# Deriving and explaining the weighted cost of capital

The Wikipedia article for the "weighted cost of capital" (WACC) defines the WACC as "the rate that a company is expected to pay on average to all its security holders to finance its assets." What exactly does this mean? Does this mean that this is supposed to be the expected returns on the company's stocks as well as the expected return on the company's bonds? Also, how are the two equations, given in the Wikipedia article, derived? For convenience, the two equations are the following. The first is $$\text{WACC} = \frac{\sum_{i=1}^N r_i \cdot MV_{i}}{\sum_{i=1}^N MV_i}, \tag{1}$$ where $N$ is the number of sources of capital (securities, types of liabilities); $r_i$ is the required reate of return for security $i$; and $MV_i$ is the market value of all outstanding securities $i$. The second is $$\text{WACC} = \frac{D}{D+E}K_d + \frac{E}{D+E}K_e \tag{2},$$ where $D$ is the total debt, $E$ is the total shareholder’s equity, $Ke$ is the cost of equity, and $Kd$ is the cost of debt.

EDIT: Thanks for the answers below. "Derivation" was too strong of a word to use, which was clear after the explanations given. It was just a little unclear what the objects were in practice. The discussions below clarified things. Also, I found this paragraph from a website linked in one of the answers helpful:

To understand WACC, think of a company as a bag of money. The money in the bag comes from two sources: debt and equity. Money from business operations is not a third source because, after paying for debt, any cash left over that is not returned to shareholders in the form of dividends is kept in the bag on behalf of shareholders. If debt holders require a 10% return on their investment and shareholders require a 20% return, then, on average, projects funded by the bag of money will have to return 15% to satisfy debt and equity holders. The 15% is the WACC.

If you are asking "Is the WACC the amount that the company expects to earn on the stocks and bonds that it holds.." then the answer is no.

The WACC, in very simple terms, is the amount of money a company pays to obtain financing for projects. These types of financing are clearly listed in the wikipedia article and clearly extend beyond stocks and bonds issued by the company.

A company can calculate the WACC to determine whether or not undertaking a project will be profitable. That is, a company can determine that it must earn some minimum return on a project before that project becomes profitable.

You, as a consumer, possibly have experience with cost of capital. Consider the interest paid on a car loan. That interest is the cost you incur for financing the purchase of your vehicle by borrowing funds from a financial intermediary.

As far as derivation goes - I'm not sure how to simplify these formulas much more. However, you can think simply about the first as the ratio of the interest paid out on sources of financing and the total market value of those sources of financing (so...some ration < 1) and the second is a more simple case of the first.

http://www.investopedia.com/walkthrough/corporate-finance/5/cost-capital/wacc.aspx

I think this website gives a decent walkthrough of the concept. Hope this helps some.

The concept of $\text{WACC}$ seems pretty straightforward... it is a weighted average percentage, calculated in principle as equation $(2)$ in the question shows. If we have two sources of financing each demanding a different interest rate and with given percentage contribution each to the total funds we want to borrow, then what would be the single interest rate that would lead to the same total interest cost, if we borrowed the sum of funds from some third source? And this easily generalizes to $n$ sources of financing.

The concept stops being so straightforward the moment one realizes that the property rights and the transaction clauses characterizing the "cost of equity funds" and the "cost of debt funds" respectively are totally different (and this is why this is the first grand partition), and also, that "the future won't necessarily repeat the past".

Related to Debt funds, Debt owners have the full power of the Law behind them to seek payment of interest and principal, against the company's assets. Also, the interest rate is specifically agreed to a certain level, and even if there are provisions that may make it variable, these usually create few scenarios one would need to evaluate. So calculating the (unitary) "cost of debt" appears relatively transparent and certain.But: a distinction has to be made related to past "cost of debt" and the prospective one: the current debts of the company may have a certain interest rate agreed upon, but the interest rate that the company must pay in order to get more debt may be different. This essentially tends to make the concept a marginal weighted average, and the company must conduct a market search in order to see what interest rates prospective debt owners would demand for new debt. But for an outsider, this information is usually not available -he has to rely on published data on debt and interest rates agreed in the past from the company's financial statements.

Turning to Equity funds, the equity holder has no power whatsoever over the company to seek payment of any rate of return, or of his funds (in cases of fraud, gross negligence etc, the equity holder may demand compensation from the persons that were involved, managers etc, but not from the company as a legal entity). In other words, there is no "legally committed" interest rate here, nor "obligatory" rate of return (funds that appear as equity but are characterized by such clauses are not considered equity but debt, under both Finance practices and Accounting principles). So here the (unitary) "cost of equity" is the rate of return that prospective equity holders would want to be persuaded that the company will generate using their funds. In two-steps: I want $\text{xx}$% rate of return, and in order to invest my funds to you as equity, you have to "persuade" me that you will manage to pay me that rate of return. Persuade how? By past performance, convincing analysis of future prospects, and charm.
So how do we calculate this component of $\text{WACC}$? We usually look at how the company is performing relative to its sector. Why the sector and not the whole market, has to do with increased risk that correlates with inexperience: investors that for some historical reasons have concentrated their investments in a sector, will be in a better position to judge the validity of prospective projects in this sector, than to some other they don't have a clue about. So some combination of company/sector rates of return provides an estimation of the (unitary) cost of equity (this again brings in the distinction between a "historical $\text{WACC}$" against a "marginal" $\text{WACC}$).

And finally, how do we calculate the relative weights of the two sources? Again, time series data from the company's financial statements can show how this mix has evolved over the years for the company.

• I wouldn´t personally use these formula for anything important, and it would be very interesting to see the derivation jmb is asking for. There´s obvious problems - rate of return for an dividend paying equity for example is independent of market value, as you point out. wrt to debt funds - I´m afraid you´re incorrect, the status of the fund wrt to the law is country and financial instrument dependent. See for example the difference between non-recourse and full-recourse lending that played a significant role in the 2007 debacle.
– Lumi
Jan 7 '15 at 15:36
• @Lumi "Debt" and "equity" are used as economic categories, not as legal nomenclature. They are fundamental contracts into which funds are to be categorized, by looking at the economic essence of each legal contract, irrespective of how each country/law etc baptizes the contract. A "lending" contract under which the debtor has various "get away with it" clauses, is not really debt -and it may not be exactly equity. Indeed, many financial instruments may be hybrids, which just make matters more complex -but the "formula" as a guiding concept is used all over the world to calculate $WACC$. Jan 7 '15 at 15:51
• & economics currently has a bad reputation around the world because these kinds of elementary mistakes are not only tolerated but defended. The real world is not a special case, and the difference between recourse and non-recourse debt has clear, and identifiable economic consequences.
– Lumi
Jan 7 '15 at 15:56
• @Lumi Yes the "non-recourse debt" is not debt in the same way that a full-recourse debt is. Which is exactly what I am saying all along. And it is Economics that a) makes the distinction and b) has the tools to quantify the difference. So Economics (in contrast to law, politicians or business executives) should actually be praised on the matter. Jan 7 '15 at 16:00
• If we agree that both are forms of debt - that's the point I'm making - however you wrote, "a lending contract under which the debtor has various... is not really debt". In my experience, lawyers and politicians are guided by economists, so it's not fair to blame them - and competant business executives are I imagine quite happy for their competitors to use formula such as this.
– Lumi
Jan 7 '15 at 16:05

The first equation is dollars time interest over total dollars.

For example, if a company wants to finance a project, issues \$1M in equity with an expected ROI to the investors of 6% and \$4M in bonds at 4%, it's WACC is:

$$\frac{(4\%*4,000,000 + 6\% * 1,000,000)}{4,000,000+1,000,000}$$

which for simplicity we can say is

$$\frac{(4\%*4 + 6\% * 1)}{4+1}$$

or:

$$\frac{22\%}{5} = 4.4\%$$

As mathtastic pointed out, it means the project needs to exceed an ROI of 4.4% to be viable.

The second WACC formula is assuming an average for debt and an average for equity, and then expressing the percentage of financing as debt times the average interest rate on debt, and the percentage of financing that's equity times the expected or needed ROI on equity to secure financing.

They derive from each other because $r_i$ and $MV_i$ don't differentiate between debt and equity.