Recall that the net present value is equal to the sum over all of the years that you
anticipate getting cash flow in divided by some factor, and this factor is a discounting
factor, so if we're earning $100,000 five years from now, we know that it's going to
be $100,000 in that year.
It's not $100,000 today, which is more then, it's going to be $100,000 in year 5.
And we want to calculate the present value, that's what the PV here is, the present value
of that $100,000 in today's dollars.
So we take that $100,000 that we'll be earning in a future time, and we have to divide that
by 1 plus, and if we have 15% here which is a little ambitious, that's what the $100,000
we'll be getting in 5 years is worth in today's dollars, not because of inflation but because
we could just take that money and invest it at 15%.
It ends up being something like $49,700.
Another way to look at this is if we take $49,700 today, invested it at 15%, in 5 years
it would be $100,000.
So that's what this term is.
We do that for each year we're going to be earning cash flow, and then we can add in
when at the end of the project, let's say it's 10 years from now, we would then salvage
our equipment, and sometimes you can use or pass on that working capital, the raw materials
and so on, to somebody else in the company or whatever, and you can get credit for that,
then we subtract what you actually put into the investment or the venture at the beginning,
at time zero.
So let's go over an example, and now we're going to calculate the net present value of
a 15 year venture, and it's being constructed from 2007-2009, it's not going to start operating
until 2010, and we're going to use instead of straight line depreciation, we're going
to use something called the MACRS, which is the Modified Accelerated Cost Recovery Schedule,
and we want to calculate the NPV, 15% cost of capital, that's the interest rate.
For the first two years we're only operating at 50% and 75%, so that's after we've built
the plant, so 50% of capacity, 75%, and cost of production excluding depreciation will
be $55 million and $78 million in those first two years, and then actually it doesn't say
here, but it goes up to $100 million, the production cost for the rest of the lifetime,
the rest of the 15 years.
This table kind of shows you an outline of year, what we invest, this is the total depreciable
capital, and in this scenario, we're taking the $90 million that we invest and over the
first three years that we build it, we're actually sinking $30 million each of those
three years, and we have parentheses here to indicate that it's negative.
So in the third year we request the working capital of $40 million, and you'll see here
we have depreciation, so that's D, we have the operating or the production cost, so the
first year of production is going to be $55, then we go to $78, and then for the remaining
time it's $100 million.
S is the revenue, so for the first year we can assume 50% because we're not quite operating
at full capacity.
This $113 is 75%, and then for the remaining time we rake in $150 million in sales.
Now to compute the net earnings, the net earnings, this is after tax earnings, our sales, $75
million in that year, in 2010, minus the production cost in that year is $55, that's our total
profit.
We haven't taken into account depreciation yet.
Depreciation in the first year is 18, so I subtract 18, and that will actually equal
this is only 2 million, and that's before taxes, so after taxes, we're going to assume
a 40% tax rate, so that equals 1.2, so that's how we get that number.
Now we have our discounted cash flow, so kind of the sum of everything for that year.
So for example, the first year, second year and third year, we just sum everything so
that's, we have our investment, our capital fixed cost, this is our fixed cost for the
third year plus our working capital that we request, so that year actually has quite a
large negative cash flow, then you'll see that we have 19.2 for the first year.
Our net earnings were 1.2, but our cash flow not only includes net earnings but it also
includes depreciation, because we can write that off on our taxes, that's 19.2.
And you can do this for all of the other ones, for example, this, let's say 2019, we're spending
100 million, that's our operating cost, we're getting 150 in sales, so 150 million minus
100 million is 50 million.
We tax that, and you'll notice that we don't have any depreciation because our equipment
has already depreciated to nothing, you can't claim, using this schedule you can't claim
that.
If you wanted to you could use maybe a 7-year or another schedule, but then during your
first couple years, your cash flow would actually be less, and there's various standards that
govern what you can use and what you can't, but we have 150 million minus 100 million,
that's 50 million, times 60% that we can get after taxes, that's our net earnings.
Our discounted cash flow is earnings plus depreciation, depreciation is zero in this
year, and that's how we get the discounted cash flow.
Now, present value cash flow is a little more difficult to calculate.
For the first year, this 26.09 is actually taking cash flow of that year and dividing
by 1 plus the cost of capital, I think we're assuming 15% here, yeah, 15% is the nominal
interest rate, to the first power.
And that equals 26.1.
And we can do the same thing, like in the cash flow that we get from 2019, the cash
flow is 30, so we have 30 million, we take 30, divide that by 1.15 to the 11th year,
that's how we can get the 5.61.
What this is saying is that the present value of the earnings, the cash flow in 2019, is
only worth about 5.6 million in today's dollars.
The last thing is the cumulative present value, and that's just done by adding everything
else before it, so you'll see that the 56.1 is just the sum of these two, 109 is the sum
of the cumulative plus that year's, so it's kind of a running tally and that's how we
do this, because if you have a negative cash flow one year and a positive cash flow the
next year, they sort of cancel out in net terms, and at the end of this we see that
the present value of our venture in 2021 in today's dollars is worth about 21.4 million.
We can also plot this, so here we have cumulative discounted cash flow in millions of dollars
as a function of year, and we see that the cumulative cash flow, even though we might
be getting positive cash flows in year 4 and beyond, the cumulative, because we have to
add that to the previous years, doesn't really make a profit until year 2018.
It's definitely making a profit even in the first 4 or 5 years, it's just the net present
value compares that value of the venture to what you could do in the current market.
So if we took this money and instead invested it at 15%, we're actually comparing the value
of this venture to what we could get if we put it at a nominal interest rate of 15%.
Are you making as much profit as you could be by putting this money in the bank?
The last thing is something known as IRR, which is Investor's Rate of Return, or also
called Internal Rate of Return.
It's a rate for which the net present value of a venture is equal to zero.
Think of it as the after-tax interest rate at which the investment is repaid by proceeds
from the project, and alot of times, this is the most important thing for a long-term
venture.
They want to know, investors want to know, and your company might want to know, what
is the internal or investor's rate of return.
And at the current cost of capital, so right now the cost of capital let's just say is
pretty low at 0.05, may 0.08 if you can get into something productive, but if the current
cost of capital is less than the internal rate of return, that means that the net present
value is positive, and the venture is profitable.
So let's say we end up with an IRR of 17%, and currently the cost of capital is let's
just say 0.05.
Then this venture is definitely worth it because essentially what we're doing is we're saying
the cost of capital would have to be 17%, so the interest rate would have to be 17%
in order for us not to make any profit, and that's when the NPV would be zero.
To calculate IRR, you sort of just set up a spreadsheet in Excel is the easiest way.
You calculate the net present value, and all we're doing is we're setting the net present
value equal to zero, and the variable here is your cost of capital.
So that's what you're actually changing, and maybe use solver in Excel or goal seek.
For our example that we worked with, and I plotted it here, this is the cumulative discounted
cash flow, so net present value, we adjust it so that at the end of your venture, the
net present value is equal to zero, and again we've done this by increasing i, because in
the previous example we had a positive net present value, but if we keep increasing i,
what that does is it decreases the net present value, and when we do that for this example,
we get an internal rate of return of about 18.5%.
Essesntially this is telling us that this is the equivalent rate of return or interest
rate that whoever invested in this project will get.
So let's say for example, the economy is doing really really well and the interest rate is
20%, if we come back and say the internal rate of return is 18.5, then the investors
might say, well you know what, let's just take this money, we're just going to become
investment bankers, we're just going to put this into the bank.
But if the interest rate is 5%, then obviously 18.5 is way better than 0.05.
Put it in the bank, you can make some money, invest it in this project, you can make alot
more.