✏️ Does leverage change the value of the entrepreneur's portion?

You are considering starting a business with some friends from work. To fund the startup, you need to raise 3 million. Daymond John is willing to provide you with three million in initial capital in exchange for 20% of the unlevered equity in the firm.

A. What is the total market value of the firm now?

This is essentially a post-money valuation calculation:

• If 20% of the firm is worth $3M, then 20% of the market cap must be $3M
• In formulas:
  • $20\% * \text{Market Cap} = \$3M$
  • Market Cap = 3/.2 = $15M ← divide both sides by 20%

Because the firm is currently unlevered, this is the unlevered value of the firm. Ie. $V_U=\$15M$

B. Suppose you borrow 1 million. According to​ MM, what fraction of the​ firm’s equity will you need to sell to raise the additional money you​ need for the $3M initial investment?

According to MM1:

$$VL = D + E = VU$$

We know VU = $15M. Therefore,

$$D + E = \$15M$$

We are borrowing $1M of debt at a fair market rate. That means that when we borrow it, it is worth exactly $1M. Therefore, D=$1M. This gives us:

$$\begin{aligned} \$1M + E &= \$15M \\ E &= \$14M \end{aligned}$$

Ie the value of the levered equity is $14M and the value of the debt is $1M, so the value of the levered firm is $15M. $V_L = D + E = V_U$

We’ve raised $1M of debt capital through a bank loan or a debt offering. We need to raise an additional $3M -$1M = $2M more. We raise this by issuing equity.

If the total equity is worth $14M and we need to raise $2M, then we need to sell 2/14=14.29% of the company. That leaves 12/14=85.71% of the company for the original owners (you).

A. What is the value of your share of the​ firm’s equity in cases ​A and B?

In case A, you own 80%=(100%-20%) of equity worth $15M, so your portion is .8*15 = $12M

In case B, you own 12/14 of equity worth $14M, so your portion is worth 12/4*$14M = $12M

A second similar example

✏️ You are starting a firm. You must raise $9,000 to start the firm. You can either sell 30% of the firm to raise the $9,000 or you can borrow $5,000 in the debt markets and sell equity to raise the rest. How much will your stake in the company be worth either way? Assume that the MM assumptions hold.

✔First, we must establish the value of the company. If 30% of the company is worth $9,000, then the company is worth
30% * MC = $9,000 ⇨ MC = $9,000/30% = $30,000

In this case you would own 70% of a $30,000 company, so you would own 70%*$30,000 = $21,000.

Next, suppose the firm borrowed $5,000. MM1 says that

$V_L = E + D = V_U$
Therefore, we can rearrange this to find the value of the levered equity:
$E=V_U − D \\ \quad = \$30,000 - \$5,000 \\ \quad =\$25,000$

A summary to illustrate MM1:
VU = Unlevered Equity	$30K
E	$25K
D	$5K

So how much equity must you sell? The firm has raised $5K from its debt sale. It still needs to raise $9K-$5K=$4K more. How much of the firm must you sell to raise the $4k?

As above , we do

%x * $25K = $4K ⇨ %x = $4K/$25K = 4/25 = 16.6666666666 %

This leaves you with 100%-16%=84% of the firm. How much is your stake worth?

84% * $25,000= $21,000

You conclude, drat, that under the MM1 assumptions, the total value of your stake in the company is not dependent on the balance between equity and debt in the capital structure of the firm. Either way, your stake is worth $25,000.