Imperfect replication

The imperfect replication is a method for evaluating uncertain payments based on its risk-value model and thus for determining the value directly based on the risk of the payment, expressed by a risk measure such as the standard deviation or value at risk (e.g. for Company valuation in an imperfect capital market ).

The capital market-oriented Capital Asset Pricing Model (CAPM), based on the assumption of perfect markets, uses historical capital market data of the valuation object (stock returns) to derive the cost of capital , whereby information on the valuation-relevant risks of future payments ( cash flows and income) is ignored. In contrast to this, the imperfect replication assumes the expected value and risks of the uncertain cash flows or income to be assessed (e.g. determined by risk analysis and risk aggregation ). Accordingly, the assessment method of "imperfect replication" (or "incomplete replication") represents an alternative to the CAPM for determining the value of e.g. B. of companies or real estate as well as the cost of capital (but includes their valuation results as a special case).

Basic idea of ​​replication

"Appropriate evaluation equations for various risk measures and also in the case of imperfect diversification, i.e. if unsystematic risks are relevant to evaluation, can be derived by means of replication." Replication comprises the generation of a series of payments in a replication portfolio ( duplication ). The series of payments of this "replication portfolio" corresponds to that of the valuation object with regard to essential properties. The basic idea of ​​replication is to replicate the payment to be valued using a suitable combination of financial instruments. H. to replicate. For reasons of simplicity, only two properties are required for replication in imperfect replication: the expectancy and the risk equivalence. This means that the expected value and risk measure (e.g. standard deviation or value at risk) should match at all times. In the case of complete replication, however, all the properties of the payment series match those of the valuation object.

Assessment by means of imperfect replication

The central assumption, i.e. H. the basic assumption is: two payments at the same point in time have the same value if they have the same expected value and correspond in the chosen risk measure (e.g. standard deviation). The CAPM includes this central assumption as a special case if the standard deviation is selected as the risk measure (and other assumptions correspond to the CAPM, e.g. with regard to the diversification of the subject of the assessment). ${\displaystyle W}$ ${\displaystyle E}$ ${\displaystyle R}$

${\displaystyle E({\widetilde {Z}}_{t}^{1})=E({\widetilde {Z}}_{t}^{2})}$ and ${\displaystyle R({\widetilde {Z}}_{t}^{1})=R({\widetilde {Z}}_{t}^{2})}$ ${\displaystyle \Rightarrow }$ ${\displaystyle W({\widetilde {Z}}_{t}^{1})=W({\widetilde {Z}}_{t}^{2})}$

General replication approach to scoring unsafe payments

In this case, replication means replicating the payment to be assessed using a portfolio of capital market instruments that are accepted as alternative investment options (e.g. risk-free government bonds and a stock market index). The value of the capital market instruments in this "replication portfolio" is known (the price). However, full replication is not required for the simulation, so that a match in essential features is sufficient, which is why the name of the approach as "imperfect replication" is justified. These characteristics are the expected value and the risk expressed by a risk measure . Thus, two investment options are sufficient for replication. Here this should be the investment in a market portfolio with an uncertain return and the investment with an interest rate . Accordingly, the value of the uncertain payment is the sum of the two investments (assets) in the replication portfolio and : ${\displaystyle Z}$ ${\displaystyle E}$ ${\displaystyle R}$ ${\displaystyle x}$ ${\displaystyle r_{M}}$ ${\displaystyle y}$ ${\displaystyle r^{f}}$ ${\displaystyle W}$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle W({\widetilde {Z}}_{A})=x+y}$

The basic assumption of replication with the requirements that (1) the expected value of the repayment of the investment in x and y is the expected value of the payment and (2) that the risk of the investment in the replication portfolio should correspond to the risk of the payment is made by means of the following two Equations fulfilled:

(1) ${\displaystyle E({\widetilde {Z}}_{A})=E(x\cdot (1+{\widetilde {r}}_{M}))+E(y\cdot (1+r^{f}))=x\cdot (1+E({\widetilde {r}}_{M}))+y\cdot (1+r^{f})}$

(2) ${\displaystyle R({\widetilde {Z}}_{A})=R(x\cdot (1+{\widetilde {r}}_{M})+y\cdot (1+r^{f}))}$

"The risk [ ] is measured by a suitable risk measure , for example standard deviation, value-at-risk, conditional value-at-risk or an LPM measure." ${\displaystyle R}$ ${\displaystyle R({\widetilde {Z}}_{A})}$

By solving the two equations, x and y and thus the value W can be derived (as a security equivalent that can be derived solely from the risk-return profile of the alternative investment opportunities without knowledge of a utility function ). It is also possible to derive cost of capital rates .

The equations for the evaluation of multi-period uncertain payments can be derived with the method of imperfect replication on the basis of the same axioms .

literature

• Gregor Dorfleitner / Werner Gleißner: Valuing streams of risky cash flows with risk-value models, in: Journal of Risk, Issue 3/2018, pp. 1–27
• Werner Gleißner / Marco Wolfrum: Equity costs and the valuation of unlisted companies: relevance of degree of diversification and degree of risk . In: Finanzbetrieb, 9/2008, pp. 602–614.
• Klaus Spremann: Valuation: Basics of modern company valuation . Oldenbourg Wissenschaftsverlag, 2004.

Individual evidence

1. ↑ Jochen Drukarczyk / Dietmar Ernst: Industry-oriented enterprise valuation . 3. Edition.
2. ↑ Werner Gleißner: Risk analysis and replication for company valuation and value-oriented company management . In: Wirtschaftswwissenschaftliches Studium 7/2011, pp. 345–352.
3. ↑ Werner Gleißner: Risk analysis and replication for company valuation and value-oriented company management . In: Wirtschaftswwissenschaftliches Studium 7/2011, pp. 345–352.
4. ↑ Gregory Dorfleitner / Werner Gleißner: Valuing streams of risky cash flows with risk-value models, in: Journal of Risk, Issue 3/2018, pp 1-27