Ihab Saad – Construction Economics
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The speakers discuss the value of money and the importance of considering the time value of equipment economics when buying equipment. They explain the impact of payments on the value of money and provide examples of how to convert a current value to a future value using a factor like F. They also discuss the use of a cash flow diagram and alternative options for replacing equipment, including the possibility of leasing a similar tractor for a fixed monthly rate. The present value is the one with the most favorable return, and the cost of ownership for the equipment is estimated using a cash flow diagram and alternative options for replacing equipment. The rate of return is an annual interest rate at which the sum of investment and expenditures equal the total income from the investment.
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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.