# Intuitive explanation of levered Beta formula, anyone?

HF)

on 4/15/09 at 2:35am
I am sorry if this question seem stupid/simple, but still I'd greatly appreciate your help.

Levered Beta = Unlevered Beta * [1 + (1 - Tax Rate) * Debt / Equity ]

Unlevered Beta = something we see on Yahoo Finance

Could someone please take a shot and explain why we multiply by [1 + (1 - Tax Rate) * Debt / Equity ] and not something else? Was this formula discovered intuitively based on previous finance formulas, or was it more of "This is the law, learn it"?

## I'm not sure if this is what

I'm not sure if this is what you're looking for, but its essentially because of the idea of the "Tax Shield" nature of Debt, due to the fact that issuing debt and accruing interest allows you to offset some revenue, which saves on taxes (lower profit, therefore lower tax expense). A more intuitive way to think of the formula may be:

Levered= Unlevered + Unlevered*(1- Tax Rate)*(D/E)

I know this seems stupid, but its easier to see the effect of leverage on Beta. You can then see that Levered Beta is Unlevered + another factor that is related to the Tax Rate and the D/E ratio. Essentially Debt magnifies the Cash Flows (increases Beta). It also takes into account the Tax Shield discussed earlier (which lowers the beta in the way that the larger the Tax Rate, the lower the Beta). Does that make any sense? Its 3am and my brain is tired.

Jack: They’re all former investment bankers who were laid off from that economic crisis that Nancy Pelosi caused. They have zero real world skills, but God they work hard.

-30 Rock

## Thanks Revsly, formula makes

Thanks Revsly, I was able to follow your ideas and whole levered = unlevered + ... now makes more sense. But why Debt/Equity? Why not Debt ^ 2 / Equity, etc.?

I guess my math background always taught me that new formulas are based on previous findings/research. In reality the entire math (integrals, derivatives, topologies, etc.) based on simple axioms.

Edit: actually it makes sense now. Assume Tax = 0. If we rise debt equal to equity/market cap, then levered beta = double unlevered, which makes sense because now company is "twice as risky"

## I thought it was Modigliani

I thought it was Modigliani and Miller, though I could be wrong. I know they worked with Value of Firms unlevered and levered, and it would make sense that beta would be a natural application. Yeah, you've got the idea.

Haha as a Math major too, I see where you're coming from (though its not quite as easy to believe in the complex plane lol)

Jack: They’re all former investment bankers who were laid off from that economic crisis that Nancy Pelosi caused. They have zero real world skills, but God they work hard.

-30 Rock

## Why do we use Levered

Why do we use Levered instead of Unlevered in cost of equity formula?

Cost of Equity = Risk-Free Rate + Equity-Risk Premuim * Levered + other premiums

My wrong intuition tells me that equity (market cap) is tied up with Unlevered Beta hence we should use unlevered beta.

After some research: "The cost of equity is the minimum rate of return that a business or organization must offer investors or owners to offset their wait for a return on investment and for assuming some level of risk." Do we use levered beta because we need to account for "some level of risk"? As soon as debt is issued, there is risk involved.

## Because

PussInBoots:Why do we use Levered instead of Unlevered in cost of equity formula?

Cost of Equity = Risk-Free Rate + Equity-Risk Premuim * Levered + other premiums

My wrong intuition tells me that equity (market cap) is tied up with Unlevered Beta hence we should use unlevered beta.

After some research: "The cost of equity is the minimum rate of return that a business or organization must offer investors or owners to offset their wait for a return on investment and for assuming some level of risk." Do we use levered beta because we need to account for "some level of risk"? As soon as debt is issued, there is risk involved.

As you get more debt on your balance sheet, your cost of equity goes up. Think about this intuitively. Your equity investors are going to require a greater return if you have more debt, thus the cost of equity increases. This is reflected by beta in the formula mentioned above.

## Optimal Capital Structure Theory

Also, as you have more debt, the financial risk of the company is greater - more interest payments to be made and repay principal.

So, equity investors will demand more return to bear this risk....

Think of it this way: As the company takes on more debt, theoretically the risk of it not repaying the debt increases and if it happens (not repay) and the company goes bankrupt, you lose ALL your equity.

To sum up, as you increase leverage, your financial risk is greater (more interest payments + generate cash flow to repay principal) and equity holders will demand "extra" return/reward to bear this risk - greater discount rate / CAPM (where leverage beta is one component).

That's why leveraging a company reduces its WACC up to a certain point. After that, the use of debt will increase your WACC (and the leverage beta is the variable that will drive this increase in CAPM and consequently WACC).

## "Why this formula" answer

I realized I actually didn't answer your original question...

"Why we multiply by [1 + (1 - Tax Rate) * Debt / Equity ] and not something else? "

1. Debt/Equity = the most traditional way of measuring leverage (basic corp finance theory)

2. (1 - Tax rate) = as mentioned above, debt has a tax shield, so this item accounts for that

3. [1 + .... = anything in finance that is a rate and you want to turn it into a factor, you've got to add 1. Try to do this formula without summing up 1, it's going to give you a proportion of unlevered beta (because you would be multiplying a number to a percentage, whereas when you multiple a number by a factor, you obtain a number sensitiized by that factor)

Hope that helps.

## 3 internets to lui for

3 internets to lui for helping poor monkey out :)

## This levering formula that

This levering formula that the OP has suggested is the most widely accepted.

However, most people seem to forget that it is only valid under two assumptions, which explain why it looks like that:

1) Debt is assumed to be risk free.

2) The Tax Shield has the same risk as the Debt.

In our current economic climate, with sovereign crises and constant downgrades, these assumptions might not be valid anymore and my guess is we will see a strong shift toward the use of more complete formulas.

The current formula will then not be accurate anymore and analysts will have to use something else.

I recommend reading Berk & DiMarzo for more on this topic.

## Following UChicago methods,

Following UChicago methods, unlevered beta = levered beta * (MV of Equit/Enterprise Value)

where Enterprise Value = MV Equity + MV Debt - Excess Cash.

## What does MV stand for?

What does MV stand for? Marginal Value?

## must be market value.

must be market value.

## humble_dude wrote: must be

humble_dude:must be market value.

Ha, of course. I feel slow now.

## Sry just stumbled on this

Sry just stumbled on this thread, was just thinking about this formula..

Basically, unlevered(asset) beta represents pure business risk. It does not incorporate financial risk at all, using unlevered beta you pretend that the firm is entirely equity financed, and all cashflows are due only to the equity holder(s).

Say I'm the sole owner of my own large corporation, which has risk equal to the market risk i.e. a beta of exactly 1. The corporation will provide me with the expected return of the market, on the equity which I have invested in it, via the aftertax free cash flows accruing to me. In turn these cashflow's discounted back at the required rate of return (in this case the mkt return) shall lead us back to the market value of the corporation, lets say its $1000MM, the standard deviations of the cashflows from my corp will equal that of the mkt. Now say I decide to mix up the capital structure, I borrow $500MM in the name of the corp and pocket it, essentially the corp now has 500MM of debt on its BS and has bought half my equity stake with the debt leading to a D/E ratio of 1. The business risk/unlevered beta/asset beta remains the SAME as it always was, nothing about the corp's operations have changed. The standard deviations of the operatating CF's are still the SAME. But relative to my equity stake now, the standard deviations of the CF's are TWICE bigger, since my equity stake is twice smaller. This is where the 1 + D/E comes in, in our case 1+D/E = 2. (we'll get to taxes later)

Another way to think of this is that if half the cashflows during the life of the corporation are now due to bondholders, then I will demand that the firms expected cash flows must now double in value, for me not to get ripped off.

So if a company is replacing equity with debt or issuing new debt, I expect to get repaid for my equity stake in the case of the former, and expect the company's FCF to double in the case of the latter( if beta = 1 then I would expect 2 x mkt return relative to my stake in the company, the company is actually earning mkt return on capital, but relative to where im sitting with half the capital, beta = 2 for me)

Thats why we have betaL = betaU(1 + D/E)

what about (1-T)? from an equity prespective, I will incure inflows from tax savings over the life of the corporation. Since D is the PV of these cashflows to debtholders, rest assured that (1-T)(D) in taxes will be saved, as long as the corporation retains same cap structure and continues to rollover debt.

Therefore if a company incurse a 1 D/E ratio out of the blue, I will first demand that its FCF double, but then i will realize that I will earn (1-T)D in cash over the life of the company, therefore I will demand that its cashflow increase by betaU(1+(1-t)D/E)) or betaU + betaU - betaU(t)(D/E), as before we'll set betaU as 1 and t as .35,

we then have 1 + 1 - 1(.35)(1) = 1.65 = BetaL,

therefore If a company with no debt takes on a debt load with D/E = 1, as an equity holder I will demand FCF increase by 2 in the case that interest is NOT tax deductible

And I will demand that FCF increase by 1.65 in the case that interest is tax deductible!!

I was thinking about this earlier and writing it out helps me understand it, sorry for such a long essay

## PondFish: Sry just stumbled

PondFish:Sry just stumbled on this thread, was just thinking about this formula..

Basically, unlevered(asset) beta represents pure business risk. It does not incorporate financial risk at all, using unlevered beta you pretend that the firm is entirely equity financed, and all cashflows are due only to the equity holder(s).

Say I'm the sole owner of my own large corporation, which has risk equal to the market risk i.e. a beta of exactly 1. The corporation will provide me with the expected return of the market, on the equity which I have invested in it, via the aftertax free cash flows accruing to me. In turn these cashflow's discounted back at the required rate of return (in this case the mkt return) shall lead us back to the market value of the corporation, lets say its $1000MM, the standard deviations of the cashflows from my corp will equal that of the mkt. Now say I decide to mix up the capital structure, I borrow $500MM in the name of the corp and pocket it, essentially the corp now has 500MM of debt on its BS and has bought half my equity stake with the debt leading to a D/E ratio of 1. The business risk/unlevered beta/asset beta remains the SAME as it always was, nothing about the corp's operations have changed. The standard deviations of the operatating CF's are still the SAME. But relative to my equity stake now, the standard deviations of the CF's are TWICE bigger, since my equity stake is twice smaller. This is where the 1 + D/E comes in, in our case 1+D/E = 2. (we'll get to taxes later)

Another way to think of this is that if half the cashflows during the life of the corporation are now due to bondholders, then I will demand that the firms expected cash flows must now double in value, for me not to get ripped off.

So if a company is replacing equity with debt or issuing new debt, I expect to get repaid for my equity stake in the case of the former, and expect the company's FCF to double in the case of the latter( if beta = 1 then I would expect 2 x mkt return relative to my stake in the company, the company is actually earning mkt return on capital, but relative to where im sitting with half the capital, beta = 2 for me)

Thats why we have betaL = betaU(1 + D/E)

what about (1-T)? from an equity prespective, I will incure inflows from tax savings over the life of the corporation. Since D is the PV of these cashflows to debtholders, rest assured that (1-T)(D) in taxes will be saved, as long as the corporation retains same cap structure and continues to rollover debt.

Therefore if a company incurse a 1 D/E ratio out of the blue, I will first demand that its FCF double, but then i will realize that I will earn (1-T)D in cash over the life of the company, therefore I will demand that its cashflow increase by betaU(1+(1-t)D/E)) or betaU + betaU - betaU(t)(D/E), as before we'll set betaU as 1 and t as .35,

we then have 1 + 1 - 1(.35)(1) = 1.65 = BetaL,

therefore If a company with no debt takes on a debt load with D/E = 1, as an equity holder I will demand FCF increase by 2 in the case that interest is NOT tax deductible

And I will demand that FCF increase by 1.65 in the case that interest is tax deductible!!

I was thinking about this earlier and writing it out helps me understand it, sorry for such a long essay

That's probably the best explanation I've ever seen, and I was a 4.0 in my finance classes. If I had an SB I'd give you one. Well done.

## PondFish: Sry just stumbled

PondFish:Sry just stumbled on this thread, was just thinking about this formula..

...

I was thinking about this earlier and writing it out helps me understand it, sorry for such a long essay

Really great post. Helped a lot. My last SB for you.

## PondFish: Sry just stumbled

## Thanks guys, I was very

## I'm new to finance too, for

## Bong bong tango: I'm new to