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In this video we're going to revisit the calculation of return on assets or ROA.

One of the things that we'll see is that the common way to measure it which is

simply net income divided by total assets is actually too

simplistic and it messes up a key feature that we want

ROA to supply when we're using this as a measure of our operating performance.

Remember that what we would ideally like this to measure is how well did we

do with our assets independent of how the assets were financed,

that is how much was debt versus how much was equity.

Our denominator in the calculation is total assets,

so that's fine because assets equals liabilities plus owners' equity.

Only the sum matters not how much is one versus the other.

The problem is the numerator of our calculation.

In the calculation of net income there is a subtraction for interest expense.

That subtraction is going to be bigger,

the more debt we have,

and so the numerator is impacted by

how much debt versus equity there is in the capital structure.

That's what we want to fix.

How to fix it?

Well, we're just going to add that interest expense back and look at

what income was before the interest expense.

The only difficulty with that calculation is interest expense also

impacted the tax expense because interest expense is tax deductible.

So, we need to add back the interest expense in a way that also adjusts for that.

And so, what we're going to do is add back the after-tax cost of interest.

This is referred to as de-levering or un-levering the firm's capital structure.

What would we have done if it was in all equity firm for example?

So, here we've got an example calculation.

One firm has no debt,

the other firm has some debt but they've got the same total amount

of assets and the same pretax, pre-interest income.

But the firm with debt is showing the subtraction for interest expense of $50.

So, if we want to un-lever that we can just add $50 to it's bottom line,

50 plus 162.5 is not going to equal 195.

This is because the interest expense also impacted the taxes.

If the tax rate is 35 percent what we have to add back

is the extra interest expense and the taxes that were saved.

So, we have to add back one minus the tax rate,

or 65 percent of the interest expense.

Once we do that,

now we're on equal footing.

So, a more improved or sophisticated version of the calculation of return on assets,

starts with income but adds back the after tax cost of interest.

Again, the idea of that is here's how much income we had before we

doled out any of it to any of the stakeholders divided by,

here's how much assets we had independent of how

much of it supplied by debt holders versus equity holders.

This is what we're really trying to get at.

Coming up with how big the interest expense was is relatively straightforward,

the tricky part of this is what's the right tax rate to use.

Now, what does this more sophisticated version of ROA do

to our understanding of how ROA and ROE are different?

Well, it changes it a little bit.

Now, ROA is income before interest.

That's not the same denominator that we're doing in ROE because

shareholders have to pay out some of that to the debt holders in the form of interest.

So, the numerator is different and the denominator is

different because in ROE we're just looking at shareholders investment,

ROA we're looking at the total investment.

So, how does leverage impact those two things?

In a similar way that we talked about before, but slightly different.

And here's the intuition for how leverage works.

Suppose we went out and we borrowed some money at six percent.

We took the money and we invested it in

something and we actually earned 10 percent with our investment.

Well, with the 10 percent return we could pay off what we owe to debt holders,

and we still have four percent left over.

Who gets that four percent?

Well, we do. And that adds to the return that we

generated without adding any to the investment that we made,

and so our return on our investment goes up.

The more debt we had,

the happier we would be as long as we can

generate a bigger return than what we have to pay out on the debt.

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But what about if the opposite happens?

If we're paying out six percent we invest in something that earns less than six percent.

Well, who makes up the difference? We have to.

So, that's coming out of our own earnings,

our own pocket, that's going to reduce the return on our investment.

So again, the notion of leverages and amplifier still holds,

it amplifies good times,

but also amplifies bad times,

it's just that the definition of good times and bad times is a little bit different.

This is the key new insight.

What's important is do you earn a higher return

on your assets than what you're paying out on the debt?

If you are, then leverage helps.

But if you earn a lower rate of return on

your assets than what you're paying out on your debt,

then leverage is going to hurt you.

We can illustrate that with a graph.

So, this graph on the horizontal axis is plotting

out different rates of return that our assets are earning pre-tax.

And then the vertical axis is what rate of

return do shareholders get of that rate of return on the assets as a whole?

The red line is graphing what that looks like for a low leverage firm,

the green steeper line is for a higher leverage firm.

In each case the firm is paying out 10 percent on their debt.

So, that's going to be the pivot point for the graph.

If the firm can earn more than 10 percent

on it's assets then leverage is going to be helpful,

and we see the green line is above the red line.

The more leverage firm does better.

If on the other hand,

the firm earns less than 10 percent on it's assets then

leverage hurts and you can see that the green line is lower than the red line there.

You can also see with this graph why leverage is an important indicator of risk.

If there is any volatility in what the assets earn,

it gets magnified by more with the green line than the red line.

Also the green line starts in a deeper hole.

There's more interest to cover,

and you've got to earn a higher rate of return to get to

zero for your ROA with the more highly leverage firms as well.

If you like equations we can express the relationship that way as well.

Here we've got a formula that relates ROE to ROA as a function of how much debt

versus equity there is and also what are you earning on

your assets versus what you're paying out on your debt.

If there's no debt in your capital structure note that ROE and ROA are the same thing.

If there is debt in the capital structure it amplifies the term in the brackets.

If the term on the brackets is positive,

return on assets is bigger than the after tax cost of debt,

then leverage is helping with respect to ROE,

but if the opposite is true then leverage hurts.

Here's ROA versus ROE for our case firm.

So, we're calculating income, interest expense,

tax rate, to figure our more sophisticated new version of ROA.

Note that this is a little bit higher than our simplified version

because we've added back the after tax cost of interest.

ROE is higher than ROA in each case.

This says that the firm is using leverage to it's advantage.

It's earning more on it's assets than it's paying out on it's debt.

But the level of ROA is dropping each year.

If this continues to fall and ROA falls below the after tax cost of interest,

then leverage is going to start hurting.

So, we've taken another look at the calculation of ROA,

our more sophisticated version tries to adjust for

the after tax cost of interest to come up with

a much better measure of the true operating performance of the company.

That is how well are we doing in

the absence of knowing anything about the capital structure,

in the absence of how this specific assets were financed with debt versus equity.

If you see somebody calculating ROA in this more sophisticated fashion that

should give you some more assurance that

they are reasonably financially sophisticated person.

If you see ROA in the more simplified version then you

might want to be a little bit more skeptical with respect to what they're doing.