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-: We know that the internal rate of return or IRR
is the rate of return when the net present value or NPV
of a series of cash flows is zero.
Let's consider project A,
where the NPV is negative $50,000
and the IRR is 8%.
As the NPV is negative,
then we know that the discount rate is higher than the IRR.
This is because a higher discount rate
makes the present value of future cash flows smaller.
So because the IRR is when the present value equals zero,
then because the NPV is less than zero,
the discount rate must be higher than the IRR.
Now let's consider project B,
where the NPV is $150,000 and the IRR is 15%.
As the NPV is positive,
then we know that the discount rate is lower than the IRR.
This is because a lower discount
doesn't have as much of an impact
on the present value of future cash flows.
So if we know that the NPV would be zero
using the IRR of 15%,
then because the NPV is greater than zero,
then we know that the discount rate used is less than 15%.
If these two projects were being assessed together
by the same organisation,
we would know that the discount rate
must be between 8% and 15%.
In this situation,
we could use the trial and error approach
or financial calculator or spreadsheet
to determine the actual discount rate.
When we know the NPV of a project
as well as the value and timing
of its underlying cash flows,
we can reverse engineer the NPV formula
to calculate the discount rate.
This is effectively what we need to do
to calculate the internal rate of return of a project.
This is because the IRR represents the rate of return
when the NPV is zero.
So to calculate the IRR,
we can use the trial and error approach
or a financial calculator or spreadsheet,
which essentially does the trial and error for you.
Let's think about a project
with cash flows of $100,000 cash outflow in year zero,
$50,000 cash inflow in year one,
$25,000 cash inflow in year two
and a $50,000 cash inflow in year three.
If we want to know the IRR for this project
using the trial and error method,
we need to identify the appropriate discount rate
that achieves an NPV of zero.
In this situation,
we might start with the discount rate of 10%
and see what the result is.
Using a rate of 10%, the NPV is positive $3,681.
This suggests that the IRR is higher than 10%,
as a higher discount rate
reduces the present value of future cash flows.
We would then trial another discount rate
higher than 10%, say 15%.
This results in an NPV of negative $4,742.
So because we know that lower rates
have less of an effect on the present value,
we know that the IRR is going to be less than 15%.
Based on this information
then we know that the IRR is between 10% and 15%
and probably slightly closer to 10%
because the NPV at 10% is closer to zero
than the NPV at 15%.
We then trial another discount rate of say, 12%,
which results in an NPV of positive $162.
From this result,
we know that the IRR is just above 12%
and a spreadsheet model confirms
that the actual IRR is 12.095%.
It is also possible
to use a financial calculator or spreadsheet
to quickly calculate the IRR.
However, it is good practise to know and understand
the trial and error method
which involves estimating discount rates
and calculating the resulting NPVs
until the NPV of the project equates to zero.