to be calculated. It can be noted as $A/P_{i,n}$ .

P		A=?	A=?		A=?	A=?	0
	0	1	2	...	n-1	n

Figure 1-7: Capital-Recovery Factor,$A/P_{i,n}$

Equation 1-5 can be rewritten for A (as unknown) to solve these problems:

Equation 1-6 determines the uniform series of equal investments, A, from cumulated present value, P, as $A=P\left[i{\left(1+i\right)}^n\right]/\left[{\left(1+i\right)}^n-1\right]$. The factor $\left[i{\left(1+i\right)}^n\right]/\left[{\left(1+i\right)}^n-1\right]$ is called the “capital-recovery factor” and is designated by A/P_i,n. This factor is used to calculate a uniform series of end of period payment, A that are equivalent to present single sum of money P.

Note that n is the number of time periods that equal series of payments occur.

Please watch the following video, Capital Recovery Factor (Time 3:37).

Example 1-6:

Calculate uniform series of equal investment for 5 years from present at an interest rate of 4% per year compound annually which are equivalent to 25,000 dollars today. (Assume you want to buy a car today for 25000 dollars and you can finance the car for 5 years with 4% of interest rate per year compound annually, how much you have to pay each year?)

P=$25,000		A=?	A=?	A=?	A=?	A=?	0
	0	1	2	3	4	5

Using Equation 1-6, we will have:
$$\begin{array}{l}A=P*A/P_{i,n}=P\left[i{\left(1+i\right)}^n\right]/\left[{\left(1+i\right)}^n-1\right]\\ A=P*A/P_{4\%,5}=25,000*\left[0.04{\left(1+0.04\right)}^5/\left[{\left(1+0.04\right)}^5-1\right]\right]\\ A=25,000*0.224627=5615.68\end{array}$$

So, having $25,000 at the present time is equivalent to investing $5,615.68 each year (at the end of the year) for 5 years at annual compound interest rate of 4%.

Table 1-11: Capital-Recovery Factor, A/P_i,n

Factor	Name	Formula	Requested variable	Given variables
$A/P_{i,n}$	Capital-Recovery Factor	$\left[i{\left(1+i\right)}^n\right]/\left[{\left(1+i\right)}^n-1\right]$	A: uniform series of equal investments	P: Present value of uniform series of equal investments n: number of time periods i: interest rate

Note that

$$A/P_{i,n}=A/F_{i,n}*F/P_{i,n}=P\left[i{\left(1+i\right)}^n\right]/\left[{\left(1+i\right)}^n-1\right]$$

Using these six techniques, we can solve more complicated questions.

Example 1-7:

Assume a person invests 1000 dollars in the first year, 1500 dollars in the second year, 1800 dollars in the third year, 1200 dollars in the fourth year and 2000 dollars in the fifth year. At an interest rate of 8%:
1) Calculate time zero lump sum settlement “P”.
2) Calculate end of year five lump sum settlement “F”, that is equivalent to receiving the end of the period payments.
3) Calculate five uniform series of equal payments "A", starting at year one, that is equivalent to above values.

P=?		1000	1500	1800	1200	2000	F=?
	0	1	2	3	4	5

1) Time zero lump sum settlement “P” equals the summation of present values:

$$\begin{array}{l}P=1000*(P/F_{8\%,1})+1500*(P/F_{8\%,2})+1800*(P/F_{8\%,3})+1200*(P/F{8}_{\%,4})+2000*(P/F_{8\%,5})\\ P=1000*0.92593+1500*0.85734+1800*0.79383+1200*0.73503+2000*0.68058\\ P=5884.03\end{array}$$

2) End of year five lump sum settlement “F”, that is equivalent to receiving the end of the period payments equals the summation of future values:

$$\begin{array}{l}F=1000*\left(F/P_{8\%,(5-1)}\right)+1500*\left(F/P_{8\%,}{}_{(5-2)}\right)+1800*\left(F/P_{8\%,}{}_{(5-3)}\right)+1200*\left(F/P_{8\%,}{}_{(5-4)}\right)+2000\\ F=1000*\left(F/P_{8\%,4}\right)+1500*\left(F/P_{8\%,3}\right)+1800*\left(F/P_{8\%,2}\right)+1200*\left(F/P_{8\%,1}\right)+2000\\ F=1000*1.36049+1500*1.25971+1800*1.1664+1200*1.08+2000\\ F=8645.58\end{array}$$

Please note that in the factor subscript, n is the number of time period difference between F (the time that future value has to be calculated) and P(the time that the payment occurred). For example, 1800 payment occurs in year 3 but we need its future value in year 5 (2 year after) and time difference is 2 years. So, the proper factor would be:$\left(F/P_{8\%,}{}_{(5-3)}\right)\text{ or }\left(F/P_{8\%,2}\right)$ .

3) Uniform series of equal payments "A" can be calculated from either P or F :
$$A=5884.03*A/P_{8\%,5}=5884.03*0.25046=1473.7$$ or
$$A=8645.58*A/F_{8\%,5} =8800.71*0.17046=1473.7$$

Example 1-8: repeat your calculations for the following payments:

P=?	800	1000	1000	1600	1400		F=?
	0	1	2	3	4	5

1) Time zero lump sum settlement “P” equals the summation of present values: $$\begin{array}{l}P=800+1000*\left(P/F_{8\%,1}\right)+1000*\left(P/F_{8\%,2}\right)+1600*\left(P/F_{8\%,3}\right)+1400*\left(P/F_{8\%,4}\right)\\ P=800+1000*0.92593+1000*0.85734+1600*0.79383+1400*0.73503\\ P=4882.44\end{array}$$

2) End of year five lump sum settlement “F”, that is equivalent to receiving the end of the period payments equals the summation of future values: $$\begin{array}{l}F=800*\left(F/P_{8\%,5}\right)+1000*\left(F/P_{8\%,4}\right)+1000*\left(F/P_{8\%,3}\right)+1600*\left(F/P_{8\%,2}\right)+1400*\left(F/P_{8\%,1}\right)\\ F=800*1.46933+1000*1.36049+1000*1.25971+1600*1.1664+1400*1.08\\ F=7173.9\end{array}$$

3) Uniform series of equal payments "A" can be calculated from either P or F:
$$\begin{array}{l}A=4882.44*A/P_{8\%,5}=4882.44*0.25046=1222.84\\ \end{array}$$ or
$$A=7173.9*A/F_{8\%,5}=7173.9*0.17046=1222.84$$

The first step in conducting an economic evaluation analysis is to understand the concept of “cash flow.” “Cash flow” represents the net inflow or outflow of money during a given period of time that can be month, quarter, or year. Cash flow can be reported as before-tax cash flow (BTCF) and after-tax cash flow (ATCF).

Operating Profit or EBITDA = Gross Revenue or Savings – Operating Expenses
Before tax Cash Flow = Operating Profit or EBITDA – Capital Expenditure
After tax Cash Flow = Before tax Cash Flow – Income Tax Expenditure

Which is formatted as:

Gross Revenue or Savings

– Operating Expenses

_____________________________

Operating Profit or EBITDA

– Capital Expenditure

_____________________________

Before tax Cash Flow

– Income Tax Expenditure

_____________________________

After tax Cash Flow

EBITDA : Earnings before interest, taxes, depreciation, and amortization

Example 1-9:

Assume an investment project for which you need to invest 20 and 15 million dollars in year 0 and year 1 (you can think of it as 20 million dollars now and 15 million dollars next year) to build a facility. In year 2, the plant will start producing and you can make revenue by selling the products. Each year, starting from year 2, operating costs and tax have to be paid. Project net cash flow can be calculated as:

Table 1-1: Cash Flow in Millions of Dollars

	Year 0	Year 1	Year 2	Year 3	Year 4	Year 5	Year 6	Year 7	Year 8
Revenue			18	20	22	24	26	28	30
Operating Cost			-4	-4	-4	-5	-6	-8	-10
Capital Cost	-20	-15
Tax Cost			-3	-4	-5	-6	-7	-8	-9
Project Cash Flow	-20	-15	11	12	13	13	13	12	11

Each column stands for a time period (that can be year, quarter, month, …) and each cell shows the inflow or outflow of money. Investment cash flow in any year represents the net difference between inflows of money from all sources, minus investment outflows of money from all sources. The cash flow for this project for all years is calculated in the last row.

As you can see, all the costs (Capital Cost, Operating Cost, Tax, ...) are entered with the negative sign in the table, and then summation of each column gives the net cash flow in that year. The negative cash flow incurred in years 0 and 1 will be paid off by positive cash flows in years 2 through 8.

Discounted Cash Flow (DCF)

If future cash flow is discounted, we can have cash flow in terms of present value, which is called discounted cash flow (DCF). As explained before, DCF considers the time value of money and applies it to the inflow and outflow of money occurred in the future. DCF is a tool that enables us to compare the future cash flow with the present value of money.

Different investment projects have different cash flows that happen in different time intervals in the future and DCF can give an assessment to decide which project is more profitable. DCF brings the future amounts to a same base that is easily understandable for decision makers. For example, assume you have two options: investing your money in Project A that gives you 1000 dollars every year from 2025 to 2035 or investing in Project B that gives you 1500 dollars every year from 2030 to 2040. Which project will you choose? DCF is a tool that can help you finding the answer. DCF can also be used to estimate the value of a company based on its future performance.

Example 1-10:

Please calculate the discounted cash flow from Example 1-9 assuming:

1) Discount rate = 10%
2) Discount rate = 12%
3) Discount rate = 15%

Assuming discount rate = 10%:

$$\begin{array}{l}\text{Cash flow in year 0: }-20\left[1/{\left(1+0.1\right)}^0\right]=-20\\ \text{Cash flow in year 1: }-15\left[1/{\left(1+0.1\right)}^1\right]=-13.6\\ \text{Cash flow in year 2: }11\left[1/{\left(1+0.1\right)}^2\right]=9.1\\ \text{Cash flow in year 3: }12\left[1/{\left(1+0.1\right)}^3\right]=9.0\\ \text{Cash flow in year 4: }13\left[1/{\left(1+0.1\right)}^4\right]=8.9\\ \text{Cash flow in year 5: }13\left[1/{\left(1+0.1\right)}^5\right]=8.1\\ \text{Cash flow in year 6: }13\left[1/{\left(1+0.1\right)}^6\right]=7.3\\ \text{Cash flow in year 7: }12\left[1/{\left(1+0.1\right)}^7\right]=6.2\\ \text{Cash flow in year 8: }11\left[1/{\left(1+0.1\right)}^8\right]=5.1\end{array}$$

We can repeat the same procedure for discount rate = 12% and 15%. Table 1-2 shows the results.

Table 1-2: Cash Flow in millions of dollars

Year	0	1	2	3	4	5	6	7	8
Project Cash Flow	-20	-15	11	12	13	13	13	12	11
DCF (discount rate = 10%)	-20	-13.6	9.1	9.0	8.9	8.1	7.3	6.2	5.1
DCF (discount rate = 12%)	-20	-13.4	8.8	8.5	8.3	7.4	6.6	5.4	4.4
DCF (discount rate = 15%)	-20	-13	8.3	7.9	7.4	6.5	5.6	4.5	3.6

Net Present Value (NPV)

Now, all the DCFs in Table 1-3 have the same base, which is present value, consequently it’s possible to add them together and create a new criterion for project evaluation. The criterion which represents this summation is called net present value (NPV). NPV is the cumulative present worth of positive and negative investment cash flow using a specified rate to handle the time value of money.

Example 1-11:

Please calculate the NPV for the cash flow in Example 1-9 assuming:

1) Discount rate = 10%
2) Discount rate = 12%
3) Discount rate = 15%

Discount rate = 10%:

$$NPV=\left(-20\right)+\left(-13.6\right)+9.1+9.0+8.9+8.1+7.3+6.2+5.1=20.1 \text{million dollars}$$

Assuming discount rate = 12%:

$$NPV=\left(-20\right)+\left(-13.4\right)+8.8+8.5+8.3+7.4+6.6+5.4+4.4=16 \text{million dollars}$$

Assuming discount rate = 15%:

$$NPV=\left(-20\right)+\left(-13\right)+8.3+7.9+7.4+6.5+5.6+4.5+3.6=10.8 \text{million dollars}$$

As you can see, the discount rate has a substantial effect on the project NPV, higher discount rates give lower NPV of the cash flow. The other important factor is the time. The closer the money is to present time, the higher present value it has, which affects the NPV.

Example 1-12:

Assume you have two alternative projects to invest your 600 dollars. The cash flow in Project A and Project B are shown in Table 1-4. Which project do you choose if the discount rate is 10%?

Table 1-3: Cash flow in dollars

Year	0	1	2	3
Project A Cash Flow	-600	500	300	200
Project B Cash Flow	-600	200	300	500

Please note that two projects have similar numbers for cash flow but they happen in different times. DCFs are displayed in following table.

Table 1-4: DCF in dollars for discount rate is 10%

Year	0	1	2	3
DCF for Project A	-600	454.5	247.9	150.3
DCF for Project B	-600	181.8	247.9	375.7

$$\begin{array}{l}\text{NPV for project A: }\left(-600\right)+\left(454.5\right)+\left(247.9\right)+\left(150.3\right)=252.7dollars\\ \text{NPV for project B: }\left(-600\right)+\left(181.8\right)+\left(247.9\right)+\left(375.7\right)=205.4dollars\end{array}$$

This example shows how time affects the NPV of an investment project. As displayed in Table 1-5 and NPV calculations, Project A which has higher positive cash flows in closer time has higher NPV and it is a better alternative for investment than Project B.

Minimum Rate of Return

The terms “minimum rate of return," “hurdle rate," “discount rate," “minimum discount rate," and “opportunity cost of capital” are interchangeable with the term “cost of capital” as used in this course and in common practice. These terms should not be confused with the “financial cost of capital,” which is the cost of raising money by borrowing or issuing a bond, debenture, common stock or related debt/equity offerings. When the usual situation of capital rationing exists, the “opportunity cost of capital” generally is larger than the “financial cost of capital."

Microsoft Excel is a useful, convenient and widely used software for financial calculations and analysis that you will learn in this course. So, you are expected to learn and use required skills to utilize such tools.
If you do not have access to a commercial-grade spreadsheet program (such as Excel or OpenOffice), you can find free Spreadsheet applications available through Google Drive or a similar online tool. Following links include tutorials for Google Spreadsheet.

• Google Spreadsheet Tutorial from Google [1]
• Google Spreadsheet Tutorial from YouTube [2]

And also if you search online for “Google Spreadsheet Tutorial”, you can find some other good tutorial websites and videos.

Microsoft Excel

If it is the first time you are using Excel, please refer to the following video for a tutorial of Microsoft EXCEL 2010. (Time 10:00)
Please note that you need to open this video in YouTube [3]. (transcript [4])

You can follow the tutorial step by step to be a master of Excel 2010, which is a very powerful tool in the industry, business, and academia.

Tutorial for calculating present Value using Microsoft Excel (Time 7:35):

And also, these two following links (Times: 5:15 and 7:50):

Tutorial for calculating FutureValue using Microsoft Excel (Time 3:58):

For practice, I strongly recommend you to come back and solve the Lesson 1 examples in Excel and compare your results.

Summary

Table 1-12 summarizes the material that we learned in Lesson 1.

Table 1-12: Compound Interest Formulas

Factor	Name	Formula	Requested variable	Given variables
F/Pi,n	Single Payment Compound-Amount Factor	$${\left(1+i\right)}^n$$	F: future value of a single sum	P: present single sum of money n: number of time periods i: interest rate
P/Fi,n	Single Payment Present-Worth Factor	$$1/{\left(1+i\right)}^n$$	P: equivalent present value of a single sum	F: single future sum of money n: number of time periods i: interest rate
F/Ai,n	Uniform Series Compound-Amount Factor	$$\left[{\left(1+i\right)}^n-1\right]/i$$	F: Future value of uniform series of equal investments	A: uniform series of equal investments n: number of time periods i: interest rate
A/Fi,n	Sinking-Fund Deposit Factor	$$i/\left[{\left(1+i\right)}^n-1\right]$$	A: Uniform series of equal end-of-period payments	F: cumulated future value of investments n: number of time periods i: interest rate
P/Ai,n	Uniform Series Present-Worth Factor	$$\left[{\left(1+i\right)}^n-1\right]/\left[i{\left(1+i\right)}^n\right]$$	P: Present value of uniform series of equal investments	A: uniform series of equal investments n: number of time periods i: interest rate
A/Pi,n	Capital-Recovery Factor	$$\left[i{\left(1+i\right)}^n\right]/\left[{\left(1+i\right)}^n-1\right]$$	A: uniform series of equal investments	P: Present value of uniform series of equal investments n: number of time periods i: interest rate

To master all the knowledge to do your homework, you also need to go through the first two chapters of the textbook. Also, to finish your homework, you will need to know how to use Excel.

Reminder - Complete all of the Lesson 1 tasks!

You have reached the end of Lesson 1! Double-check the to-do list on the Lesson 1 Overview page [5] to make sure you have completed all of the activities listed there before you begin Lesson 2.