The Black and Scholes option pricing theory (OPT) offers a clue as to how the equity in a firm may be valued. Suppose we recognise the fact, that an equity investor in a geared firm with limited liability has a call option on the underlying assets of the firm. In that case, we have, potentially, a method for valuing the business.
Conceptually this is an influential theory; the use of option pricing methodology in the valuation of a business creates difficulties in estimating the necessary input parameters into the model.
Using the real options methodology, one approach is to simulate the future cash flows of a firm given realistic current conditions and estimates of the volatility of key input variables.
With this, we can generate an overall estimate of the future volatility of the business and then using a specified set of assumptions about the terminal value of the company creates an option value for the business. This modelling approach has many refinements but essentially provides both methods and insights into the valuation of all firms that are financed partly by debt and, in particular, highly leveraged, fast-growing start-up companies.
The limits on the value
Traditionally, the value of the firm in the hands of its investors will have a lower limit equal to the break-up value of the firm, less all external claims on the business (the sum of its short and long term liabilities). Generally, it was argued that once the present value of the firm’s future cash flows (when discounted at the equity investors rate of return) falls below this value then it would be rational for the investors to cut their losses, liquidate the firm and salvage what value they could. However, this relatively simple analysis is tempered in the light of options theory. From an options theory perspective, the equity investors in a geared firm have a call option on the value of the firm’s assets over and above the value of the debt. If the value of the assets should fall below the value of the debt then given limited liability the equity investors could put the firm into members’ voluntary liquidation and walk away leaving the debt holders to bear the loss. Thus in a geared firm, the equity value of the business is the value of a call option on the firm’s net assets. In an ungeared firm, the option value does not exist. Thus the value of the firm to the equity investors is simply the present value of the net cash flows anticipated over the lifetime of the business.
This line of reasoning suggests that valuing a firm depends upon the existence of gearing and that the value of the firm is not simply the present value of its assets in current use minus the amount of its outstanding debt.
If the firm is not geared, the critical numbers are: (i) the realisable value of its assets and (ii) the present value of the firm’s assets in continued use. The greater of these is its equity value. In the presence of gearing the critical numbers are: (i) the present value of the firm’s assets and (ii) the value of the firm’s outstanding debt. The value of the firm, in this case, will be the value of the option to continue in business.
This perspective on the value of a firm suggests that the following variables are critical:
(i) The present value of the firm’s assets in use. Generally, the greater this value, the greater will be the value of a call on those assets at exercise.
(ii) The value of the outstanding debt (the exercise value of the firm) – generally the lower this value, the more valuable the call becomes until at the limit of zero leverage the call value equals the present value of the firm’s assets in use.
(iii) The term to maturity of the debt – the longer the time, the greater the value of the equity call.
(iv) The risk-free rate of return used to discount the exercise value multiplied by the probability of exercise. Given the inverse relationship between exercise value and firm value, this would suggest that the higher the rate, the greater the call value. However, this is unlikely to be the case overall given that the present value of the firm’s assets (i) will be determined in part by the discount rate and that this in its turn is partly influenced by the risk-free rate.
(v) The expected volatility of the present value of the firm’s assets, which leads to the somewhat paradoxical result that the higher the uncertainty about future cash flows, the more valuable is the call option on those cash flows.
Valuing the firm as an option
In the more general valuation context, a firm’s equity should not be traded, or we may have reason to believe that the correct valuation is considerably different from that revealed by the share price.
Schwartz and Moon (2000) developed a procedure for the contingent valuation of equities using option pricing and simulation methods.
They used as their case study Amazon.com which at that time had been in business for just over three years. The company was still not profitable in the conventional sense but was growing its market and its revenues at a rapid rate.
In March 1996 the quarterly sales of Amazon.com were $0.875 million. By September 1999 its sales had risen to $355.8 million. Here, in outline, is the procedure Schwartz and Moon followed:
A stochastic model was created of the firm’s revenue-generating process and its cost structure. This model contained a drift term which reflects the expected rate of growth in its revenues and a stochastic period reflecting the degree of uncertainty about that growth rate. The expenditure model reflected not only the company’s fixed and variable cost structure but also the impact of taxation upon the company’s profits.
Refinements of this stochastic model included a mean-reverting process to the estimated long-run rate of revenue growth as well as a procedure for carrying forward losses for tax purposes from one period to the next.
A bankruptcy condition was imposed were given a starting amount of cash bankruptcy was defined as the point when the amount of cash and other monetary assets reached zero.
A time horizon was defined for the simulation of the firm’s future cash flows, and a terminal value invoked. In their study, Schwartz and Moon, set the final value as ten times EBITDA. Another approach would be to take the net cash flow figure and to capitalise the following year’s projected earnings at the risk-free rate of return less the terminal growth rate of profits. However, the time horizon should be such that by that time, the equity holder’s option is so far into the money that there is zero default risk.
A simulation is then undertaken to generate a large number of cash flow paths. In the simulated series of quarterly cash flows for Amazon assuming a starting revenue of £356million per quarter, a growth rate of 11 per cent a quarter and a starting volatility of revenues of 10 per cent per quarter. The factors were built-in, and the firm’s starting balance of cash resources available was assumed to be £200 million.
The key point to note is that a number of the price paths had been generated one of which shows default in period 5. Indeed, depending upon the software used, a model can be built which will permit many hundreds of such price paths to be generated. The model can be refined to reflect a wide range of different circumstances: different patterns of growth and the decline in growth as the firm matures, other cost structures, the correlation between variables, differing tax regimes and different initial conditions. From the simulation, the volatility of the cash-flow projection can then be determined, and the likely default in each time period determined. This volatility, expressed as the standard deviation of the company’s future cash flows, can then be adjusted to a continuous-time basis and a valuation of the options component of the company’s valuation determined.
The challenge with this approach of valuation lies in estimating the initial volatilities of the future revenue growth and, in later variants of the model, the volatilities of future costs.
Nevertheless, the technique does offer a potential route for valuing companies that are in their early stages of growth and which rely upon substantial investment in intangible assets.