Optimal Financing Mix: Cost of Capital Approach

Question

According to Cost of Capital approach to optimal financing mix we can calculate Cost-of-Capital-minimizing $\frac{D}{E}$ ratio as follows:

$\frac{D}{E}_{opt} = argmin_{\frac{D}{E}}WACC$,

where

$WACC = \frac{E}{D+E}r_{e} + \frac{D}{D+E}r_{d}(1-T)$

Cost of Equity ($r_{e}$) is determined by CAPM model (with given levels of riskfree rate $r_{f}$ and market premium $r_{m}$):

$r_{e} = r_{f} + \beta_{L}r_{m}$

Levered value of $\beta$ is determined by Hamada's equation:

$\beta_{L} = \beta_{UL}(1+ \frac{D}{E}(1-T))$

Cost of Debt ($r_{d}$) is determined using bond rating approach, where each interval of $\frac{D}{D+E}$ (in this case 0%-10%, 10%-20% and so on) is assigned with a certain interest rate on debt.

This gives us:

$WACC = \frac{E}{D+E}(r_{f} + \beta_{UL}(1+ \frac{D}{E}(1-T))r_{m}) + \frac{D}{D+E}r_{d}(1-T)$

Assume $\frac{D}{E} = X$ After all simplifications we have:

$WACC = \frac{1}{X+1}(r_{f} + \beta_{UL}r_{m}) + \frac{X}{X+1}(1-T)(r_{d} + \beta_{UL}r_{m})$

Now all we have to do to minimize is $\frac{\text{d}WACC}{\text{d}X} = 0$

Since $r_{d}$ is a function of $X$ and we would have to take $\frac{\text{d}r_{d}}{\text{d}X}$, we will instead seach the minimum on each interval (0%-10%, 10%-20% and so on), where $r_{d}$ will be constant.

Taking the derivative:

$\frac{\text{d}WACC}{\text{d}X} = \frac{(1-T)(r_{d} + \beta_{UL}r_{m}) - (r_{f} + \beta_{UL}r_{m})}{(X+1)^{2}}$

Assuming X is nonnegative, the only solution to

$\frac{(1-T)(r_{d} + \beta_{UL}r_{m}) - (r_{f} + \beta_{UL}r_{m})}{(X+1)^{2}} = 0$

is

$(1-T)(r_{d} + \beta_{UL}r_{m}) - (r_{f} + \beta_{UL}r_{m}) = 0$

All the variables here are constant and independent of $X$, which gives us no answer about the optimal $\frac{D}{E}$ ratio. Am I missing something here?

Answer 1

The point is exactly that, $r_d$ depends on $X$, meaning that: $r_d(X)$. So in practice you will have an answer $X=f(r_d)$. Where X is a function of $r_d$. Also, if you change your capital structure, given your Hamada's equation, $r_m$ will also be a function of $X$ and therefore in fact your optimal $X$ should be : $X=f(r_d,r_e)$. You cannot solve this problem analytically given your assumptions.

The best way is to use a minimization algorithm (solver in excel should do the trick) where you minimize:

$WACC = \frac{E}{D+E}(r_{f} + \beta_{UL}(1+ \frac{D}{E}(1-T))r_{m}) + \frac{D}{D+E}r_{d}(1-T)$

subject to the constraints:

(1) $r_{e} = r_{f} + \beta_{L}r_{m}$

(2) $\beta_{L} = \beta_{UL}(1+ \frac{D}{E}(1-T))$

(3) $r_d$ as a function of the buckets you defined

This should be very straightforward to compute.