Forever

A very simple model of the firm's value is the dividend growth model, perhaps better called the dividends-with-growth model.  In it, a firm's value, market capitalisation, P, is given as next year's cash dividend divided by an appropriate discount rate minus a growth rate.  P=D/(r-g).

Separately, no longer in the world of equity but in the world of fixed income, if I chose to lend somebody money with a view never to get the principal back, only annual coupons, then the value of that infinite stream of coupons is a perpetuity, whose value is given by C/r, the coupon divided by an appropriate discount rate.

I just want to find a set of simplifying assumptions about the firm which unite these two valuations.  Imagine there are N companies in the world, and the average company produces a dividend stream D with an average corporate cost of capital (weighted average of all sources of funding for the average company) was r.  Let's further pretend that all N companies actually have this D,r set-up - i.e. you momentarily pretend there's no variation in the N companies.  In those N companies there'd be financial institutions too.  So they too have a weighted average cost of capital of r and a dividend yield of D.

Let's further pretend that the only line of business these financial institutions were in was making a market for perpetuities.  Namely that they only made loans or took money on the basis that they'd never get/give the capital back, merely that they participated in a perpetuities market.

You have some money and you want a return.  You look at the bank's perpetuity market making.This bank would not be wise, given the modern theory of value creation and destruction, to engage in any lending or borrowing activity which somehow didn't return at the very least the average corporate WACC, r.  So, ignoring bid ask spread, and the imposition of healthy profit margins, you'd expect your principal investment to be worth the equivalent in perpetuity coupon terms  of C/r, namely that you'd receive a regular annual payment, in today nominal terms, of C each year.

If companies in general were perfectly efficient then the growth component, g, of the dividend discount model might reduce in the limit to the expected long term inflation rate.  That is to say, in a fictitious super-competitive corporate environment, the only dividend growth would be inflation based.  So let's live in an inflation free world, where g=0 and no innovation occurs (at least with respect to these perpetuity banks).

The dividend yield is a fraction of capital made, on an annualised basis, by the firm.  So the only way in which D can be made is through the one line of business, namely perpetuity market making.  So in general it must make D on the bid ask spread, that is, it must transact a sufficient number of deals.  Assuming If no defaults ever happen and it lends out all of its capital it must charge in aggregate D as the perpetuity coupon. The tighter the actual bid ask spread, the wider must be its balance sheet.