# Roll-back procedure

Roll-back formulas

The roll-back process is a computational technique used to determine company value , with which future cash flows can be discounted without the need for mathematical iteration . The company value is determined based on the value of the perpetual annuity, period for period backwards to the valuation date (recursive company value determination). The circularity problem that frequently occurs in the determination of company value is resolved in this process by formal equivalence transformations of the valuation equations. The ease of use of the roll-back process leads to a significant reduction in complexity compared to the iteration process. No mathematical iteration is necessary due to the formal equivalence transformations of the evaluation equations. The required calculation steps can be managed with a simple pocket calculator.

Different beta adjustment formulas can be derived due to different premises when adjusting beta factors . Corresponding roll-back formulas can be derived. An overview of the roll-back formulas commonly used in practice and their application requirements can be seen in the figure. A distinction is made between the detailed planning phase and the perpetual annuity.

## Calculation example

Information for the calculation example:

Information
Growth in retirement (g) 2.0%
Risk Free Interest Rate (i r ) 4.0%
Beta through no fault (ß u ) 0.8
rEK u 8.4%
FK costs (r FK ) 6.0%
Tax rate (s u ) 25%

The specified flows to equity and the market value of the borrowed capital for the respective period can be found in the following table:

PLAN PLAN PLAN PLAN PLAN
t 0 t 1 t 2 t 3 t 4 t 5
FTE 50.0 80.0 75.0 95.0 96.9
MW FK 660.0 700.0 750.0 600.0 650.0

In a first step, the shareholder value is determined at the end of the detailed planning period (t 4 = T), i. H. the market value of the equity of the perpetual annuity determines. The Harris / Pringle formula is used for this calculation, which assumes that the risk of the tax shield corresponds to the risk of the non-indebted company. The corresponding roll-back formula for the perpetual annuity is:

${\ displaystyle EK_ {T} = {\ frac {FTE_ {T + 1} - (r (EK) _ {u} -r_ {FK}) \ cdot FK_ {T}} {r (EK) _ {u} -G}}}$

Based on the example, the following market value of the equity at the time EK t4 + results :

${\ displaystyle EK_ {t4 +} = {\ frac {96 {,} 9- (8 {,} 4 \, \% - 6 {,} 0 \, \%) \ cdot 650} {8 {,} 4 \ , \% - 2 \, \%}} = 1 \, 270 {,} 3}$

In a further step, the market value of the equity at time t 3 is determined by using the corresponding roll-back formula for the detailed planning period:

${\ displaystyle EK_ {t-1} = {\ frac {FTE_ {t} + EK_ {t +} - (r (EK) _ {u} -r_ {FK}) \ cdot FK_ {t-1}} {1 + r (EK) _ {u}}}}$

Based on the example, the following market value of the equity at the point in time EK t3 + results :

${\ displaystyle EK_ {t3 +} = {\ frac {95 + 1 \, 270 {,} 3- (8 {,} 4 \, \% - 6 {,} 0 \, \%) \ cdot 600} {1 +8 {,} 4 \, \%}} = 1 \, 246 {,} 2}$

If the computer technology shown is used recursively up to t 0 , a market value of the equity is calculated at t 0 i. H. of 1,108.4 (see table).

Summary of results:

PLAN PLAN PLAN PLAN PLAN
t 0 t 1 t 2 t 3 t 4 t 5
MW EK 1,108.4 1,167.4 1,202.2 1,246.2 1,270.3

## literature

• Alexander Enzinger / Peter Kofler: The roll-back procedure for company valuation - Circularity-free company valuation with autonomous financing policy using the equity method In: Valuation Practitioner No. 4, 2011, pp. 2-10. [1]
• Alexander Enzinger / Peter Kofler: DCF method: Adjustment of beta factors to achieve consistent evaluation results, in RWZ 16/2011, 52–57. [2]
• Christopher Casey: New aspects of the roll-back process in company valuation , in ZfB 74th year 2004, issue 2, 139–163.
• Bernhard Schwetzler / Niklas Darijtschuk: Company valuation with the help of the DCF method - a comment on the "circularity problem , in ZfB 69th year 1999, issue 3, 295-318.

## Individual evidence

1. cf. Alexander Enzinger / Peter Kofler: The roll-back process for company valuation - circularity-free company valuation with autonomous financing policy using the equity method In: Valuation Practitioner No. 4, 2011, p. 3.
2. cf. RS Harris / JJ. Pringle: Risk-Adjusted Discount Rates Extensions form the Average-Risk Case In: Journal of Financial Research 1985, pp. 237-244.