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An investment formula

by Arnold Kling
August, 1995

I frequently use a shortcut formula in many ways, including:

The purpose of this article is to explain the thinking behind the formula. The formula is something that an economist might use to determine the value of an asset. An asset is something that will last for a long time, such as a fruit-bearing tree (most economists love fruit-bearing trees, but I'm allergic to the ones near where I live).

An asset will yield "rents" (the fruit from the tree) and will enjoy price appreciation (I may be able to sell the tree for more than the original price I paid). The formula is,

profitability = rental rate + appreciation rate - interest cost
What I mean by profitability is the expected annual profit, expressed as a percent of the price of the asset. The asset could be a house, some shares of stock or of a mutual fund, or our fruit tree.

The rental rate is the ratio of the first year rent to the purchase price. The first-year rent would be the rent on an equivalent house, the dividends from the stock, or the proceeds from selling the fruit.

The appreciation rate is the rate at which the price increases, expressed as an annual percentage rate. Much of this price increase could be due to general inflation. In the late 1970's, inflation in the U.S. reached 10 percent and over. More recently, inflation has been closer to 2.5 percent.

Some of the price increase may be specific to the particular market. In housing, over long periods of time prices go up at the same rates as rents in an area. Increases in stock prices are called capital gains. Fruit tree prices will tend to be correlated with the price of their fruit.

The interest cost is the cost of financing the asset purchase. With housing, most people think of this as the mortgage interest rate. With stocks or mutual funds, many individuals do not borrow. However, they could have put their money in CD's or bonds and earned interest, and it is this foregone interest (or "opportunity cost") that should be used as interest cost. Whether we borrow to buy the tree or finance the tree with our own funds, there is an interest cost to tying up our money in the tree.

Below is a sample spreadsheet to illustrate these concepts. We buy an asset (fruit tree) at the end of 1995, using funds borrowed at a 12 percent interest rate. Until we sell the asset, the amount that we owe will increase at 12 percent per year, minus the amount that we can pay off from the proceeds of harvesting the fruit. :

Assumptions:
  Buy asset at end of 1995 for $100,000
  In 1996 fruit from tree worth $7000
  Rental rate .07    ($7000/100000) 
  Interest rate .12   (12 percent annual interest rate, 
                       expressed as a decimal)
  Appreciation  .06   (6 percent annual appreciation, 
                       expressed as a decimal)

Approximation Formula: 
           annual profitability = rental rate plus appreciation 
                                  minus interest rate

                                = .07 + .06 - .12 = .01
(The asset will earn 1 percent profit per year)

(The calculations below serve to confirm that the formula is 
approximately correct:)

Year    rent     debt      price     net worth    profitability

1995     $0    $100000   $100000        $0           --
1996   $7000   $105000   $106000      $1000          1.0 %
1997   $7420   $110180   $112360      $2180          1.1 %
1998   $7865   $115536   $119102      $3565          1.2 %
1999   $8337   $121064   $126248      $5184          1.4 %
2000   $8837   $126754   $133823      $7069          1.5 %

The formulas for the cells are:  
  rent in 1997 = rent in 1996 times (1 + Appreciation)
  debt in 1996 = debt in 1995 times (1 + Interest rate)
                   minus rent in 1996
  price in 1996 = price in 1995 times (1 + Appreciation)
  net worth in 1996 = price in 1996 - debt in 1996
  profitability in 1997 = 100 times (net worth in 1997 
                          minus net worth in 1996)/price 
                          in 1996

The spreadsheet shows cash flows in complete detail. The point, however, is that the simple approximation of taking rental rate plus appreciation minus interest rate gets you to the right answer. It is exactly right for the first year, and only starts to be a little off later on because it misses the effect of the debt paydown.


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