# Ihab Saad – Construction Economics

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Steve, Hello again. Today we're going to talk about the time value

of money and equipment economics. So as we know,

the value of money changes with time, so how does that affect our

estimate on the equipment economics, whether it's going to

be better to buy the equipment today, or buy it in a couple of

years, or how we're going to pay for it. This is basically what

we're going to be discussing in this lecture. So in the context of

equipment management, a piece of equipment costs you $50,000 to

purchase. You could also lease it or rent it. Which option should

you choose

on one project, you need to excavate 10,000 cubic yards of

earth. How much does it cost you?

Equipment a costs you $20,000 with a useful life of five years, while

equipment B costs you $30,000 with a useful life of seven years.

Which one is a better value for your money? All of these are

sample questions that any equipment manager or any project

manager is going to be subject to, and we'll have to answer them

based on an educated decision making process.

So what's the real cost of money? Money is a valuable commodity that

changes value with time. An increase would be evaluation, or a

decrease would be devaluation, and it's a super resource, which means

it's used to buy any other resources or to acquire all the

other resources. Because if you remember our definition of a

resource, it's something that you need for the project, you have the

ability to manage it, and that you pay money to acquire it. If it

does not meet all of these three conditions, it is not considered a

resource. The issues of costing and financing, which are two

different issues, can make or break a construction operation or

even a construction organization. So most of the construction

companies that fail or declare background bankruptcy, it's

usually due to financing problems they have not been able to provide

the required amount of money at the proper period or point in time

to meet their obligations for the project. So money has a time value

which increases with time. Since money increases as we move forward

from the present to the future,

it also decreases in value if you move backward from the future to

the present.

The other concept is the concept of interest, or usury. What's

called interest is a fee assessed to use borrowed money. The

percentage and amount of the fee will depend on the amount of money

borrowed, the length of time it's borrowed, and the prevalent

interest rate at the time of borrowing. So these are the three

elements that control the amount of debt, percentage or debt

interest. Interest rate is the percentage rate charged against

the borrowed capital or principal. So we need to know how it is it,

how is it calculated, and how is it going to affect our decision in

buying or renting or leasing equipment.

So what's the time value of money? We have to talk about something

called the concept of equivalence. Payments that differ in magnitude

but are made at different time periods may be equivalent to one

another. In plain English, what does that mean any sum of money

recorded as receipts or inflows received money, money that we get

are called Cash in, and any sum of money recorded as disbursements or

outflows, money that we pay out is called cash out. The difference

between cash in and cash out at any point in time represents the

cash flow at that point. So again, this is, as I said before, this is

the major reason why most construction companies goes go

bankrupt, because they did not cater for that difference between

the cash out and the cash in. They might need certain sum of money at

a certain point in time. Failing to provide that sum is going to

result in bankruptcy.

So an example on cash flow. For example,

if you buy a car today for $4,500

you have an outflow cash out of 4500 which is represented by this

negative 4500 and then at the end of year one, you you perform

regular maintenance on that car to keep it in a good shape, good

operating condition. So you spend, on average, $350

per year of ownership of that car. So the maintenance cost per year

for year one was three, $350

for year two, the same thing. For year three, the same thing. And by

the end of year four, you also pay that same amount to be able to

sell the car. And then when you sell the car, you sell it for

$2,000 so as you can see here, all all of this, all the negatives are

cash out, which are represented by negative sign. And then you get a

positive inflow, or cash in of two.

$1,000 when you sell the car. So this is an example of cash flow.

The question now is, are these $350

equivalent? Are they the same? Since you spend one

in year one, one in year two, year three and year four, obviously

it's not going to be the same, because $350

today is different from $350 a year from now, from two years from

now, and so on.

So in order to graphically represent the cash flow, because

once you are able to draw the problem, to draw the cash flow,

that's basically half of the solution. If you understand

properly how that cash flow is is disbursed, you can very easily

solve the problem. The horizontal axis represents a timeline of the

analysis, marked off in equal increments, whether they are

months, years, weeks, whatever period equal period, receipts are

represented by arrows directed upward. So the cash in is going to

be moving upward. This burden disbursements, or cash out, or

payments, are represented by arrows directed downward. That's

the negative. The arrow length is drawn to a scale proportional to

the magnitude of the cash flow. So it's relatively gone. These arrows

are going to be relatively drawn to scale. Let's have a let's look

at an example here. So here's the same example that we just

discussed about the car that you purchased. You bought it for

$4,500 you paid $4,500 so that's a cash out 4500

by the end of year one, 350 end of year two, 350

end of year three, 350 end of year four, 350, and then you sell the

car for 2000 which means a cash in that's why it's pointing upward of

$2,000

so once we do understand that, we can visualize graphically the cash

flow of this property.

So now we're going to start talking about the different

payments and the impact of time on these payments. Single payments

may occur either today or at some time in the future, like when you

bought the car, you paid $4,500

that's a single payment, but it was made today, so it has a

present value of $4,500

so P is used to indicate a sum bed or received today at the present,

whereas the $2,000 that you received four years from now,

it's $2,000 future, because you're going to receive it in the future.

So it's going to be represented by F, which is used to indicate a

future sum

i is used to indicate the interest rate. So if you're going to borrow

money and you're going to have to pay interest, what's the interest

rate? The future value of the present sum invested at an

interest rate I for a period of n years, is represented as the

future value of the money that you're paying today is equivalent

to p, which is the present value times one plus i to the power of

n1,

plus i to the power of n i here is going to be represented as a

percentage. So 5% is going to be point o5,

10% is going to be point one and so on. Therefore, if you got, if

you're going to invest some money today, let's say $1,000 today for

five years at an interest rate of 10%

we wish so at interest rate of 10% for five years, we want to know,

what are these $5,000

What's this $1,000 that I'm gonna pay today gonna be worth in five

years? So F, that's the future value that we don't know about is

equal to P, which is 1000 times one, plus I point one, the 10% to

the power of n, which is five years. And that gives us the

future value based on a known present value today and based on a

known interest rate and known number of increments or number of

years. The term one plus i to the power of n, is called Single

payment compound amount factor, which is used to determine the

future worth of a present sum of money. So to convert present to

future, we use one plus i to the power of n.

The present worth factor

is actually the opposite of that. I have a future value. I know that

I'm going to receive $5,000 in five years. And I want to know

what's their current value today? What's their present value today?

So the reciprocal

which is one over x, so it's one over one plus i to the power of n

of the single payment compound amount factor is called the single

payment present worth factor.

100 it's used to determine the present worth of a future sum of

money. So to convert present to future, I use one plus i to the

power of n to convert from mutual to present. I divide by one plus i

to the power of n.

So let's look at an example. Here, a contractor plans to purchase a

pickup truck in five years. He's going to purchase it in five

years. So the purchase price we need to calculate is the one in

five years, not today. So it's a future value. How much should he

invest at a 6% interest rate today to have the $30,000 needed to

purchase the truck at the end of five years. Let's start thinking

about it. Let's start drawing it. What are we looking for? Are we

looking at for a present value, or are we looking for a future value?

Which one do we know? What are the knowns in this problem we have? N

equals five. Which is the five years, we have i, which is equal

to point, oh, six, the 6%

and we have a future value, which is the value of that car

in five years.

That's $30,000 so I want to know how much money does that

contractor have to invest today to get to 330, $1,000 in five years.

So the purchase price in this problem is a known future value,

f, 30,000 and the unknown is the present worth, which is p. In this

case, P is equal to f over one plus i to the power of n.

So 30,000 divided by one plus point oh six, which is 106 1.06

to the power of five. That gives us that the contractor has to

invest today $22,417

that's going to be equivalent to 30,005 years.

Another example, a contractor bought a 15,000 pump, dollar pump

with an expected service life of 10 years. So he bought it today

for 15,000 that's a P its salvage value at the end of 10 years would

be $4,000 now at the end of 10 years, which means it's a future

value. What is the total cost to the contractor for owning the pump

if the annual interest rate is 8%

again, here we have a present value. We have a future value. And

we have n we have I. So we have all the ingredients we need to

calculate the total cost at the end of

these 10 years.

So the purchase price is P, known the salvage value is a future

value F. We can either calculate all of this cost of ownership

today, or we can calculate it in 10 years. So we can calculate it

in terms of p, or we can calculate it in terms of f. Doesn't matter,

provided that we know what is the point of time, or point in time

where we are making this calculation. So if we make this on

the basis of p, is going to be the 15,000 which is the purchase price

today, minus the future value that you're going to see 10 years from

now, brought back to today's prices, which is a future value,

brought back to a present value. So we divide by one plus i to the

power of n. I is 8% n is 10. So we divide by 1.08 to the power of 10,

which makes the $4,000 in 10 years equivalent to 1800 52 today.

Therefore, the total cost of ownership is going to be the

15,000 minus 1852 which is 13,001 48 today, we could have solved the

same problem in future to calculate what's the value at the

end of the 10 years as well.

Another example here, or another form of calculation is going to be

based on a uniform series of payments. So if you're paying for

something in installments, let's say you bought a car and you pay

let's say $500 every month for four years, the $500 for the first

month is different from the $500 for the second month is different

from the $500 for the 48th month at the end of the four years. So

it's sometimes necessary to determine the present worth P, or

the future value, future worth F, of a uniform series of payments,

these annuities or this regular amount that you're going to be

paying every month or every year is called an annuity, uniform

series of payments or receipts over a certain period of time. In

other situations, it's necessary to determine the series of equal

payments or receipts to equal a present value to.

A future value f. So it's either we're going to convert the P or

the F into annuities uniform series, or we're going to

calculate, what are these uniform series of payments equivalent to,

either today, at the present value or in the future at the future

value?

So this uniform series, compound amount factor is used to determine

the future worth of a series of equal payments or receipts. It's

represented as one plus i to the power of n minus one divided by I.

That's the factor

the uniform series, present worth factor is

used to determine the present worth of a series of equal

payments or receipts represented as one plus i to the power of n

minus one divided by I times one plus i to the power of n. Let's

look at an example which is going to make these things much easier

to understand.

So the uniform series sinking fund factor is used to determine a

series of equal payments or receipts that's equivalent to a

required future sum. So if I say, for example, at the end of four

years, I should have had $20,000

how does that translate into annual payments

and uniform series. Capital recovery factor is used to

determine a series of equal payments or receipts that's

equivalent to a present worth some if I pay 20,000 if I have to pay

$20,000 today, if the price of that car is $20,000 a day, if I

want to make that in four years on four installments, how is that?

Are these in equal installments gonna amount to

so let's look at the example here, which going to make things much

clearer. A contractor is investing $5,000 per year in saving

certificates at an interest rate of 6%

and plans to continue the investment program for six years

in order to be able to pay the down payment for new equipment.

What will be the value of the investment at the end of six

years? What do we have here? We have the contractor paying 5000

per year. So this is an annuity. This is a repetitive amount. It's

not just the present value. Today is going to be today, a year from

now, two years from now, etc. What we want to calculate is F, a

future value at the end of six years. So what we have is n, is

six, and we have i is also, in this case, six, which is a 6% so

based on a which is the annuity 5000

I, 6%

n6, what we need to calculate is F, the future value at the end of

these six years.

Therefore, here's the equation. The annual investment is an annual

uniform series, and the unknown is the future worth. Therefore, we

will use the formula for uniform series, compound amount, factor f

is equal to A, which is the 5000 times one plus i to the power of n

minus one divided by i, which is 5000 times 1.06 to the power of

six minus one divided by point oh six, and that gives f the this

uniform series of $5,000 over six years at the interest rate of 6%

are going to be equivalent to $34,875

Another example, let's do the reverse. A contractor has

purchased a new truck for $125,000

and plans to use it for six years. After six years of use, the

expected salvage value will be $30,000

what is the annual cost, or any annual uniform series for the

truck at an interest rate of 10% so here we're gonna use P, F and

A,

P is under 25,000 that's down a down arrow of under $25,000

which is cash out. N

is six years.

I The interest rate is 10% and then we have a future value, which

is a salvage value at the end of these six years of $30,000 so what

are these equivalent to? In annual increments, equal annual

increments.

So in this problem, the purchase price or the present value is

known as well as the salvage value or future value. The unknown is a

series of equal annual payments. So we're going to convert the

present value into annuities by using the equation, and we're

going to convert the future value into annuities also by using the

equation. But these are going to have two different signs, because.

Us, this is negative and this is positive, or this is positive and

this is negative because they point into do two different

directions. So

that amounted to 150,000 times point two, three, which is all of

this stuff here, minus 30,000 times point divided by point o3,

which point one three, which is this amount here, which makes the

annual payment to account for all of this transaction of $24,850

okay, now we're gonna look at alternative analysis when two or

more alternatives are capable of performing the same function, the

economically superior alternative will be the one with the least

present worth cost. So if I pay for this equipment

in cash today, or if I pay for it in installments over five years or

six years or whatever. Let me make the comparison at the same point.

Either convert both of them into annuities or convert both of them

to a present value. And that would be the better way so that I

compare present to present today, the option that's going to be more

financially appealing and feasible is going to be the one that has

the lower present value. So the present worth comparison, however,

should be used only for assets with the same lifespan. I cannot

now compare two pieces of equipment, one that's going to

serve me for five years and another is going to serve me for

seven years or eight years. It has to have the same end number of

years or lifespan. For assets with different lifespans, annual cost

method of comparison should be used so it's going to be based on

annuities. How much is it going to cost per year? The annual cost

method assumes that the asset will be replaced by an identical

replacement at the end of the useful life. And this is basically

the concept, when we talked about the tax breaks and so on for the

construction equipment and depreciation, this is basically

the concept, why do we depreciate the equipment, basically to have

some savings that enable us, at the end of the service life of the

equipment, to replace it.

So the first step to compare alternatives is to construct the

cash flow diagram for each a common basis for comparison is

selected, whether it's going to be present value, future value or

annuity, and an equivalent sum of uniform annual series is

determined for each. Then the alternatives are compared to

select the one that is most favorable.

The contractors usually use the minimum attractive rate of return

to perform cash flow analysis, which means, if I had invested

this money in another enterprise, it would have given me a return on

investment. What would be the minimum acceptable return on

investment, or interest rate that I would accept to use that money

for this to buy this equipment. So for example, if I invested this

money in the stock market, it would have given me a 6%

return on investment,

I should not buy equipment that's going to give me a return on

investment of only 5% it's going to be more profitable to invest in

the stock market, but if buying that equipment is going to get me

a return on investment of 7% now it's more profitable than

investing in the stock market, and that can make me make my decision

is used as the effective interest rate in cash flow analysis.

So again, examples make things much easier to understand. A

contractor is considering the purchase of a used tractor

$480,000

that could be used for 10 years n is 10 years, and then sold for an

estimated salvage value of $10,000 so f is 10,000

annual maintenance and repair costs are estimated at 15,000 per

year. So this is an annuity, a

and an alternative. So this is the first part to be considered. The

alternative is the contractor could lease a similar tractor for

$4,000 a month. With the lease, he doesn't have to pay a high

purchase price at the beginning. He's not going to have a salvage

value at the end. And we know that the annuity that's going to be pet

per month is fixed. Annual operating cost is the same for

both alternatives, fuels, filters, lubricants, etc, is going to be

the same. So we're going to disregard it in both cases, the

minimum attractive rate of return is 12% which option should the

contractor select? So what we have to do here is to draw two

different cash flows and come.

Pair these two and then calculate the present worth or the annuity

or the future value for both of them, and look at which one is

going to be more appealing.

So for the first option,

we're going to pay $180,000

at the beginning.

We're gonna receive $10,000 at the end salvage value, and then in the

interim, for 10 years, we're gonna be paying $15,000 per year in

maintenance or whatever. Since the monthly rental rate is known,

$4,000

we will compare the alternative on an annual cost basis. So the

annual cost for the rental alternative. Now remember that

this is monthly, and the other one here, this is annual, so we have

to bring it to the same

increment. So we're going to assume, for simplicity, that

$4,000 times 12, which is going to be the 40,040 $8,000 this is what

you have to pay as an annuity to be compared to this 15 and the

eight, 180 and the 10. So for the purchase alternative, the cash

flow will look like that, 180,015

1515, until the 10th year, and then you receive 10,000 back.

So for the purchase option, the annual costs can be determined

using the formula, we have an interest rate

of 12% for 10 years. So

the purchase price, which is a P times the formula to convert it to

a plus the A's the 15,000

per year, minus the 10,000 which is the future value, converting it

into annuities, which is going to amount to $46,290

per year. So this is cost of ownership. Okay, we have just

mentioned that the cost of lease or rental is going to be $48,000

per year. So we are comparing 48,000 for the lease option to 46

290 for the ownership option. Obviously this is less expensive,

therefore owning the equipment is going to be more economical than

renting it or leasing it.

Now this is a summary of all the equations that we've discussed.

The single payment compound, the single payment present worth etc.

So to convert from P to f,

this is the form that we use.

So f is equal to P times one point plus i to the power of n. To do

the opposite, to convert an F to A p. So what we know is F and we

want to get p. So p is equal to f times one plus r1, divided by one

plus i to the power of n to convert F to A. This is the

equation to convert P to a, to convert A to F and to convert a to

p. Now I'm not going to ask you to memorize all of these equations,

so on the test, I'm going to allow you to have a simple note card

with these equations so that you don't have to memorize them.

Contractors often want to estimate the prospective rate of return of

an investment, or compare anticipated rate of return for

several alternative investments, just as we did, is it better to

buy the equipment? Is it better to lease the equipment? Is it better

to invest in this project or another project, or just the stock

market? So we're going to look at different rates of return. The

rate of return is an annual interest rate at which the sum of

investment and expenditure equal the total income from the

investment. It involves setting receivables equal to expenditures

and solving for the interest rate. In

many cases, the interest rate can be obtained from a trial and error

method of solution. So two or more interest rates are assumed,

equivalent present worth or annual costs are calculated, and the rate

of return is found by interpolation. Let's look at an

example, which, again, is going to make things much easier.

So a contractor plans to invest $300,000

in used construction equipment, the estimated annual maintenance

and repair cost. Now the first one is a present value. Here we have

annual maintenance and repair cost annuity, $60,000

expected annual income, also an annuity. 115,000

the expected profitable service life of the equipment is eight

years. So n is eight years. What is the prospective rate of return

on these investment? What is i? So what we're looking for here in

this equation is I,

let's draw the cash flow again. So.

We have a present value that we're gonna spend of 300,000

and then we're gonna spend each year $60,000 in operating cost in

return. We're gonna have a return on investment of $115,000

every year in return. So the problem can be solved based on the

present worth P, or the annual cost a.

The annual cost formula can be abbreviated as now based on the

table that we have looked at A and P and I and N. So to get the A

based on the p1,

115,000 is already known,

is equal to P,

is equal to another A, which is

the 60,000 in the other other direction, plus converting the P

into annuities.

So basically we're gonna put this on the other side. So here we have

55,000

is equal to 300,000

and then the equation of converting a p into an A, the only

unknown is the i,

which is going to give us point 183, solution of this equation is

going to give us point 183,

through trial and error and using interest tables, we can start with

9% which gives point what a 181

trying 10% gives point 197, these tables, by the way, you can find

them in any

economics book, or you can find them Online, even so, the 9% gives

point 181, the 10% gives point 197, what we're looking for is

1.183

which is closer to the 9% so by interpolation, we're going to find

that the rate of return is 9.3%

which is somewhere between the nine and

10. That's basically our lecture on construction economics. We

learned in this lecture about how to convert a present value into a

future value, how to convert the future into present, how to

convert either a present or a future into annuities, and how to

have combinations, if the problem includes present value annuities

and future value at the same time, how to bring it into single terms,

whether it's converting everything to future or present or annuities

and then calculating for the rate of return. I'll see you next time

in another lecture you.