## Abstract

Standard discounted cash flow approaches suffer from a rudimental modeling of the possibility of a default, as the main characteristics such as the default probability and potential bankruptcy costs are commonly disregarded. This paper aims at providing a tractable extension of the well-known WACC approach for both default risk and bankruptcy costs. The corrected WACC discount rate reveals that default risk results in a systematically higher WACC because the tax component is scaled by the survivorship probability and an aditional component for bankruptcy costs must be added. This difference between the classical WACC discount rate and the simple modified WACC rate can be remarkable especially for firms from businesses with high bankruptcy costs and a relevant default probability.

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## Notes

- 1.
- 2.
Noteworthy exceptions for DCF approaches dealing with bankruptcy are Cooper and Nyborg (2008), Molnar and Nyborg (2012). In these approaches the tax shield consequences for defaultable debt are quantified. However, the explicit default probability and bankruptcy costs are not considered and the resulting discount rate does not exhibit a tractable form.

- 3.
Moosa and Li (2012) provide empirical determinants of firms’ capital structure choice.

- 4.
The nominal interest rate

*c*might contain a risk premium for default risk and bankruptcy costs. However, regardless of the size of*c*, the company cost of capital*k*_{ V }within the DCF approach ensure a consistent pricing of the firm value (see Fernandez (2012), topic 3) for the treatment of nominal interest rates. - 5.
In the case in which only a default and a survivorship with full use of tax shields occurs, the expected level of tax shields is equal to the survivorship probability

*p*and the expected level of bankruptcy costs equal to the default probability 1 −*p*.

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## Appendix: WACC formula for multi-state case

### Appendix: WACC formula for multi-state case

To account for a very general case of tax shields, in which the firm does not default but only obtains a part of the full tax shield, we can extend the model framework as follows:

Let the tax shields at time *t* + 1:

where \(\phi_{\tau}\in\left[ 0;1\right] \) denotes the fraction of the full tax shield \(\tau\cdot D_{t}\cdot c\) that is in effect at time *t* + 1. In the case of positive earnings, *ϕ*
_{τ} is obviously one, and in the case of a default in which the tax deductions cannot be used,* ϕ*
_{τ} amounts to zero. In other cases,* ϕ*
_{τ} can take values between zero and one which can be interpreted as a firm that is currently able to satisfy its interest payments but does not have positive earnings in order to take full advantage of their feasible tax shields. Hence, \(\phi_{\tau}\cdot\tau\cdot D_{t}\cdot c\) is the present value of tax shields that is obtained from carrying forward the tax deductions \(\tau\cdot D_{t}\cdot c\) to future periods. Technically speaking,* ϕ*
_{τ} is a random variable from the perspective of time *t* and realizations are observed at time *t* + 1. We impose the assumption that the distribution of the realization of* ϕ*
_{τ} at time *t* + 1 is known at time *t*. Hence, we no longer have two states in which* ϕ*
_{τ} can only be zero or one but now we can incorporate arbitrarily many cases with arbitrary distributions for* ϕ*
_{τ}.

We can apply a similar notion for the bankruptcy costs. Let us denote the bankruptcy costs at time *t* + 1 as:

where* ϕ*
_{
α
} denotes the fraction of bankruptcy costs that is in effect at time *t* + 1. Apparently, without a default* ϕ*
_{
α
} is zero and in the case of a default it amounts to one. However, in the case of a reorganization, in which both outcomes a liquidation and a recovery are thinkable,* ϕ*
_{
α
} can be between zero and one. In line with the modelling of the discount factor* ϕ*
_{τ} for tax shields, we also assume that the statistical distribution for* ϕ*
_{
α
} is known one period in advance.

When modifying Eq. (6) with the notation introduced in this appendix, we now obtain for the expectation of total wealth from holding a firm alive at time *t* for one more period:

The second equation is a straightforward consequence of the linearity of the expectation operator \({\mathbb{E}_{t}\left( \cdot\right) . }\) Combining this representation with the representation for the WACC discount factor in Eq. (5), we obtain:

As a result, the WACC discount is the company cost of capital *k*
_{
V
} reduced by a tax shield component and increased by a bankruptcy component. The tax shield term equals the well-known term \(\tau\cdot\frac{D}{V}\cdot c\) multiplied by the expected level \({\mathbb{E}_{t}\left( \phi_{\tau}\right). }\) The same is true for the bankruptcy term.

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Koziol, C. A simple correction of the WACC discount rate for default risk and bankruptcy costs.
*Rev Quant Finan Acc* **42, **653–666 (2014). https://doi.org/10.1007/s11156-013-0356-x

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### Keywords

- Firm valuation
- Discounted cash flow (DCF)
- Default risk
- WACC approach

### JEL Classification

- G12
- G31
- G33