Cost of capital

For a financing title , its cost of capital is the expected interest rate at which the creation of the financing title is only possible. ${\displaystyle f}$ ${\displaystyle r\left(f\right)}$

There are

• the cost of equity for , if an equity security is, ${\displaystyle f}$ ${\displaystyle f}$
• the borrowing cost rate for , if a receivable title is, ${\displaystyle f}$ ${\displaystyle f}$
• the total cost of capital for when a business is. ${\displaystyle U}$ ${\displaystyle U}$

Since financing titles can be “packed together” (for example in a fund ), there is also a cost of capital rate for each compound financing title .

• The equity of a company can be understood as a composite financing title of all equity securities of the company.
• The debt capital of a company can be understood as a composite financing title of all receivables of the company.

Total cost of capital

Be

• ${\displaystyle U}$a company whose liabilities consist only of senior debt and subordinated equity,
• ${\displaystyle EK\left(U\right)}$the company's equity , ${\displaystyle U}$
• ${\displaystyle FK\left(U\right)}$the company's borrowed capital , ${\displaystyle U}$
• ${\displaystyle r\left(EK\left(U\right)\right)}$the company's cost of equity , ${\displaystyle U}$
• ${\displaystyle r\left(FK\left(U\right)\right)}$the company's borrowing cost rate , ${\displaystyle U}$
• ${\displaystyle r\left(U\right)}$the company's total cost of capital , ${\displaystyle U}$
• ${\displaystyle V\left(EK\left(U\right)\right)}$the market value of the company's equity , ${\displaystyle U}$
• ${\displaystyle V\left(FK\left(U\right)\right)}$the market value of the company's borrowed capital , ${\displaystyle U}$
• ${\displaystyle V\left(U\right)}$the market value of the company . ${\displaystyle U}$

Then because of : ${\displaystyle V\left(U\right)=V\left(EK\left(U\right)\right)+V\left(FK\left(U\right)\right)}$

• ${\displaystyle r\left(U\right)={{r\left(EK\left(U\right)\right)\cdot V\left(EK\left(U\right)\right)+r\left(FK\left(U\right)\right)\cdot V\left(FK\left(U\right)\right)} \over {V\left(U\right)}}}$ respectively

${\displaystyle r\left(U\right)={{r\left(EK\left(U\right)\right)\cdot V\left(EK\left(U\right)\right)+r\left(FK\left(U\right)\right)\cdot V\left(FK\left(U\right)\right)} \over {V\left(EK\left(U\right)\right)+V\left(FK\left(U\right)\right)}}}$

• ${\displaystyle r\left(U\right)\cdot V\left(U\right)=r\left(EK\left(U\right)\right)\cdot V\left(EK\left(U\right)\right)+r\left(FK\left(U\right)\right)\cdot V\left(FK\left(U\right)\right)}$

Modified cost of capital

If the assumption of perfect capital markets is dispensed with, capital cost rates are derived directly from measurable risk information of the payment series (according to planning). The capital market only needs to determine the market price of the risk, but not the determination of the risk measure (e.g. the equity requirement). Such approaches thus take into account the availability of superior information about the series of payments (e.g. in corporate management versus the capital market) and, if necessary, also the assessment relevance of non-diversified company-specific risks.

The following formula is used to calculate the cost of capital (WACC) depending on the equity requirement as a risk measure (Gleißner, 2005):

${\displaystyle WACC^{\text{mod}}=k_{\mathrm {EK} }{\frac {\text{Eigenkapitalbedarf}}{\text{Gesamtkapital}}}+k_{\mathrm {FK} }{\frac {{\text{Gesamtkapital}}-{\text{Eigenkapitalbedarf}}}{\text{Gesamtkapital}}}\cdot (1-s)}$

An increasing risk with a higher need for “expensive” equity leads to increasing total cost of capital (WACC) ( ). The equity requirement (EKB) as a risk measure and the equity costs depend on the maximum probability of default p accepted by the lenders . In order to aggregate the individual risks - systematic and non-diversified unsystematic - of a company into equity requirements, they must first be described by probability distributions and then assigned to those positions in corporate planning where they can lead to deviations from the plan. With the help of simulation methods (Monte Carlo simulation), a large representative sample of possible risk-related future scenarios of corporate development is then evaluated, which allows conclusions to be drawn about the extent of risk-related losses. ${\displaystyle k_{\mathrm {EK} }>k_{\mathrm {FK} }}$

A “suitable” cost of equity rate must be calculated for the equity requirement, which also depends on p . A simple estimate is possible using the following method, which assumes an investment of equity in the market portfolio (shares) as an alternative investment to the company.

It calculates the expected return on investment in a stock portfolio (market portfolio) if it had the same probability of default due to the use of outside capital (opportunity costs). Depending on the expected return on the market portfolio ( ), the standard deviation of this return ( ) and the accepted probability of default p , the following rating-dependent (p-dependent) costs of equity are obtained. ${\displaystyle r_{m}^{e}}$ ${\displaystyle \sigma _{m}}$

${\displaystyle r_{\mathrm {EK} ,p}^{e}={\frac {r_{m}^{e}-(1-a)\cdot k_{\mathrm {FK} ,p}}{a}}}$

With

${\displaystyle a=-\left(r_{m}^{e}+q_{p}\cdot \sigma _{M}\right)}$

${\displaystyle k_{\mathrm {EK} ,p}=r_{\mathrm {EK} ,p}^{e}={\frac {r_{m}^{e}-(1+(r_{m}^{e}+q_{p}\cdot \sigma _{M}))\cdot k_{\mathrm {FK} ,p}}{-(r_{m}^{e}+q_{p}\cdot \sigma _{M})}}={\frac {r_{m}^{e}\cdot (1-k_{\mathrm {FK} ,p})-(1+q_{p}\cdot \sigma _{M})\cdot k_{\mathrm {FK} ,p}}{-(r_{m}^{e}+q_{p}\cdot \sigma _{M})}}}$

Here, a is the equity share in the portfolio (EKB as a percentage of the investment), so that the probability of default p is achieved. In addition, the value of the inverted distribution function of the standard normal distribution at the confidence level and the expected return on debt with an accepted probability of default is p (not the debt interest rate). ${\displaystyle q_{p}}$ ${\displaystyle \alpha =1-p}$ ${\displaystyle k_{\mathrm {FK} ,p}}$

Literature and references

1. ↑ Werner Gleißner (2006): New ways for company valuation and value-oriented company management in an imperfect capital market. In: Finance and Accounting. Yearbook. 2006, ZDB -ID 2064250-7 , pp. 119-154.

See also

• Capital Asset Pricing Model (CAPM)