Cost of Capital

BUSI 721: Data-Driven Finance I

Kerry Back, Rice University

Overview

• Equations for efficient portfolio with cash
• Risk premium formula based on beta with respect to efficient portfolio
• Capital Asset Pricing Model (CAPM)
• Weighted Average Cost of Capital

Efficient portfolio with cash

minimize

$$\frac{1}{2} w'\Sigma w$$

subject to

$$\mu'w + \left(1-\sum_{i=1}^n w_i\right)r_f= r$$

where $r=$ target expected return and $r_f=$ borrowing and saving rate.

Target expected return constraint

• Expected return is $$\mu'w + (1-\iota'w)r_f = r_f + (\mu-r_f\iota')w$$
• Equals target expected return $r$ if and only if $$(\mu-r_f\iota)'w = r-r_f$$
• So, minimize $$\frac{1}{2} w'\Sigma w$$

subject to

$$(\mu-r_f\iota)'w= r = r_f$$

Marginal benefit-cost ratios

• Benefit = risk premium = $(\mu-r_f\iota)'w$
• Cost = half of variance = $(1/2)w'\Sigma w$
  • Minimize cost subject to benefit $=r-r_f$
  • The optimum is such that the ratio of marginal benefit to marginal cost is the same for all assets.

$$ \frac{\partial \text{Benefit}/\partial w_i}{ \partial \text{Cost}/\partial w_i} = \text{constant depending on $r-r_f$}$$

Related problem

• Outputs 1, ..., n. Quantities = $x_i$
• Output prices $p_i$
• Efficient production plan miminizes cost given the revenue it produces: $$ \text{minimize} \quad \frac{1}{2}C(x_1, \ldots, x_n) \quad \text{subject to} \sum_{i=1}^n p_ix_i = \text{revenue target}$$
• Optimum is such that $$ \frac{p_i}{\partial C/\partial x_i} = \text{constant depending on revenue target}$$
• as long as the $x_i$ are $\ge 0$ at the solution of these equations.

Why marginal benefit-cost ratios the same?

• Suppose two outputs both have marginal cost = 1. One has price = 1 The other has price = 2.
• Reduce production of the low-price output by $\Delta$.
• Increase production of the high-price output by $\Delta$.
• Cost goes down from reducing production of one and up from increasing producing of other, and these offset due to equal marginal costs.
• Make more revenue.

Back to finance

• Marginal benefit is the risk premium of asset $i$ = $\mu_i - r_f$.
• To calculate marginal cost, assume only two assets for simplicity.

$$\text{Variance} = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2 \rho\sigma_1\sigma_2 w_1 w_2$$

• Differentiate with respect to $w_1$ for example:

$$\frac{\partial \text{Variance}}{\partial w_1} = 2w_1\sigma_1^2 + 2\rho\sigma_1\sigma_2 w_2$$

$$\frac{\partial \text{Cost}}{\partial w_1} = w_1\sigma_1^2 + \rho\sigma_1\sigma_2 w_2$$$$ = w_1\text{cov}(r_1, r_1) + w_2 \text{cov}(r_1, r_2)$$$$ = \text{cov}(r_1, w_1r_1) + \text{cov}(r_1, w_2r_2)$$$$ = \text{cov}(r_1, w_1r_1 + w_2r_2)$$

• With $n$ assets, the marginal cost of asset $i$ is $$\text{cov}\left(r_i, \sum_{i=1}^n w_ir_i\right)$$ where $(w_1, \ldots, w_n)$ is the efficient portfolio.

• Call the constant $k$
• So, marginal benefit / marginal cost $=k$
• So, marginal benefit $=k \times \text{marginal cost}$
• So, for each asset, $$ \text{risk premium} = k \times \text{cov}(\text{asset return}, \text{efficient portfolio return})$$
• This provides some hope for estimating the return investors expect from a stock (expected return = risk premium $\;+\;r_f$)

A few more steps

• The risk premium formula also holds for portfolios of assets, so use it for the efficient portfolio return: $$\text{efficient portfolio risk premium} = k \times \text{var}(\text{efficient portfolio return})$$
• This is a formula for $k$: $$k = \frac{\text{efficient portfolio risk premium}}{\text{var}(\text{efficient portfolio return})}$$
• Substituting this, we get, for each asset, $$ \text{risk premium} = \frac{\text{cov}(\text{asset return}, \text{efficient portfolio return})}{\text{var}(\text{efficient portfolio return})}$$ $$\qquad \qquad \times \; \text{efficient portfolio risk premium}$$

• A covariance to variance ratio like this is the slope in a regression.
• Let $r_p$ denote the efficient portfolio return.
• We are dealing with risk premia, so compute excess returns $r_i-r_f$ and $r_p-r_f$.
• Run the regression: $$r_i - r_f = \alpha_i + \beta_i (r_p-r_f) + \varepsilon_i$$
• Covariance to variance ratio is $\beta_i$. So, for each asset, $$\text{risk premium} = \beta \; \times \text{efficient portfolio risk premium}.$$

• All of this is tautological and so far doesn't help us.
• To find the efficient portfolio by calculation, we already need to know all of the risk premia.

• But suppose we could guess a efficient portfolio. Then,

  • We can run regressions to estimate betas.
  • We can estimate the efficient portfolio risk premium using hopefully a very long data series of returns.
  • We get a formula for the return investors expect from a stock.

  $$\text{expected return} = r_f + \beta \times \text{efficient portfolio risk premium}$$

Capital Asset Pricing Model (CAPM)

• Assume the market portfolio of stocks is an efficient portfolio.
• Why? People should hold efficient portfolios - maximize the Sharpe ratio.
• A weighted average of portfolios on the maximum Sharpe ratio line is also on the line.
• So the market is efficient if everyone holds an efficient portfolio.
• CAPM:

$$\text{expected return} = r_f + \beta \times \text{market risk premium}$$

Intuition for the CAPM

• Market risk is the biggest risk diversified investors hold
• Therefore, the relevant risk of an asset is how much it contributes to market risk
• This depends on its covariance with market risk, captured by beta
• So, the risk premium of an asset should depend on its beta

Weighted average cost of capital (WACC)

• Compute equity / (equity + debt) and debt / (equity + debt) on a market value basis.
• Compute the expected return on your stock using probably the CAPM = cost of equity.
• Determine the coupon at which you could issue debt at face value (market rate for your debt) = cost of debt.
• Determine the marginal tax rate. It matters because interest is tax deductible but dividends are not.

$$\text{WACC} = \text{pct equity} \times \text{cost of equity}$$$$\qquad \qquad\text{$+$ pct debt} \times (1-\text{tax rate}) \times \text{cost of debt}$$