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

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


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

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

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


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

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?


1 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.

  • $\begingroup$ Obtaining a numerical solution is fairly straightforward here. I just want to know what's wrong with my attempt of obtaining an analytical solution. $r_{e}$ is obviously dependent on $X$, however it is not present in the final equation due to the substitutions I've performed, so we don't have to account for that. Moreover, rather than minimizing the whole function, we search the minimum on each interval of $\frac{D}{D+E}$ so that on each of the intervals $r_{d}$ is independent of X and constant. Then we just take the minimum of all those minimums we've found. $\endgroup$ Dec 8, 2015 at 10:36
  • $\begingroup$ I agree that $r_e$ was correctly eliminated. The issue is that there is no minimum of those minimums! If on a certain interval $rd$ is constant, then trivially the firm just wants maximum leverage on that interval. Notice that your last condition is always bigger than zero given $r_d > r_f$. $\endgroup$
    – phdstudent
    Dec 8, 2015 at 10:46
  • $\begingroup$ While it is not always bigger than zero (since one part is multiplied by $(1-T)$ and the other is not), the part about maximum leverage on each interval makes perfect sense, thanks. $\endgroup$ Dec 8, 2015 at 10:56
  • $\begingroup$ Usually it also holds that $(1-T)r_d > r_f$ . Except for very particular cases such as apple of berkshire. $\endgroup$
    – phdstudent
    Dec 8, 2015 at 11:05

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