We use the BPM to determine the values of a levered firm’s debt and equity securities, given the current value of the firm (V), its upstate and downstate future values (or ), and the promised payment on the debt, X.
I. The Assumed Distribution of a Firm’s Future Value in the Binomial Model The binomial distribution provides the simplest statistical model of risk. Applied to a firm’s assets, values of the assets are modeled over a single period, which extends from time 0, the current date, to time 1, a future date. The future value of the assets can take on only two possible values, which are defined relative to the assets’ current value. Denoting the current value of the firm’s assets as V, the future value of the firm’s assets can take on only one of two possible values, or , where >V and <V. That is, over the single period involved, the value of the firm’s assets can either rise to or fall to . Our choices of the values of and define the riskiness of the firm’s assets. Appropriate values for and depend on three factors: (i) the value of V; (ii) the actual riskiness of the assets of the firm that we are attempting to model; and (iii) the span of time involved in the model’s single period. To address factor (i), it is generally assumed thatand represent proportional ‘up’ and ‘down’ jumps relative to V. Specifically, the proportional up jump is denoted as u, where u>1, and the down jump is denoted as d, where d=1/u<1. Thus, = uV (2.19) and = dV. (2.20) Regarding factors (ii) and (iii), we generally wish to model risk as a function of time, where risk increases with the length of the period. To do so, we must address factors (ii) and (iii) simultaneously. For instance, to model the riskiness of the assets of a particular firm, if the length of the period as a month, we would choose a particular value of the risk parameter u, while if the period is a year, another, larger value of u should be specified. Cox, Ross, and Rubinstein (1979) provide a formula for the parameter u that produces an approximation to the riskiness of the firm, in terms of the per annum standard deviation of return, as if the returns were normally distributed. The formula is: (2.21) where is the per-annum standard deviation of the firm’s assets, and T is the length of the period, in years. This translation allows us to more easily specify reasonable values for u.
A numerical example The following numerical example illustrates the binomial distribution as defined above. Suppose the current value of a firm is V=100, and the annual standard deviation of returns on the firm’s assets is =20%. If we specify the period of the binomial model as T=1 year, equation (2.21) can be used to calculate the appropriate value of the risk parameter u:
=1.2214.
Thus, d=1/u=0.8187. Given these values for u and d, equations (2.19) and (2.20) can be used to calculate the future values of the firm in the up and down states: = uV = 1.2214(100) = 122.14 and = dV = 0.8187(100) = 81.87.
That is, the 1-year return on the firm’s assets is either +22.14% or –18.13%.
II. The Binomial Model and the Valuation of the Debt and Equity of a Levered Firm If the value of the assets of a levered firm follows the binomial distribution specified above, we can determine the no-arbitrage values of the firm’s debt and equity. We define the firm’s debt as a promise to pay debtholders the amount X at time 1. Assume the general case that the firm’s debt is risky (that is, >X. In the up state, bondholders will receive the promised amount of X, so =X, and equityholders will receive = -X. In the down state, the firm will default, and bondholders will receive =, while equityholders receive nothing (=0). We will initially value the firm’s levered equity. We can do so by creating a riskless portfolio
