How do I determine the ideal selling price for a cash flow that rises with inflation?

Question

Let's say I own a parking space. I have two options:

• I can rent out this parking space for $1,000/month. I am assuming that the rent will keep pace with inflation, which we'll call 2% over the long term, but otherwise remain fixed in real terms. I am also assuming for now that it's always occupied (i.e. no vacancy) and that there is no tax owed on the income.

• I can sell the space, and use the proceeds to pay down debt that I would otherwise carry throughout my lifetime. Let's say my borrowing rate on that debt is 4.5% over the long term, and again ignore tax effects (e.g. deductions).

I'm trying to figure out the breakeven price above which it makes sense for me to sell instead of rent.

I thought you solve this by calculating the present value of these ongoing cashflows. But I'm having some trouble figuring that out. For example, if I treat it like a "growing perpetuity" (given the presumed annual inflation adjustments), then that would put the sales price at:

P = $\frac{CF}{d - g}$ = $\frac{12000}{.045-.02}$ = $480,000

But that seems... wrong. So what's the right way?

(As an ancillary question: It seems that using my own discount rate here would not output the optimal selling price, but merely my own minimum bar to clear over renting. If those future cash flows are worth even more to someone else, then I should certainly sell at their higher price - i.e. if the market discount rate is lower than mine, then I should use that rate in the calculation instead of my own?)

Answer 1

Perpetuities are enormously difficult to intuitively understand because they go on for infinity. It get's more complicated when you add a growth element.

If you do accrual accounting, which is essentially what you're doing by accounting for the negative carry, it's going to be even more tricky to understand intuitively. But this is the lens you seem to view the problem through, so let me try:

Let us simplify the problem statement. Let us take the same parameters, g=0.02, c=1200, r=0.045. But this time assume the time horizon is finite at 50 periods.

At 50 periods, the value is 336,967 USD if you use the growing annuity formula with 50 periods as opposed to the perpetuity.

Let us assume you borrow 336,967 USD at 4.5% interest in order to finance the transaction.

In the first year, you earn 12,000 USD in income. You pay interest on the 336,967 USD which comes out at 15,164 USD.

15,164 USD is more interest than the 12,000 USD income, so you have a negative carry of 3,164 USD. This negative carry is added to the original loan amount so that in the next period you owe 340,131 USD. You pay 15,306 USD on this new total debt, and your income grows 2% to 12,240 USD, so you have a negative carry of 2,808 USD for the second period, which is added to to the existing debt/capital position.

If you keep doing this recursively throughout all 50 periods, you'll find that your breakeven point is the 50th period. At that point, your debt position is going to be zero. Why? Because eventually the growth rate of the 2% on the income is going to outpace the growth rate of the debt, so that eventually the carry becomes positive and you start paying off the debt.

Rate of growth of debt in first year: 343,197/340,131 - 1 = 0.9% Rate of growth of income = 2%. Eventually the growth of income outpaces the accumulating debt, and you start paying off the debt, and eventually it converges to zero at exactly 50 periods.

If you extend this to 500 periods, you guessed it, the breakeven point is reached in the 500th period. If you extend this to 5000 periods, your breakeven point is reached in the 5000th period. And if your time horizon is in perpetuity, well guess what, your breakeven point is in "forever" years time.

I encourage you to try a few numerical examples with finite periods to see this for yourself, but I will warn you that the numbers involved quickly become obscene, and the terminal debt starts deviating from literal zero due to accumulating float/rounding errors. But you get the point.