DOI link for §6-5.
DOI link for §6-5.
There remains, however, a signiﬁcant difﬁcultywith implementing the above formula. This stems from the fact that we know from §6-2 above that νt = (1+ r)−t , and so the cost of capital r appears as an argument on both sides of the formula given at the end of §6-4 above. This means that we need to substitute the actual value of r into the right-hand side of the above formula before we can determine r from the formula; in other words, we need to know r before we can determine r through the above formula. Fortunately, there is a fairly simple way around this problem. We can illustrate this by deﬁning the function
in which case our problem is to ﬁnd the value of r for which
r = g(r) Now let rn be an approximation to the solution r of this latter equation and consider the iteration procedure under which rn is used to obtain the following updated approximation to this solution:
rn+1 = g(rn) We can expand g(rn) as a Taylor series about the solution r in the above equation, in which case we have
rn+1 = g(r)+ (rn − r)g′(ξn) where ξn is a number that lies between rn and r. Moreover, since r = g(r) is the solution we are looking for, it necessarily follows from the above equation that
rn+1 − r = (rn − r)g′(ξn) Noting that the error in the approximation at the nth iteration is en = rn − r, we can restate the above result in the following equivalent form:
en+1 = eng′(ξn) We now let n = 0, in which case the error at the ﬁrst iteration will be
e1 = e0g′(ξ0) Likewise, when n = 1, the error at the second iteration will be
e2 = e1g′(ξ1) = e0g′(ξ0)g′(ξ1) When n = 2, the error at the third iteration will be
e3 = e2g′(ξ2) = e0g′(ξ0)g′(ξ1)g′(ξ2)
Continuing with this procedure shows that the error at the (n+ 1)th iteration will be
en+1 = e0 n∏
Hence, provided that ∣∣g′(ξj)∣∣< 1 for all j, the error in the above approximation procedure will
become progressively smaller as we increase the order of iteration; that is, as we let n become larger and larger, the error in the above approximation procedure will become progressively smaller. Fortunately, it can be shown that the condition
∣∣g′(ξj)∣∣< 1 will normally be satisﬁed, and so the procedure outlined here will usually provide a convergent approximation to the cost of equity. We can demonstrate the application of this approximation procedure in terms of the following simple example.