Most investors are obsessed with growth. And the right type of growth can indeed produce mind blowing investment returns over the years.

But not all growths are created equal. Growth has a cost as well as a benefit, and the wrong type of growth can in fact destroy shareholder value.

Take a restaurant chain with a capital base C = $100 million producing distributable earnings E = $20 million. Earnings E are distributable in that they are already net of any capital expenditures necessary to restore the value of the ongoing capital base. They could be fully distributed to shareholders every year without impacting earnings power in future years.

Such business has a return on capital ROC = 20% (20/100), that is, it generates $20 of distributable earnings for each $100 invested in the business.

Value without growth

If our opportunity cost of capital as investors is R = 8%, then the intrinsic value of such business under a policy of full distribution of earnings would be:

V0 = E/(1+R) + E/(1+R)^2 + ... = E/R = ROC/R * C = 20/8 * $100 million = $250 million.

We would expect shares to sell at about 2.5x book value because the underlying business produces 2.5x the returns that investors demand at that level of risk.

This expression also shows why, unless returns on capital grow in proportion to interest rates, a business is worth less the higher the interest rates (a higher R is demanded the higher interest rates).

Value of one-time growth

But what if the restaurant chain could plug some of the distributable earnings back into the business to expand its capital base and increase future earnings?

Say the business wants to invest I = $10 million to open 10 new restaurants, expanding the restaurant base from 100 to 110 sites, or the capital base from $100 million to $110 million, assuming an asset base of $1 million per site.

If the characteristics of the new restaurants are comparable to those of the current restaurant portfolio, 20% ROC is to be expected.

Hence, the I = $10 million expansionary capital expenditure would increase future annual distributable earnings by ROC * I = 20% * $10 million = $2 million.

The net present value (NPV) of such investment is given by:

ΔV = ROC*I/(1+R) + ROC*I/(1+R)^2 + ... - I = ROC*I/R - I = (ROC/R - 1) * I = (20/8 - 1) * $10 = $15 million

The investment generates shareholder value because it is done at returns in excess of cost of capital (ROC > R). Had the investment returned ROC = 8%, the investment would have been worthless in that the NPV from increased earnings (8% * $10 million or $800k/year) would have been completely offset by the initial $10 million outflow. And in fact, at ROC<R, the investment would have destroyed shareholder value.

The capital base after the investment is C*(1+g), where g = I/C is the rate of growth, 10% in this case. Thus, we can express the value generated by one-time growth as ΔV = (ROC/R - 1) * g*C.

The percentage increase in the intrinsic value of the business due to the one-time expansion of the asset base by g is given by:

% Increase EPV = ΔV/V0 = [(ROC/R - 1) * g*C] / [ROC/R * C] = (1 - R/ROC) * g = (1-8/20) * 10% = 6%.

An implication of this equation is that if the economics of the reinvestment are as for the ongoing business (same ROC), the increase in intrinsic value is smaller than the reinvestment. In other words, for a fixed ROC, value grows slower than assets, and slower than revenues and earnings, assuming fixed asset turnover and margin.

Growth is not free. Higher revenues and earnings require a larger asset base, which is not possible without retaining a portion of earnings that would have been otherwise distributed to shareholders.

In the restaurant chain, expanding the asset base by g = 10% increases intrinsic value by 6%.

Put another way: after the reinvestment, the capital base is C*(1+g), but intrinsic value is not ROC/R * C*(1+g), but ROC/R * C*(1+g) - g*C, since an investment of g*C was required to increase the asset base to produce that growth.

Value of one-time growth at different returns

That said, rarely are the economics of the expansion the same as for the ongoing business.

When a business operating in a very profitable niche diworsifies outside that niche, the core business ROC is not extrapolable to the diversification initiatives, and so intrinsic value will increase by even less than indicated in the above equation. In fact, the value may be destroyed if investment returns fall below opportunity cost.

Conversely, business with operating leverage exhibit increasing marginal ROC. For those businesses, intrinsic value creation will be larger than indicated in the equation. That is likely to be the case with the restaurant chain, as fixed central costs will be spread over 10 more restaurants after the investment.

Indeed, if the return on capital on the investment is ROC' = ΔE/I = 30%, higher than the average ROC = 20% before the investment, the percentage increase in intrinsic value is given by:

% Increase EPV = ΔV/V0 = (ROC' - R)/ROC * g = (30-8)/20 * 10% = 11%,

which is actually larger than asset growth g = 10%.

Value of perpetual growth

We have shown before that, if the ROC on new investments is expected to be the same as for the overall business, the percentage increase in intrinsic value resulting from a one-time percentage capital expansion g is (1 - R/ROC)*g.

But what if the business keeps expanding its asset base at an annual rate g every year for the foreseeable future? What is the value of that growth?

The theoretical percentage increase in intrinsic value resulting from annual percentage capital expansions g in perpetuity can be shown to be:

% Increase EPV = [1 - (g/R) / (ROC/R)] / [1 - g/R].

This expression is of limited quantitative utility because it relies on predictions of uncertain metrics in perpetuity.

It is however of immense qualitative importance in that it reveals necessary conditions for a multi-bagger, that is, and investment with 10x, 100x, 1000x and higher potential returns over a number of years.

We shall call it...

Theorem of Multi-baggics

An investment has multi-bagger potential if it can expand at returns above cost of capital in an industry with immense TAM.
(Proof: if ROC>R and g->R in the equation above, then intrinsic value tends to infinity).

The other necessary condition is of course that the investment be entered at a sufficiently attractive price.