# Cost of capital

Cost of capital is a term used in business administration and describes costs that a company incurs as a result of using equity capital for investments or obtaining outside capital for them. In practice, companies often evaluate their business activities according to whether the expected return is sufficient to cover the necessary capital costs (see also business value contribution ). The cost of capital is a value driver in "value-based management".

Lutz Kruschwitz / Andreas Löffler recommend to define the term cost of capital as a (safe) conditional expected return and to define it according to the following equation: ${\ displaystyle k}$

${\ displaystyle k_ {t} = {\ frac {E \ lfloor {\ tilde {Z}} _ {t + 1} + {\ tilde {W}} _ {t + 1} \ mid F_ {t} \ rfloor } {W_ {t}}} - 1}$

where:

${\ displaystyle W_ {t}}$= Company value at time t
${\ displaystyle F_ {t}}$= Information available at time t
${\ displaystyle {\ widetilde {Z_ {t}}}}$= Insecure payment at time t

Only in a one-period model do the expected returns and discount rates match - but not in a multi-period model.

## Borrowing costs

Borrowing costs are the costs that the company has to pay to a credit institution or other lender, primarily interest costs for loans or bonds, running costs arising from guarantees for borrowed funds or from pension debts. Ground rent or administrative cost contributions also count as borrowing costs . These costs are usually contractually regulated and known. Their amount and other conditions (term, repayment, etc.) are negotiated between the capital provider and capital user on the capital market .

A distinction must be made between the cost of debt and the contractually agreed interest rate on debt. The difference depends on the rating or the probability of insolvency, as well as the losses that creditors suffer in the event of insolvency. The contractually guaranteed interest corresponds to the contingent return of the lender in the event that the company is able to meet its contractual obligations during the term of the loan.

The cost of debt , on the other hand, reflects the expected return on debt, which is why the probability of default (p) must also be taken into account. The following relationship applies: ${\ displaystyle k_ {FK}}$

${\ displaystyle k_ {FK} = (1-p) \ times (1 + k_ {FK} ^ {0}) - 1}$

Homburg, Stephan and Weiß (2004, p. 277) explain:

"From the point of view of a partially debt-financed company, the expected return on debt represents the cost of debt that corresponds to the discount factor of the expected payments to the lender in the context of the company valuation."

## Cost of equity

The cost of equity is not a matter of actual costs, but of the expected distribution of company profits to the equity providers, e.g. the shareholders of a stock corporation . They expect a portion of the company's earnings commonly referred to as the return on investment or interest. The equity is served from the company's net income after taxes. Since the level of profit distribution fluctuates, equity investors often claim a risk premium over the possible interest rate on an investment that they have not made in fixed-income investments (see opportunity costs ). In addition, unlike borrowing costs, equity costs cannot be taken into account for tax purposes. These points mean that the cost of equity is usually set higher than the cost of debt.

Since the profit to be distributed on equity cannot be determined in advance, many companies use an imputed interest rate . In addition, the cost of equity can be determined using the capital goods price model (CAPM), which takes into account alternative investment opportunities for equity investors and a company-specific risk factor.

## Cost of capital as a control instrument

If a company is foreign capital providers no adequate return can offer, it is not viable. Therefore, every company must at least generate the cost of capital in its business activities. Is it the desired self non-capital interest rate yield, it is not considered competitive on the capital market. For investors, the cost of capital thus represents the minimum risk-based requirement for the expected return.

## Possible incorrect steering and misinterpretations in the cost of equity factor

The accounting terms lose their analytical precision, as “profit” is defined as “costs”.

Controlling investments based on opportunities on the capital market makes the development of new products more difficult, since their profitability threshold is raised above cost recovery.

In addition, investor interests gain a disproportionate weight in the business triangle of entrepreneurs, employees and investors. The fact that the entrepreneur is usually also a shareholder and thus benefits from this rebalancing sufficiently explains why this view is widespread in orthodox business administration.

## Cost of capital in accounting

In external accounting, for example in accordance with IFRS / IAS , borrowing costs must be capitalized if it is a so-called "qualified asset". For example, the provision interest can be activated with the asset if the creation of the asset takes a longer period of time.

Equity costs cannot be capitalized in accounting according to the German Commercial Code (HGB) or the International Financial Accounting Standards, as they only represent imputed costs.

## Total cost of capital

The WACC approach ( Weighted Average Cost of Capital ) is often used to calculate the total cost of capital . It results from the sum of the equity and debt costs - weighted according to their respective share in total capital.

${\ displaystyle {\ text {WACC}} = {\ frac {E} {G}} \ cdot k_ {E} + {\ frac {F} {G}} \ cdot k_ {F} \ cdot (1-T )}$

With:

${\ displaystyle E}$: Equity
${\ displaystyle G}$: Total capital (equity and debt)
${\ displaystyle k_ {E}}$: Cost of equity rate
${\ displaystyle F}$: Outside capital
${\ displaystyle k_ {F}}$: Borrowing cost rate
${\ displaystyle T}$: Tax rate ( tax shield )

The level of indebtedness is important in two ways. On the one hand, the gearing determines the weighting, and on the other hand, the level of the cost of equity itself. Due to the leverage effect, the expected equity costs of a debtor company depend on the gearing as follows:

${\ displaystyle k _ {\ text {EK}} ^ {v} = k _ {\ text {EK}} ^ {u} + (k _ {\ text {EK}} ^ {u} -k _ {\ text {FK} }) (1-s) {\ frac {{\ text {FK}} ^ {M}} {{\ text {EK}} ^ {M}}}}$

where denotes the cost of equity of an indebted company and the cost of equity of a non-indebted company. ${\ displaystyle k _ {\ text {EK}} ^ {v}}$${\ displaystyle k _ {\ text {EK}} ^ {u}}$

In a Modigliani / Miller model world, every change in the level of debt leads to a corresponding change in the cost of equity, so that an increase in the proportion of cheap debt beyond the tax advantage from debt financing does not lead to falling total cost of capital and rising company values. This result can only be transferred to the real world with moderate debt and neglect of bankruptcy costs.

${\ displaystyle k _ {\ text {WACC}} = k _ {\ text {EK}} ^ {u} \ left (1-s {\ frac {{\ text {EK}} ^ {M}} {{\ text {EK}} ^ {M} + {\ text {FK}} ^ {M}}} \ right)}$

Applicable only with autonomous financing, i.e. with constant borrowed capital over time. In principle, an assessment is also possible in the case of a variable level of borrowed capital, provided that this remains at least certain over time.

In the case of value-based financing, i.e. when adjusting the debt capital to the market value of the equity capital, the Miles – Ezzel adjustment is decisive. The following applies:

${\ displaystyle k _ {\ text {WACC}} = (1 + k _ {\ text {EK}} ^ {u}) \ left (1-s {\ frac {r_ {f}} {1 + r_ {f} }} {\ frac {{\ text {EK}} ^ {U}} {{\ text {EK}} ^ {U} + {\ text {FK}} ^ {U}}} \ right) -1}$

## Calculation problems

When dealing with the cost of capital, some problems or misunderstandings keep coming up:

1. A company's cost of capital is time-dependent and can be influenced by entrepreneurial measures (e.g. risk management), which, however, is not recorded in many value-based management systems. It changes when changing from
• Financing structure,
• Scope of risk as well
• Changes in the risk-free interest rate and the market risk premium.
2. As a benchmark for the expected return, a specific cost of capital rate is to be used, which is to be derived on the basis of the respective (additional) risks relevant to the valuation. The use of uniform cost of capital rates to assess all business activities, investments or business areas leads to serious wrong decisions.
3. Borrowing costs are lower than the contractual borrowing rates, as the former describe the expected return from lenders ( ) and thus capture a possible bankruptcy of a company (which is expressed in the rating). As in the case of insolvency, lenders no longer get back the (full) use, but only the recovery rate (RR abbreviated from English recovery rate ), (borrowing costs) at a contractually agreed borrowing rate (calculated as the expected debt yield ) and a failure probability (p) the following borrowing cost rate ( ): ${\ displaystyle r_ {FK} ^ {e}}$${\ displaystyle k_ {FK} ^ {0}}$${\ displaystyle k_ {FK}}$
${\ displaystyle r_ {FK} ^ {e} = k_ {FK} = (1-p) \ cdot (1 + k_ {FK} ^ {0}) + p \ cdot RR-1}$
4. The actual risks should be reflected in the cost of capital according to the underlying planning. This is not guaranteed if one derives the cost of capital from capital market information (especially that in the capital good price model CAPM), because the information advantage of the company itself with regard to its planning is not used.${\ displaystyle \ beta}$
5. The future expected returns of the market portfolio relevant to the valuation are lower than the historically average share returns. The reason for this so-called “Equity Premium Puzzle” is an increase in the valuation level of stocks over the past 50 years, primarily as a result of falling inflation rates and interest rates. The future long-term stock returns can be estimated from fundamental economic data as the sum of the dividend yield of a stock portfolio (approx. 3%), the long-term expected inflation rate (approx. 2.5%) and the long-term real economic growth rate (approx. 2.5%) that determine the long-term profit development. The resulting approx. 8% expected return on equity is considerably lower than the investment cost rates used in many companies. Empirical studies over the last 200 years also only show real returns on equity investments of around 6%, also because the dividend growth rate lags behind that of national income. The overestimation of the cost of capital has the consequence that many value-adding investments are wrongly omitted.${\ displaystyle r_ {m} ^ {e}}$

Caution should therefore be exercised when deriving the cost of capital (based on the CAPM capital good price model), which is still common in practice. These models assume that the capital market has the same information as corporate management, that there are no bankruptcy costs and that all investors have perfectly diversified portfolios in which company-specific risks do not play a role (and are therefore not recorded). Accordingly, capital cost rates based on the CAPM (at best) show the opinion of the capital market with regard to the risks of a company - but not the actual risk situation. Because of these weaknesses, it is not surprising that the CAPM has been refuted almost consistently in empirical studies for around 15 years. ${\ displaystyle \ beta}$

## Possible solutions

There are now new methods for estimating the cost of capital:

1. In addition to the beta factor ( ), further systematic risk factors are taken into account. In the three-factor model by Fama and French (1992), a variant of the APT, these are the book value-price ratio and the company size.${\ displaystyle \ beta _ {i}}$
2. Instead of the statistical analysis of historical returns, a future-oriented cost of capital estimate is made. These are calculated as the internal interest rate at which the stock market price is derived from the future income forecast by financial analysts.
3. In addition to the standard deviation and the beta factor, other risk measures are also used which, because of people's aversion to loss, give greater weight to the possible negative plan deviations (the value-at-risk, the CVaR and LPMs)
4. With the help of the method of replication, a way of determining the value of uncertain payment series is taken that does not require a valuation model or cost of capital rates. In order to determine the value of the uncertain series of payments , it is modeled from series of payments up to , the price of which is known (arbitrage-free capital markets).${\ displaystyle {\ overset {\ sim} {Z}}}$${\ displaystyle {\ overset {\ sim} {Z}} _ {1}}$${\ displaystyle {\ overset {\ sim} {Z}} _ {n}}$
5. So-called “ad hoc factor models”, which are based on econometric studies, take into account any determinants that are not interpreted as risk factors when explaining expected returns. You are giving up the principle that higher expected returns can only be justified by higher risks.
6. 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. (see cost of capital )

## Income risk-dependent equity costs

Cost of capital (k) can be derived directly from the company's earnings risks (e.g. standard deviation of cash flows); H. without evaluating historical capital market data. This makes use of the fact that the risk of future payments , i.e. the scope of the possible deviations from the expected value , can be taken into account in two ways. In the more common risk surcharge method, a risk surcharge is added to the risk-free interest rate in order to obtain a discount rate (approximate cost of capital) for discounting the future expected payments . is the standardized assessment-relevant risk scope (in% of the value), i.e. a "return risk measure" (such as the standard deviation of the stock return or the beta factor derived from it), which can be determined depending on the risk scope of the property and the diversification factor (d) of the valuation subject . λ is its "price": ${\ displaystyle ({\ widetilde {Z_ {1}}})}$${\ displaystyle (E ({\ widetilde {Z_ {1}}}))}$${\ displaystyle (r_ {z})}$${\ displaystyle (r_ {f})}$${\ displaystyle (k = {r_ {f} + r_ {z}})}$${\ displaystyle R ({\ tilde {Z}})}$${\ displaystyle E \ left ({\ widetilde {Z_ {1}}} '\ right)}$

${\ displaystyle W ({\ tilde {Z}} _ {1}) = {\ frac {E ({\ tilde {Z}} _ {1})} {1 + k}} = {\ frac {E ( {\ tilde {Z}} _ {1})} {1 + r_ {f} + r_ {z}}} = {\ frac {E ({\ tilde {Z}} _ {1})} {1+ r_ {f} + \ lambda _ {RZ} \ cdot R ({\ tilde {Z}} _ {1} ^ {'})}} = {\ frac {E ({\ tilde {Z}} _ {1 })} {1 + r_ {f} + \ lambda _ {RZ} \ cdot R ({\ tilde {Z}} _ {1} ^ {\ text {object}}) \ cdot d}}}$

Equation 1

However, if the risk surcharge is uniform for positive and negative payments, this procedure leads to evaluation errors. Because of risk aversion, the aim of discounting uncertain payments is to assign a lower value than safe payments. However, this is exactly what is not achieved when discounting (possibly) negative payments: with discounting, the value increases (becomes less negative). It is therefore advisable to use the risk discount or security equivalent method shown in equation 2, which provides correct evaluations and whose evaluation equation can be derived based on less restrictive assumptions: the same expected value and risk level of a payment (at time t) leads to the same value.

${\ displaystyle W ({\ tilde {Z}} _ {1}) = {\ frac {S {\ ddot {A}} ({\ tilde {Z}} _ {1})} {1 + r_ {f }}} = {\ frac {E ({\ tilde {Z}} _ {1}) - \ lambda _ {S {\ ddot {A}}} \ cdot R ({\ tilde {Z}} _ {1 })} {1 + r_ {f}}} = {\ frac {E ({\ tilde {Z}} _ {1}) - \ lambda _ {SA} \ cdot R ({\ tilde {Z}} _ {1} ^ {\ text {object}}) \ cdot d} {1 + r_ {f}}}}$

Equation 2

The amount of risk involved in a payment is recorded with a discount in the counter. is the risk measure and shows the assessment-relevant scope of the risk of the payment or income to be assessed (in monetary units), e.g. B. the standard deviation of EBIT. This can be divided into the risk of the property and the diversification factor of the valuation subject. It should be noted that, under certain circumstances, the (diversified) "standardized valuation subject" bears only part of the risks of the valuation object, in CAPM only the systematic risks that can be recorded by a "risk diversification factor" (d) ("correlation" in CAPM or CCAPM). ${\ displaystyle R ({\ tilde {Z}})}$${\ displaystyle R ({\ widetilde {Z_ {1}}})}$

The risk preference and time preference (risk-free interest rate in the denominator) are clearly distinguished in this approach.

${\ displaystyle R \ left ({\ tilde {Z}} '\ right) = {\ frac {R \ left ({\ tilde {Z}} \ right)} {W \ left ({\ tilde {Z}} \ right)}}}$

The risk analysis (and risk aggregation ) of the payments (cash flows) or income to be assessed leads to planning and risk-based risk measures that are not derived from historical stock returns. Suitable risk measures can e.g. B. the deviation value-at-risk or the standard deviation also used in the CAPM.

By equating both equation 1 and equation 2 given above, a risk-adjusted cost of capital rate (or ), e.g. B. to simplify uniformly based on a representative period, it depends on the coefficient of variation V - the ratio of the standard deviation to the expected value of the result (EBIT or free cash flow) - and thus the findings of the risk analysis are used for the assessment: ${\ displaystyle k}$${\ displaystyle r_ {z}}$${\ displaystyle \ sigma _ {z}}$${\ displaystyle ({\ widetilde {Z_ {1}}})}$

${\ displaystyle W ({\ tilde {Z}} _ {1}) = \ underbrace {\ frac {E ({\ tilde {Z}} _ {1})} {1 + k}} _ {\ text { Risk surcharge method}} = \ overbrace {\ frac {E ({\ tilde {Z}} _ {1}) - \ lambda \ cdot \ sigma _ {z} \ cdot d} {1 + r_ {f}}} ^ { \ text {Risk discount method}}}$
${\ displaystyle \ Rightarrow \; k = {\ frac {1 + r_ {f}} {1- \ lambda \ cdot {\ dfrac {\ sigma _ {{\ tilde {Z}} _ {1}}} {E. \ left ({\ tilde {Z}} _ {1} \ right)}} \ cdot d}} - 1 = {\ frac {1 + r_ {f}} {1- \ lambda \ cdot V (Z) \ cdot d}} - 1}$

With

${\ displaystyle \ lambda = {\ frac {r_ {m} ^ {e} -r_ {f}} {\ sigma (r_ {m})}} = {\ frac {MRP} {\ sigma (r_ {m} )}}}$

in other words, as the Sharpe ratio, i.e. the market risk premium (MRP) divided by the standard deviation of the return on the market index. ${\ displaystyle \ lambda}$

## Rating-dependent cost of capital

Without knowledge of the earnings risks, the cost of capital can be derived, depending on the rating (the probability of insolvency) as a proxy and scope of risk. In the world of perfect and complete Capital Good Pricing (CAPM) markets, there is no bankruptcy. If you want to determine the expected return of an equity investor who accepts an insolvency probability of p based on the opportunity cost calculation, you first need a suitable replication portfolio as an alternative investment. This must show the same probability of insolvency as the investment (company) to be valued. In the following, it is assumed - similar to the assumptions of the CAPM - that the investor can (i.e. ) invest in the market portfolio and that a risk-free investment with the safe interest rate is also available. In addition, it is assumed that an investor can borrow at this interest rate, i.e. can borrow part of an investment in the market portfolio (leverage). A cost of equity rate that depends on the probability of insolvency p (the rating) can be derived from these assumptions. ${\ displaystyle \ beta = 1}$${\ displaystyle r_ {f}}$${\ displaystyle r_ {f}}$

A simple estimation of the expected return on equity (cost of equity) depending on the accepted by the creditor insolvency probability is obtained by calculating which expected return investment in an equity portfolio (market portfolio) would have if this is the same probability of failure due to use of leverage would have . This necessary share (a) of equity can be determined from the lower p% quantile ( value at risk ) of the return depending on the expected return on the market portfolio , the standard deviation of this return and the accepted insolvency probability p : ${\ displaystyle p}$${\ displaystyle p (LPM_ {0})}$${\ displaystyle r_ {m} ^ {e})}$${\ displaystyle \ sigma _ {m}}$

${\ displaystyle (1) \ qquad a = - (r_ {m} ^ {e} -q_ {P} \ cdot \ sigma _ {m})}$

It expresses the equity share in the portfolio (equity requirement as a percentage of the investment), which is necessary with a normal distribution of the return, so that the default probability just reaches p.

This gives the following rating or insolvency probability p dependent equity costs:

${\ displaystyle (2) \ qquad EK_ {p} = r_ {EK, p} ^ {e} {\ frac {{\ text {expected portfolio return}} - {\ text {borrowing costs}}} {\ text {share of Equity in the portfolio}}} = {\ frac {r_ {m} ^ {e} - (1-a) k_ {FK}} {a}}}$

thus with (1) and reshaped

${\ displaystyle (3) \ qquad r_ {EK, p} ^ {e} = {\ frac {r_ {m} ^ {e} (1-k_ {FK}) - k_ {FK} (1-q_ {p } \ sigma _ {m})} {q_ {p} \ sigma _ {m} -r_ {m} ^ {e}}}}$

It is the expected return on equity for insolvency probability (confidence level) p. In addition, it again indicates the expected return on debt (debt costs) with an accepted probability of default p. For ${\ displaystyle r_ {EK, P} ^ {e}}$${\ displaystyle k_ {FK}}$

• ${\ displaystyle p = 0 {,} 5 \, \%}$(i.e. ),${\ displaystyle q_ {P} = 2 {,} 576}$
• ${\ displaystyle k_ {FK} = r_ {f} = 4 \, \%}$
• ${\ displaystyle \ sigma _ {m} = 20 \, \%}$ and
• ${\ displaystyle r_ {m} ^ {e} = 8 \, \%}$

For example, equation (3) yields an expected return on equity of:

${\ displaystyle (4) \ qquad r_ {EK, p} ^ {e} = {\ frac {0 {,} 08- (1+ (0 {,} 08-2 {,} 576 \ cdot 0 {,} 2)) \ cdot 0 {,} 04} {- (0 {,} 08-2 {,} 579 \ cdot 0 {,} 2)}} = 0 {,} 132 = 13 {,} 2 \ \% }$

This results in the risk premium

${\ displaystyle (5) \ qquad r_ {z} = r_ {z, p} = k_ {EK, p} -r_ {f} = 13 {,} 2 \ \% - 4 \ \% = 9 {,} 2 \ \%}$

## Literature and references

1. ^ Lutz Kruschwitz / Andreas Löffler, A New Approach to the Concept of Discounted Cashflow , in: Journal für Betriebswirtschaft , Issue 55, 2005, pp. 21–36. The cost of capital depends on the risk that is captured by a risk measure (e.g. standard deviation).
2. Cf. W. Gleißner: The Influence of the Probability of Insolvency (Rating) on ​​the Company Value and the Cost of Equity - At the same time, comment on the specialist text Lobe CORPORATE FINANCE biz 3/2010, p. 179 (182). In: CORPORATE FINANCE biz. 4/2011, pp. 243-251. According to the basic model of the neoclassical finance theory for perfect markets, there is also no distinction between debit and credit interest. Companies also have the option of financing investments of any size (I) at a uniform capital market rate if the providers of debt capital are investors risk-neutral and there are no bankruptcy costs.${\ displaystyle (r_ {f})}$
3. The so-called loss given default (LGD).
4. Cf. P. Baecker, W. Gleißner, U. Hommel: Company valuation: Basis of rational M&A decisions? A selection of twelve essential sources of error from a practical point of view. In: M&A Review. 6/2007, pp. 270–277 and C. Homburg, J. Stephan, M. Weiß: Company valuation with breathing difficulties and insolvency risk . In: Business Administration. 64th year, 2004, p. 277.
5. Vettiger and Volkart explain in a similar way: “The contractually agreed interest rate on borrowed capital is offset as the cost of borrowed capital. In the case of risky borrowed capital - which is particularly the case when using high-yield bonds - the “promised” interest rate (for bonds: coupon) is more or less above the (average) expected return on the part of the creditors. ”… “ However, theoretically correct in the WACC actually the return demands, d. H. also enter into the return expectations of investors; in the case of high-risk borrowed capital, this would be ... "... " Failure to take these relationships into account leads to theoretically excessive WACC values ​​in companies with high financial leverage and correspondingly high-interest borrowed capital, which - important from a macroeconomic point of view - is also an "underinvestment problem" Can cause. ”, cf. T. Vettinger, R. Volkart: Cost of capital and enterprise value: Central importance of the cost of capital. In: The Swiss Trustee. 09/2002, p. 754. See also IA Cooper, SA Davydenko: The Cost of Debt. 2001, p. 2.${\ displaystyle (k_ {FK})}$
${\ displaystyle E (r_ {FK})}$
6. Cf. A. Löffler: Two comments on WACC. In: Journal for Business Administration. 74 (2004), pp. 933-942, cf. also Kruschwitz / Löffler (2003)
7. See V. Metz: The capitalization interest rate for the company valuation base rate and risk premium from a business point of view and from the point of view of case law. Wiesbaden 2007 and Gleißner (2010)
8. ↑ It should be mentioned at this point that with constant borrowed capital (autonomous financing), the application of the APV variant of the DCF method is generally preferable to the WACC variant. See e.g. BL Kruschwitz, A. Löffler: A new approach to the discounted cash flow concept. In: Journal for Business Administration. Issue 55, 2005, pp. 21-36.
9. See JA Miles, JR Ezzell: The weighted average cost of capital, perfect capital markets, and project life: a clarification. In: Journal of Financial and Quantitative Analysis. 15, 1980, pp. 719-730.
10. See R. Mehra, EC Prescott: The Equity Premium. A puzzle . In: Journal of Monetary Economics . Volume 15, No. 2, 1985, pp. 145-161 and R. Mehra, EC Prescott: The Equity Premium in Retrospect . In: Handbook of the Economics of Finance . Volume 1, Part 2, 2003, pp. 889-938 and Eugene Fama , Kenneth French : The Equity Premium . In: The Journal of Finance . Volume 57, No. 2, 2002, pp. 637-659.
11. See WJ Bernstein, RD Arnott: What Risk Premium Is "Normal"? In: Financial Analysts Journal . Volume 58, March / April 2002, pp. 64-84.
12. ^ Eugene Fama, Kenneth French: Common risk factors in the returns on stocks and bonds . In: Journal of Financial Economics . Volume 33, No. 1, 1992, pp. 3-56 and C. Ulschmid: Empirical Validation of Capital Market Models . Frankfurt am Main 1994 and P. Zimmermann: Estimation and prognosis of beta values . Munich 1997.
13. H. Daske, G. Gebhardt: Future-oriented determination of risk premiums and equity costs for company valuation . In: Journal for Business Research . Volume 58, June 2006, pp. 530–551.
14. ^ P. Albrecht, R. Maurer: Investment and Risk Management . Stuttgart 2005.
15. ^ Klaus Spremann : Valuation. Basics of modern company valuation . Munich 2004.
16. RA Haugen: The inefficient stock market. What pays off and why . New Jersey 2002.
17. ^ W. Gleißner: Capital costs. The weak point in company valuation and value-based management . In: Finanzbetrieb . Volume 7, No. 4, 2005, pp. 217–229 and W. Gleißner: New ways for company valuation and value-oriented company management in an imperfect capital market . In: C. Meyer, D. Pfaff (Ed.): Finance and accounting. Yearbook 2006. Zurich, pp. 119–154.
18. This value, which can vary depending on the period, can, however, be determined based on historical data. There are three operationalization variants: a) Correlation of the stock return to market return (as in CAPM), here: 0.42 on an annual basis b) Correlation of the result variable to the market return, here approx. 0.1 c) Correlation of the result variable to the cumulative results of the Companies in the market portfolio (based on the CCAPM see Jr. RE Lucas: Asset Prices in an Exchange Economy. In: Econometrica. Vol. 46, No. 6, Nov, 1978, pp. 1429ff. And DT Breeden, MR Gibbson, RH Litzenberger: Empirical test of the consumption-oriented CAPM. In: The Journal of Finance. 44, 1989, pp. 231–262 and BR Auer: Can consumption-based capital market models explain the cross-section of international equity risk premiums? In: DBW. 2/2012, pp. 159-177.
19. See Spremann: Valuation. 2004, p. 253 ff and IDW S 1 (source: WPg Supplement 3/2008, p. 68 ff, FN-IDW 7/2008, p. 271 ff) from April 2, 2008 (status).
20. See equation 2 and W. Ballwieser : The choice of the discount rate for company valuation, taking into account risk and inflation. In: BFuP. 33, 1981, pp. 97-114.
21. The parameters and match when (as in a perfect market) applies: .${\ displaystyle \ lambda _ {RZ}}$${\ displaystyle \ lambda _ {S {\ ddot {A}}}}$${\ displaystyle R ({\ widetilde {Z ^ {'}}}) = {\ dfrac {R ({\ widetilde {Z}})} {W ({\ widetilde {Z}})}}}$
22. ^
23. It is assumed that the valuation subject only regards the probability of insolvency as a risk measure. The likelihood of insolvency is dependent on the earnings risk and the risk-bearing capacity. W. Gleißner: The influence of the probability of insolvency (rating) on ​​the company value and the cost of equity - At the same time, comment on the specialist text Lobe CORPORATE FINANCE biz 3/2010, p. 179 (182). In: CORPORATE FINANCE biz. 4/2011, pp. 243-251.
24. Understood as the sum of all risky assets - not as the result of a portfolio optimization.
25. The rating-dependent costs of equity capital can be derived by means of replication (see Spremann, Valuation - Basics of Modern Company Valuation, 2004 and further on W. Gleißner, M. Wolfrum: Cost of Equity and the Valuation of Unlisted Companies: Relevance of Degree of Diversification and Risk Measure. (PDF; 503 kB ) In: Finanzbetrieb . 9/2008, pp. 602–614. For the application of the replication method to any risk measure).
26. This is shown in the following for an investment whose original risk (investment risk) corresponds to that of the market portfolio, i.e. β = 1. The changes in the scope of risk depending on the insolvency probability p result only from the associated change in the level of debt (debt financing). Of course, you can immediately expand the following calculation for other risks and returns of the “insolvency-free” basic portfolio, e.g. B. with a risk-adjusted return according to CAPM.${\ displaystyle r_ {EK} (\ beta)}$
27. LPM0 is a Lower Partial Moment of degree zero, i.e. a special downside risk measure, see e.g. B. Albrecht / Maurer, Investment and Risk Management, 2005.
28. ↑ In general, for each risk measure used for the assessment and calculated using risk aggregation, a suitable expected return (price) must be estimated from market data or economic models.
29. is the value of the inverted distribution function of the standard normal distribution at the confidence level .${\ displaystyle q_ {P}}$${\ displaystyle p}$
30. See in more depth W. Gleißner, M. Bemmann, F. Leibbrand: The risk rating - an approach to recording original company assessment. In: Risk Manager. 03/2006, pp. 10–15 and W. Gleißner: The Influence of the Probability of Insolvency (Rating) on ​​the Company Value and the Cost of Equity - At the same time, opinion on the specialist text Lobe, CORPORATE FINANCE biz 3/2010, p. 179 (182). In: CORPORATE FINANCE biz. 4/2011, pp. 243-251 especially for this reference portfolio. The following applies: In the derivation, it was assumed (for the sake of simplicity) that the systematic risk of the equity investment corresponds precisely to that of the market portfolio (ß = 1) and that an increase in the systematic risk arises solely from the external financing (leverage effect). Is there a higher systematic risk of equity from the outset (ß> 1). For an "insolvency-free" basic portfolio of replication with any of the CAPM, as a return that follows replaced in (3) .${\ displaystyle \ beta = {\ dfrac {1} {a}}}$${\ displaystyle \ sigma _ {P}}$${\ displaystyle \ beta}$${\ displaystyle r_ {p} ^ {e} = {r_ {f} + \ beta (r_ {m} ^ {e} -r_ {f})}}$${\ displaystyle r_ {m} ^ {e}}$
${\ displaystyle r_ {EK, p} ^ {e} = {\ frac {r_ {f} + \ beta (r_ {m} ^ {e} -r_ {f}) - k_ {FK} (1-q_ { p} \ sigma _ {m})} {q_ {p} \ sigma _ {p} -r_ {f} - \ beta (r_ {m} ^ {e} -r_ {f})}}}$