Tuesday, April 5, 2011

Pricing Assets In Perpetuity: the Case of Farmland

A recent article in The Progressive Farmer indicates that the price of farmland in the midwest is skyrocketing.

$11,000 per acre may seem to be pretty pricey for farmland. However, there is a way to understand the imputed market value of an acre of land.  First, recall that when we thought about the price of a share of stock of a company, we considered the present value (PV) of the long run stream of profits you could expect to earn as an owner of the company. Suppose you thought profits would be x0 this year, x1 next year, x2 the year after that, and so forth. The PV of this stream of returns over time out to time = T would be the sum of

PV = x0  +  x1/(1+r)  +  x2/((1+r)^2)  +  x3/((1+r)^3)  +  ...  +  xT/((1+r)^T)

Lets make things simple and assume for the moment that the returns in each period were some constant x (that is, we assume that x0 = x1 = x2 = x3 = ... = x).  Farmland is a special asset, one that is an "asset in perpetuity" since the owner can bequest it to her heirs, etc.  The good news is that we know for very large T, the equation simplifies to


In other words, if you expected to earn $100 per year forever, and the interest rate was .05, then the PV of that infinite series would be $100/.05  =  $2000.  That is the same thing as saying that "the discounted stream of an annual series of $100 annuities would be comparable to $2000 in the bank today" which is the same thing as saying "if you gave me $2000 today and I put it in the bank at a guaranteed annual rate of 5% forever, I could continue to take $100 out on Jan 1 of each year, forever...."

So back to our farmland story.  The same story cited above indicates that owners of farmland can simply rent the land (this is called "cash rent") for as much as $450 per acre in McLean county.  This is pure profit to the owner; the renter pays this for the right to use the land and that same renter incurs all production costs on top of this land cost and then keeps any and all profits after selling the corn.

We know the price of land ($11,000) represents the present value of the long run stream of annual profits ($450 per year) you could earn by simply keeping the land in your ownership.  Or, the $11,000 per acre today is what it would take to compensate you for giving up the $450 per acre into perpetuity.  Does that make sense?  Lets check:

Since we know the formula is:   PV = x/r   then we are asking does   $11,000 = $450/r

To make that equation work, we would need an interest rate, r, equal to 4 percent.  Is that reasonable? Well, if you look at the interest rate on long term bonds today, you see that the rate on 20 year bonds is about 4.3% which makes the $11,000 number quite reasonable.

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