I’m trying to understand how they derived this formula but spinning my wheels and not getting anywhere:
On page 93 of the Corp Finance text:
Let us define Ro as the cost of capital for a company financed only by equity. Then by MM porposition I, Rwacc = Ro, so
Rwacc = (D/V)*Rd + (E/V)*Re = Ro (3)
Recalling that D+E = V and using this to substitute for V, we can rearrange Equation 3 to solve for the cost of Equity
Re = Ro + (Ro - Rd) * D/E (4)
My algebra must be failing me, I just can’t seem to make the jump from eqtn 3 to 4 using the D+E = V substitution.
Anybody care to have a crack?
Never mind, I ended up figuring it out, though I think the formula will be easier to recall from memorisation rather than trying to re-derive it from scratch.
One thing I’m still not 100% clear on is why is the assumption of Ro made in the first place in Eqn(3). Ro is meant to be the cost of an all equity company, so why is there D/V component in the formula. Doesn’t that relationship only hold when D/V is 0?
This is one formula I’m not certain there’s need to recall or derive. Surely, one can simply use the same ol’ WACC formula and then plug in the figures, no?
Don’t we at least need to know this one?
Reqt = Ro + (D/E) *(Ro - Rd) * (1- tax)
where Ro = unlevered cost of equity and Rd = cost of debt.
The Capital Structure reading seemed very easy when I ran through it on the first pass, but now that I’m digging deeper seems a lot more complicated.
The EOC Q14-19 item set is really tripping me up so trying to get a more intuitive understanding of this reading rather than rely on the old plug and chug approach.
If you know the weighted average cost of capital formula, you shouldn’t have a need to learn or derive this formula.
Take the example on page 93 for instance.
You know that WACC = (D/V)*Rd + (E/V)*Re
You were given D = 15,000, E = 35,000, total capital =50,000, WACC = 10%, Rd = 5%. You simply have to back out Re
The situtation will often be framed as, Mr X has a company that is all equity financed, in which case the cost of equity s equal to the WACC, what will be the effect on cost of equity if Mr X increases his leverage by say 50%.
The weighted average cost of capital will remain the same, given this is only a weighted cost of capital from two sources of Finance, but the cost of equity, Re will change to reflect risks due to increasing leverage. (this is what the whole MM proposition two is about)
Hey bloodline,
Thanks for your response, your explanation made me realize I really wasn’t grasping some key concepts. I’ve had a bit more of read through the material. I agree with you that you can use the WACC to derive Reqt based on changing (D/V) when assumption of no taxes, but when taxes come into the equation I think you need to know this formula:
Reqt = Ro + (D/E) *(Ro - Rd) * (1- tax)
With no taxes, the WACC is constant (MM proposition 2), but when taxes come into play the WACC will change, so trying to solve Reqt using the WACC equation will leave trying to solve Reqt with 2 missing variables.
This is at least my understanding. Let me know if you disagree.
You will only have one missing variable at a time.
Introducing taxes have no effect on the cost of equity, but on the WACC. The weighted average cost of capital is reduced due to the tax shield.
Think about it this way;
In an All Equity company Cost of Equity = WACC.
When you introduce taxes at this point, no effect is felt on the cost of equity or WACC, the value of the firm simply decreases due to additional claim on its cashflows. i.e taxes
Now introducing leverage
Cost of Equity increases due to leverage, WACC stays the same, Cost of debt is known, we can simply solve for Cost of Equity with our good ol formula — (this explains proposition 2)
When you introduce taxes at this stage.
Cost of equity is unchanged, the value of the firm decline, but not as much as in the first case without leverage. Cost of debt is known, tax is known, the overall cost of the firm’s capital is the only thing that would have changed. We simply solve for the over cost of capital with our good ol formula.
Yup, looks like you’re right again. Just went through the numbers. It’s weird, you calculate the cost of equity with debt added but with no taxes, then apply that cost of equity in your new WACC calc with taxes (and new D/E relationship).
My mind is blown but I think I got this topic covered now.
Thanks mate.
Just in case anybody cares :
Rwacc = (D/V)*Rd + (E/V)*Re = R0 (open up the parentheses and make it a single fraction)
(D*Rd + E*Re)/V = R0 (Multiply both sides by V) ->
D*Rd + E*Re = R0*V (Solve for Re) ->
Re = (R0*V - D*Rd)/E (Convert V using V = D+E) ->
Re = [R0*(D+E) - D*Rd]/E =>
Re=(R0*D + R0*E - D*Rd)/E (Factor our D) ->
Re= [D*(R0-Rd) + R0*E]/E (Break into two fractions) ->
Re= [(R0-Rd)*D]/E + R0*E/E =>
Re = R0 + (R0-Rd)*D/E