How to Calculate the Value of a Growing Perpetuity Using a State Price Matrix?

Summary

I wish to value perpetual cash flows through state contingent claims on real consumption, where the state of the economy is assumed to follow a finite markov chain (Similar to Banz and Miller (1978)). Moreover, I wish to account for a steady growth rate in the cash flows. I have tried to give a clear explanation of the building blocks in the model being used below, in addition to the specific problem I am encountering. I have provided some of my thoughts on possible solutions as well and some questions which I hope somebody is able to answer.

Building Block 1 - State Prices

Consider the following vector of state contingent claims (prices) on one unit of real consumption in one time period

$$\textbf{v}= \begin{bmatrix} 0.43 & 0.32 & 0.22\\ \end{bmatrix}$$

where the prices correspond to the following economic states

$$\begin{bmatrix} Recession & Normal & Boom\\ \end{bmatrix}$$

If these are the only three states the economy can take on, the real risk free rate is

$$r=\ln(1/\sum_{i=1}^3v_i)= \ln(1/0.97)=3.05\%$$

where $v_i$ denotes the elements of $\textbf{v}$.

Building Block 2 - State Price Matrix

If we assume that economic states follow a finite markov chain and that state prices are the same in all initial states (and time) we have the state price matrix

$$\textbf{V}= \begin{bmatrix} 0.43 & 0.32 & 0.22\\ 0.43 & 0.32 & 0.22\\ 0.43 & 0.32 & 0.22\\ \end{bmatrix}$$

Building Block 3 - Valuation

If we have a project paying out the following cash flows (in real terms) in two time periods (where the entries in the vector correspond to the vector denoting the economic states)

$$\textbf{c}= \begin{bmatrix} 120 & 200 & 240\\ \end{bmatrix}$$

we can value the project as

$$\textbf{V}^2\textbf{c}= \begin{bmatrix} 0.43 & 0.32 & 0.22\\ 0.43 & 0.32 & 0.22\\ 0.43 & 0.32 & 0.22\\ \end{bmatrix}^2 \begin{bmatrix} 120\\ 200\\ 240\\ \end{bmatrix} = \begin{bmatrix} 163.35\\ 163.35\\ 163.35\\ \end{bmatrix}$$

where the price is the same in all initial states, as state prices were assumed to be constant across initial states.

Building Block 4 - Perpetual Valuation

Take now the same project, with the exception that cash flows are paid every time period in perpetuity. We can derive the perpetual state price matrix $\textbf{V}_\infty$ (i.e. the prices for claims of one unit of real consumption every time period in perpetuity in respective states) and value the project.

First we have

$$\textbf{V}_\infty=\textbf{V}+\textbf{V}^2+\cdots$$

Also, given

$$\textbf{S}_\infty = \textbf{I}+\textbf{V}+\textbf{V}^2+\cdots$$

we have

$$\textbf{V}\textbf{S}_\infty = \textbf{V}+\textbf{V}^2+\cdots = \textbf{V}_\infty$$

which allows us to write

$$\textbf{S}_\infty-\textbf{V}\textbf{S}_\infty = (\textbf{I}+\textbf{V}+\textbf{V}^2+\cdots)-(\textbf{V}+\textbf{V}^2+\cdots) = \textbf{I}$$

or equivalently

$$(\textbf{I}-\textbf{V})\textbf{S}_\infty=\textbf{I}$$

which we rewrite as

$$\textbf{S}_\infty=(\textbf{I}-\textbf{V})^{-1}$$

Since the rows of $\textbf{V}$ sum to less than 1, we know that $\textbf{S}_\infty$ converges. Using the fact that $\textbf{V}\textbf{S}_\infty = \textbf{V}_\infty$ we get

$$\textbf{V}_\infty = \textbf{V}(\textbf{I}-\textbf{V})^{-1}$$

Calculating the perpetual state price matrix we get

$$\textbf{V}_\infty= \begin{bmatrix} 14.33 & 10.67 & 7.33\\ 14.33 & 10.67 & 7.33\\ 14.33 & 10.67 & 7.33\\ \end{bmatrix}$$

which has an implied risk free rate of

$$r_j=\ln(1+1/\sum_{i=1}^3v_{ij})=\ln(1+1/32.33)=3.05\%$$

for each row $j$

This lets us calculate the value of our project in a similar fashion as before

$$\textbf{V}_\infty\textbf{c}= \begin{bmatrix} 14.33 & 10.67 & 7.33\\ 14.33 & 10.67 & 7.33\\ 14.33 & 10.67 & 7.33\\ \end{bmatrix} \begin{bmatrix} 120\\ 200\\ 240\\ \end{bmatrix} = \begin{bmatrix} 5613.33\\ 5613.33\\ 5613.33\\ \end{bmatrix}$$

The Problem - Perpetual Valuation with Growth in Cash Flows

Say that our cash flows grow by a fixed percentage every time period ($g$), which could be due to something like population growth perhaps, is there any way to compute the value of the perpetual cash flows adjusted for the growth? I have tried to come up with a solution, which you can see under, but I am not sure of whether or not it is consistent with the rest of the model and/or economically reasonable. My line of reasoning for arriving at my "solution" is as follows:

1) Since the cash flows after one time period, will be equal to the cash flows at the beginning of the time period accrued with the growth rate, we can write

$$\textbf{c}_{t+1}= \begin{bmatrix} e^g & 0 & 0\\ 0 & e^g & 0\\ 0 & 0 & e^g\\ \end{bmatrix} \begin{bmatrix} 120\\ 200\\ 240\\ \end{bmatrix}$$

2) The cash flows in perpetuity would be

$$\textbf{c}_{\infty}= \begin{bmatrix} e^g & 0 & 0\\ 0 & e^g & 0\\ 0 & 0 & e^g\\ \end{bmatrix}^{\infty} \begin{bmatrix} 120\\ 200\\ 240\\ \end{bmatrix}$$

3) From 2) we can see that if we sum the infinite stream of all future cash flows (which are growing) we will reach infinity, i.e. the sum will diverge. However, if the growth is used to reduce a discount rate (as in Gordon's growth formula) it might be useful anyway.

4) Since it does not matter whether we increase cash flows with growth (keeping discount constant), or reduce discount with growth (keeping cash flows constant), we can write the following

$$\textbf{V}_{\infty}^g\textbf{c}= \textbf{V}\textbf{G}(\textbf{I}-\textbf{V}\textbf{G})^{-1}\textbf{c}$$

where $\textbf{G}$ is the growth matrix. Also note that the formula means that we assume the cash flows in $\textbf{c}$ are actually at time 0)

This works as long as the growth is not higher or equal to the implied risk free rate (i.e. row sums are lower than 1). If it is however, $(\textbf{I}-\textbf{V}\textbf{G})^{-1}$ does not converge, and we cannot really say anything about the present value of the cash flows (other than the fact that it infinite).

So the implication as far as I can see, is that real growth for a project in perpetuity cannot be higher than or equal to the real risk free rate in perpetuity.

I think this would make sense if the real risk free rate was the real growth rate of aggregate consumption (or wealth if they are in one-to-one correspondence), since a project can never outgrow the world. However, this would still mean that the value is infinite when the growth rate is equal to the real growth rate of consumption. I find this slightly strange as the project/company/investment then just maintains its market share for a perpetual amount of time.