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## Abstract

Entrepreneurs and VCs often disagree on their valuation of start-ups. This article offers a new reason why. The Venture Capital (VC) method is a standard method for valuing startup firms. It requires an estimate of terminal firm value or Net Income, a comparable industry’s P/E ratio, and a target rate of return to be used as the discount rate. All of these quantities are subject to estimation risk, but the choice of the target discount rate is the most questionable. VCs typically use high target rates of return to compensate for the risk of the venture; a natural question is how to justify these high target rates of return. Discounted cash flow (DCF) methods, such as the VC method, assume that a start-up can be valued like any other standard capital budgeting project. However, start-ups have valuable embedded real asset options which are ignored by DCF methods. This article uses option pricing techniques to impute the appropriate target rate of return. It shows that the appropriate target rate depends on precisely what the VC is valuing. If the VC is valuing a stream of ordinary expected cash flows, as in the DCF approach, one target rate of return is implied. A different target rate is implied if the VC is valuing a start-up firm including its embedded real asset options. This article shows how to adjust target rates of return for insurance value and gives detailed illustrations of the magnitudes of the adjustment.

Valuing VC investments in start-ups is a difficult proposition, and entrepreneurs and VCs often disagree on the valuation. There are many explanations for this disagreement (Yitshaki [2008]). This article offers a new explanation. Because VCs typically finance growth, the main problem is capturing the economic value of the growth opportunities being financed. This usually involves specifying and estimating future growth rates in some underlying value-driving variable such as free cash flows. DCF (discounted cash flow) valuation methods for VC investments involve estimating future cash flows, their growth rates, and a horizon terminal value representing the enterprise’s value at the VC’s exit. P/E ratio (multiple) methods are essentially equivalent to DCF methods but make strong assumptions concerning the stability of P/E ratios. All such methods involve an explicit or implicit estimate of the growth rate, *g^*, in free cash flows. It is well known that such valuation methods are highly sensitive to the *g^* estimate.

Do DCF methods capture the economic value to VCs of their start-up investments? Are entrepreneurs adequately compensated for their exchange of ownership to VCs? This article addresses these issues by using option pricing techniques to rationalize VCs’ target required rates of return. Some preliminary remarks will put these issues, and this article’s approach, in context. A start-up firm has to have a business plan that delineates potential product(s) and a strategy for capturing growth over some time horizon. VCs demand this analysis and then decide whether to include particular start-ups in their typically undiversified portfolios. But, to the VC, the value of a start-up investment is not based simply on the deterministic introduction of products by the entrepreneurs nor on the expectation that subsequent free cash flows will grow at a deterministic growth rate. Nor are VC financing modes consistent with this way of viewing assets.

Since product introduction at the end of a time horizon is a risky proposition and growth is volatile, a VC is simply buying the ability (but not the obligation) to capitalize on the value of the growth in free cash flows generated by a start-up’s product introduction. Such value may turn out to be ephemeral and/or economically unjustifiable when the product introduction date approaches. No matter—the VC’s position is optional. That is, VCs are making real asset *option* investments in start-ups by financing their growth opportunities.

DCF valuation methods price assets based on the expected free cash flows generated by the one-time decision to put these assets in place or not. In the case of growth opportunities, the DCF valuation assumption is that the VC is evaluating the decision to currently buy the future expected free cash flows generated by growth opportunities. The uncertainty associated with these expected free cash flows is incorporated in the target discount rate and not by building options into the model. That is, there are no option features in DCF valuation methods. The VC, however, is well aware that growth may or may not occur and hedges his/her position by taking a call option position on the start-up. Unlike a naked position on the start-up, call option positions incorporate insurance features.

The financial options pricing literature tells us that DCF methods are not the correct procedure for valuing derivative assets (McDonald [2005]). This applies equally to real asset options. The correct procedure, first introduced by Black and Scholes, is risk-neutral, arbitrage-free option valuation. Given that DCF methods essentially fail to correctly value the growth opportunities invested in by VCs, how can one identify, create, value, and financially manage such growth opportunities? One way is heuristically by analogy to R&D. There is a very close analogy between R&D, growth opportunities, and VC financing. For example, R&D investments create *options* on growth—an important management function in a competitive environment where value-adding, risky growth opportunities are hard to find. Further, R&D is a vehicle both for *creating* and ultimately capturing potential growth opportunities. Consider the case of biotechs whose real asset base is often primarily in R&D.

To pursue and apply the analogy, R&D is normally viewed as an option on product introduction (Lint and Pennings [1998])—one introduces the product(s) if and only if the R&D option is in-the-money at the first possible product introduction date; otherwise not. The cost of introducing the product is the option’s exercise price, which must be estimated at the inception of the R&D. The time to maturity of the R&D option is the length of time until the first date at which it is technologically feasible to go into production. The option’s current value is the present value of all the R&D investment anticipated prior to the maturity date. Other key variables in option pricing models are the risk-free rate taken as the yield on the shortest term treasury securities traded in the market and the volatility of percentage rates of return on the underlying “stock” price.

One must of course define a correct analogy to a stock price which can conveniently be taken as the present value (appropriately discounted) of the expected future incremental benefit stream generated by the R&D. This is the hardest variable to measure but no harder than expected future cash flows and their growth rates. Knowing what the stock price is helps to establish the connection between target rates of return employed by the VC method to value the “stock price” and the target rate of return used by the options approach to value the VC’s call option position on the stock price. This is a hybrid valuation method incorporating both the DCF approach and an options approach and will be explained shortly. Of course, there is no economic reason to think that these target rates of return will be the same. As we shall demonstrate, they are not.

The literature on this topic is sparse, hence the contribution of this article. We give a few general references. Cochrane [2005] is a discussion of the risk-return characteristics of VC investments. Lerner [2002] discusses alternative valuation methods including the VC method. McDonald [2005] contains all the option background needed. Shane [2008] is a general reference on entrepreneurship and Yitshaki [2008] gives reasons why VCs and entrepreneurs tend to disagree on their respective valuations.

The remainder of this article is structured as follows. The first section describes the VC start-up valuation method and its implicit assumptions. It shows that the NPV of product introduction to the VC is equal to the current value of a levered stock where the stock price is given by the VC method. The VC’s target required rate of return is used to establish this stock price. The next section values the start-up as a European call option on product introduction, and uses put-call parity to impute the target rate of return implicit in this valuation procedure. We contrast the two target rates of return: the DCF target rate of return and the option-based target rate of return. In the following section, we show that the downward adjustment of the option-based target required rate of return depends inversely on the ratio of the implicit put price to the stock price. We illustrate this relationship.

**THE VC VALUATION METHOD**

We start with the VC’s current valuation of the entire start-up firm. This total firm value will be shared between the VC and the entrepreneur(s) according to the percentage ownership required for the VC to earn his target required rate of return. We assume that there will be a single round of financing at time 0, there are no dilution effects, the firm currently has no products, and a decision is made today to introduce the product at some future fixed time *T*. That is, product introduction at time *T* is not optional. Under these conditions, the VC method values the expected NPV of the incremental free cash flows associated with the product’s introduction. Further, risk is incorporated into the model primarily through a usually high, non-CAPM target required rate of return used as the discount rate. The current Net Present Value of the anticipated product introduction is given by in *NPV _{t}
*
Equation (1).

**Valuing the Stock Price by the VC Method**

The so-called VC method translates Equation (1) into a computable quantity in Equation (2) by forecasting an estimate of Net Income at time *T* (as a proxy for Cash Flow), applying an industry *P/E* ratio to it to obtain an estimate of *S _{T}
*, and then discounting by the VC’s required target rate of return,

*r*

_{target}, to get an estimate of

*S*in Equation (2). Finally, to get an estimate of

_{t}*NPV*in Equation (1) one subtracts to

_{t}*PV*(

_{t}*E*) from

_{T}*S*obtain Equation (3).

_{t}In order to compare the VC valuation method to the alternative option valuation of the start-up in the next section we assume that *E _{T}
* is discounted at the risk-free rate. Then the VC method in Equation (3) essentially values a levered stock.

**Imputing the Target Rate of Return to Value the Start-Up as A Levered Stock**

The current stock price, *S _{t}
*, and the target required rate of return,

*r*, in Equation (3) are jointly determined. One can easily impute

_{target}*r*

_{target}from the other variables by solving for it in Equation (4). This facilitates a comparison to the target required rate of return imputed by the option valuation approach in the next section.

**VALUING THE START-UP AS A EUROPEAN CALL OPTION ON PRODUCT INTRODUCTION**

As argued, a start-up’s value to the VC is primarily option value where the option is on product introduction at time *T* with exercise price *E _{T}
*. Therefore, the VC method should actually be valuing a European call option on product introduction at time

*T*. In this case, a decision is

*not*irrevocably being made today to introduce the product at some future fixed time

*T*. That decision is optional and will be made at time

*T*. This represents a much more flexible, risk-managed, and realistic approach, at least from the VC’s perspective. The “Stock Price” at time

*t*is the same value

*S*given in Equation (2) but in this case the valuation is of a European call option on the stock price. Therefore, it includes more than the stock price. Let

_{t}*C*(

*S*) denote the current European call option price which is the object to be priced. By put-call parity, the call option value differs from the right-hand side of Equation (1) by the current value of a European put option on product introduction, denoted by

_{t}*P*(

*S*). The put-call parity relationship for European options and the consequent option valuation are given in Equation (5).

_{t}We are herein modifying the plain vanilla *NPV _{t}
* in Equation (1) to become the option-adjusted
in Equation (5). This is a standard real asset options approach that values the implicit put option feature implied by the VC’s call option position on the start-up. To implement it, we must adapt the standard VC valuation method in Equation (2) and thereby Equation (3) so that it incorporates the implicit put option

*P*(

*S*) contained in the valuation of the start-up as a European call option on product introduction. To do so, replace the left-hand side of Equation (2) by

_{t}*S*+

_{t}*P*(

*S*) and search for a modified target rate of return, that equates the right-hand side of Equation (2) to the new left-hand side. This is described in Equation (6).

_{t}This is the hybrid valuation method alluded to in the introduction. It is not a pure DCF approach because it incorporates the value of the put option. It is not a pure options approach because it does not simply value the enterprise directly as a call option but proceeds indirectly by using the DCF approach and modifying the discount rate. Its main advantage is that it allows us to compare target required rates of return. Note that the same stock price *S _{t}
* given by Equation (2) (using

*r*

_{target}as the discount rate), the same Industry

*P/E*ratio, and the same estimate of expected Net Income at time

*T*are used. However, the required target rate of return, , must be lower (i.e., the valuation must be higher) to account for the additional value of the put option,

*P*(

*S*), included in the valuation of the start-up as a European call option. This put option is not included in the non-option approach expressed in the VC method in Equation (2). The correct modification of Equation (3) with the left-hand side replaced by follows by definition in Equation (7).

_{t}To implement Equation (7), first obtain the stock price, *S _{t}
*, from Equation (2) by using the VC valuation method in the normal, non-option theoretic fashion. Plug this stock price, along with the other option parameters needed, into the Black-Scholes put option pricing formula to obtain

*P*(

*S*). Then conduct the search for the modified discount rate, , as indicated.

_{t}**Imputing the Target Required Rate of Return to Value the Start-Up as an Option**

It is easy to solve Equation (6) in Equation (8) for the new lower target required rate of return, , that makes the option-modified VC valuation method work. Note that the second equality in Equation (8) follows from the definition of the stock price in Equation (2).

(8)A discussion of risk is relevant at this point. Presumably VCs adjust their required target rates of return upwards for the relatively high risks associated with financing start-ups. What are these risks? Clearly, there are liquidity risks associated with real assets. In particular, the VC may not be able to earn his target required rate of return by exiting the market through an IPO. Indeed, a viable IPO may not be possible. The risk adjustments made by VCs are much higher and inconsistent with the beta risks predicated by equilibrium capital asset pricing models. That is, diversifiable risks appear to be priced by VCs. Apparently, VCs are pricing start-ups based on the total risk of the returns on the underlying stock price, that is on *s*. This is consistent with the VC regarding start-ups as call options. ^{1} However, by valuing the enterprise using a DCF (discounted cash flow) method in Equation (2), they are under-valuing start-ups by at least the put premium. That is, even if we accept the high target rate, *r*
_{target}, used in Equation (2) and the resulting stock price, *S _{t}
*, the VC under prices the start-up. That is,

*r*

_{target}still has to be adjusted downward to in order to account for the put option value. That is the basic argument of this article.

The intuition for why
< *r*
_{target} is clear. Consider the analogy of the valuation in Equation (3) to that of a levered stock. The idea is that the right-hand side of Equation (3) is economically equivalent to borrowing the present value of *E _{T}
* and using it to partially finance the purchase of the stock at the stock price

*S*. Under this interpretation, the risk of this levered position is that at time

_{t}*T*, the stock price

*S*will be below the amount owed on the loan,

_{T}*E*. Now compare this risk analysis to that implied by the options approach in Equation (7). The call option, by put-call parity, contains an implicit put option. So if

_{T}*S*<

_{T}*E*then one doesn’t default on the loan of

_{T}*PV*(

*E*) because one simply exercises the put option and sells the stock for

_{T}*E*and uses the proceeds to cover the loan. (In practice, the VC does this by not exercising the call option at time

_{T}*T*.) A European call position is just like a levered stock position but insured against default on the leverage if the market declines. Therefore it is less risky than the naked levered position which the VC is valuing in Equation (3). Because it is less risky, the required target rate of return, <

*r*

_{target}. Further, the more valuable the insurance feature of the put, the lower must be the discount rate . In general, =

*F*(

*P*) =

_{t}*F*(

*P*(

_{t}*t*,

*E*,

_{t}*s*,

*r*,

*S*)). The conclusion is that we can partially rationalize VCs’ required target rates of return in terms of what they are valuing. Note that we do not attempt to fully rationalize VCs’ required target rates of return because we take

_{t}*r*

_{target}and the resulting stock price,

*S*, as given.

_{t}**ADJUSTING THE TARGET REQUIRED RATE OF RETURN FOR INSURANCE VALUE**

In this section we look at how one plus the option-adjusted target required rate of return, 1 +
varies as a percentage of one plus the DCF-based target required rate of return, 1 + *r*
_{target}. The relationship is derived in Equation (9) which is obtained from Equation (8).

We conclude from Equation (9) that the ratio inversely depends on the ratio of the value of the put option, *P*(*S _{t}
*), to the value of the stock price,

*S*. When the value of the put option is high (low) relative to the stock price, 1 + will be low (high) relative to 1 +

_{t}*r*

_{target}.

A detailed analysis is given in the Exhibit which considers a typical set of option parameters in the world of VC financing. These are an exercise price *E* = 100, *R _{F}
*= 7%,

*s*= 50%, and

*t*= 2 years. Given these parameters, we input alternative stock prices and compute the corresponding Black-Scholes put option prices, and the ratio . In computing this ratio, we alternatively assume

*r*

_{target}equals a high value of 60%, a mid value of 40%, and a low value of 20%. Note that whatever changes stock prices, once the stock price and the put price are determined the ratio is given in Equation (9).

The downward adjustment of *r*
_{target}to
to reflect the value of the put option clearly depends upon the ratio of the put price to the stock price in Equation (9). When the put option is out-of-the-money (above the exercise price = 100) the put is worth relatively less compared to the stock price. At *S _{t}
* = 130 the adjustment factor is 95%, requiring that

*r*

_{target}be reduced by 7%, 6%, and

*r*

_{target}6% for high, mid, and low as described above. As the stock price declines the put price becomes relatively more valuable and the required downward adjustment of

*r*

_{target}significantly increases. For example, when the stock price is $70 and

*r*

_{target}= 0.20, a downward reduction of 20 percentage points is required. At this point equals zero: there is no discounting. can even be negative as indicated in the Exhibit. How is this possible? When either the ratio in Equation (9) and/or 1.0 +

*r*

_{target}is small enough, then the product, 1.0 + can be less than 1.0, which forces < 0. For example, when

*S*= 60,

_{t}*r*

_{target}= 0.20, the ratio in Equation (9) is 0.7891 and the product equals 0.9469, which forces to equal -0.0531. Other examples are given in the Exhibit. The reasoning is that, when the stock price is so low and the put price is consequently so high, normal discounting cannot incorporate the put value. The put eventually becomes worth more than the stock (which eventually it must), and goes negative—not a good investment because the VC would expect to earn a negative rate of return. The implication is that as long as it is roughly more expensive to insure the stock price than it is worth, a VC should never finance such a start-up.

Note that this is an analysis of the ex-ante required rate of return. The modified DCF approach is not sufficiently robust to incorporate exercise of the put option. As long as *S _{T}
* <

*E*at expiration, the put will be exercised yielding a payoff of

*E*rather than

*S*. This will significantly improve the ex-post rate of return on the start-up. In the previous example, suppose that

_{T}*S*= 60 and that by expiration it goes nowhere,

_{t}*S*= 60. Then the put is exercised and the ex-post rate of return increases to 0.02. As the stock price drops by expiration, exercising the put becomes more and more valuable, and for the last entry in the Exhibit the ex-post return increases to 0.02 vs. the ex-ante rate of return of –0.59. This is the advantage of having the put option insurance.

_{T}Next, we describe the dependence of the downward adjustment of *r*
_{target} on the option parameters one at a time, all other parameters held constant. Increasing the exercise price and increasing *s* increases the put option price relative to the stock price so these parameters increase the downward adjustment. Increasing the risk-free rate makes the put relatively less valuable and so decreases the downward adjustment. Time to maturity usually, not always, increases the value of the put. However, 1/*T* occurs in Equation (9) and has the opposite effect on the ratio. The net effect will depend on the interaction of these two counter-vailing effects.

**IMPLICATIONS TO ENTREPRENEURS AND VCS**

Start-ups are risky investments. In fact, most fail (Shane [2008]). VCs are well aware of the riskiness of their investments and attempt to control the risk, in part, by diversifying their portfolios. They can also take (call) option positions on start-ups—thereby limiting the downside risk of the start-up’s implicit stock price falling below the (exercise) price of introducing the product. DCF methods value a start-up as a levered stock. Option valuation adds a put option to this DCF valuation. The valuations cannot be the same and the DCF target required rate of return must be adjusted downward to reflect the additional value of the put option. We examined this adjustment and showed that it depends on the ratio of the put price to the stock price. The higher the put price relative to the stock price the greater the downward adjustment of the target required rate of return. For some relatively low stock prices the put becomes more valuable than the stock and the target rate of return drops first below the risk-free rate and then below zero. Long-shot start-ups (*S _{t}
* « E) have deep-in-the-money corresponding puts and are not viable VC investment targets. By observing the insurance value corresponding to the stock price implicit in the start-up, VCs can better monitor potential start-ups in terms of the value of the insurance needed to protect them. This allows them to better screen their investments and ex-ante earn acceptable target required rates of return. On the other hand, entrepreneurs must be rewarded for the put options they write for VCs. Finally, through understanding what they are trading, VCs and entrepreneurs can more readily reach valuation agreement. This will make this market more liquid and generate more efficiently priced funding.

## ENDNOTES

↵

^{1}Note that the impact of risk on the valuation of start-ups depends on the valuation method. Using a DCF approach, an increase in beta risk (other variables constant) decreases the valuation. In an options approach, increasing total risk*s*unambiguously increases the value of the call position. Thus we have a contradiction here. If VCs are pricing non-diversifiable risk and thinking in terms of options, start-ups should be worth more.

- © 2009 Institutional Investor, LLC