**Bruno Husson**

Honorary partner, Accuracy

**Time and risk in the valuation of a business – The example of motorway concessions**

The value of an asset can be easily approached by the prices observed on a market where comparable assets are exchanged. This is the analogical approach to valuation. An alternative solution consists of replicating in a valuation model the way in which the market forms these prices. This is the intrinsic approach to valuation and the founding principle of the discounted cash flow (DCF) method, the prominent method associated with this second approach.

**The two key parameters of
any financial valuation: time and risk**

The starting point of the intrinsic approach to valuation is the definition of the concept of financial value. According to this concept, the value of an asset is based on the cash flows that the asset holder is likely to receive in the future. As these cash flows are spread over time and subject to risk, the valuation model must necessarily incorporate the behaviour of the investor with regard to two parameters: time and risk. Financial theory indicates how to incorporate time and risk in isolation, that is, how to incorporate time without consideration of risk and how to incorporate risk as part of a single-period model (i.e. without consideration of time).

When considering time, valuation models use the discounting technique, that is, the commonly accepted assumption that an individual has a preference for the present. On financial markets where we implicitly exchange time (i.e. a sum of money held today against a sum of money available at a later date) through the acquisition of securities considered risk-free (treasury bills or government bonds), an individual’s preference for the present is logically reflected by the existence of a positive interest rate. This risk-free interest rate crystallises a fundamental principle of finance: the time value of money (one euro today is not the equivalent of one euro tomorrow because the euro received today, deposited at the risk-free rate, will give more than one euro tomorrow). Based on this principle, the discounting technique makes it possible to aggregate cash flows (assumed to be risk-free) occurring at different dates, bringing them back to today’s date by means of the interest rate, and finally to determine the value of the asset associated with this series of cash flows.

When considering risk, the valuation model frequently used by valuers is the Capital Asset Pricing Model (CAPM). This model relies on segmenting risk into two components: (i) the specific risk (or diversifiable risk), which asset holders can eliminate by diversifying their portfolios, and (ii) the systematic risk (or undiversifiable risk), which even the investor whose portfolio is perfectly diversified must bear. According to the CAPM formula, the return required on a financial asset is equal to the risk-free interest rate plus a risk premium that depends only on systematic risk (the market does not remunerate the diversifiable risk). Thanks to CAPM, we know how to calculate the value of an asset generating a risky cash flow over a single period: it is the average cash flow (or ‘expected’ cash flow) discounted at the rate of return given by the formula. The two components of risk linked to the assets are taken into account: the specific risk through the calculation of the expected cash flow (i.e. in theory, the average of the expected cash flows in the various possible scenarios, weighted for the likelihood of occurrence of these scenarios), and the systematic risk via the discounting of the expected cash flow at a rate incorporating a risk premium.

However, in practice, it is necessary to take into account the point that the valuations might be for entities generating cash flows over several periods (or even in perpetuity). This leads valuers to step away from the theoretical context mentioned above to incorporate both the time and risk parameters in the same model (in other words, combining risk with time).

**The usual way of integrating risk in the discount
rate can lead to a significant underestimation of the entity being valued: the example
of motorway concessions**

The typical approach used by valuers to incorporate risk consists of transposing the CAPM into a multi-period framework. More concretely, the price of time and (systematic) risk are considered simultaneously over the lifetime of the entities being valued via the discounting of expected future cash flows at a single risk rate. This rate is equal to the risk-free interest rate increased by the (constant) risk premium from the CAPM formula.

The alternative approach incorporates successively (and not simultaneously) the time and risk parameters: first the risk parameter via the determination of the ‘certainty equivalent cash flows’ and second the time parameter via the discounting of these cash flows at the risk-free interest rate. The certainty equivalent cash flows incorporate the entirety of the risk and are therefore lower than the expected cash flows that only incorporate the diversifiable portion of risk.

The difficulty in the alternative approach lies in determining the adjustment coefficients to be applied to the expected cash flows in order to obtain the certainty equivalent cash flows. These coefficients can be estimated within the theoretical framework of the CAPM, but the calculation formula, which is rather convoluted, proves inapplicable in practice. It is also worth highlighting that, in the context of a business valuation, the valuer must first appreciate the level of optimism of the business plan, before even considering the risk integration mechanisms. If the valuer considers that the business plan represents the average scenario associated with the expected cash flows, he or she can then either discount these cash flows at the CAPM risk rate or determine the certainty equivalent cash flows and discount them at the risk-free interest rate. If the business plan appears rather conservative, or even pessimistic, the valuer cannot implement the usual approach without adjusting the business plan cash flows upwards; however, he or she can directly choose the alternative approach by considering that the business plan cash flows provide a reasonable estimation of the certainty equivalent cash flows.

The usual approach to integrate risk can be criticised because, by using the discounting technique to combine risk with time (though this technique is, in theory, only supposed to take into account the time value of money), it implicitly makes a significant assumption on the development of the systematic risk by assuming that this risk increases considerably with time. The alternative approach appears more solid because, by dealing separately with the issues related to the incorporation of time and risk, it does not make any assumption on the development of the risk. This allows all valuation cases to be handled rigorously and in particular the valuation of activities that benefit from good visibility over a long period (for example, infrastructure projects) and for which the assumption of a risk growing with time is particularly debatable.

By way of illustration, let us consider a motorway concession likely to generate on average an annual cash flow of €800m over a period of 30 years (inflation is assumed to be nil). Based on a (real) risk-free interest rate of 1.5%, an asset beta of 0.5 and a market risk premium of 5.5%, the rate of return given by the CAPM formula amounts to 4.25% and the value of the concession using the typical risk integration approach comes to €13,423m (value of annual cash flow of €800m discounted at 4.25%). Given the good visibility of the revenues, which despite a relatively fixed cost base grants the activity a low systematic risk (confirmed by the beta coefficient of 0.5), it seems reasonable to base the determination of the certainty equivalent cash flows on a constant abatement coefficient of 0.15. On this basis, the value of the concession using the alternative approach comes to €16,331m (value of the annual certainty equivalent cash flow of €680m discounted at the risk-free interest rate of 1.5%). The gap in the estimated values given by the two approaches amounts to around 22% and comes from the implicit assumptions made on the development of risk over time. With the alternative approach, the risk is assumed to be unchanging (the deduction on the expected cash flow for risk is 15% in any given year); with the usual approach the risk increases significantly with time (the deduction for risk thus grows from 10% in year 4 to 21%, 31%, 40% and 50% in years 9, 14, 19 and 26 respectively, i.e. a very significant increase that the risk profile of the activity cannot justify).

In conclusion, adopting the usual risk-integration approach to value activities that benefit from good visibility over a long duration is debatable and can lead to significant undervaluations.