### House price growth

The Freakwenter got mail!

Dear real person,

Subtracting inflation from growth to get real growth isn't exactly right, but it's usually better than nothing.

One source of confusion on this issue is that the averaging of growth rates can be done in two ways. Notice that a 10% growth in year 1 followed by a 20% growth in year 2 results in a cumulative 32% growth (1.1 times 1.2 = 1.32). You might think that this implies an "average annual growth" of 16%. But I'm pretty sure that the standard averaging method is to just average the individual annual growth numbers, for an average of 15% growth. (Investopedia agrees.)

A bizarre feature of this standard definition of "average annual growth rate" is that a given average growth rate does not uniquely determine the cumulative growth. A 0% growth in year 1 followed by a 30% growth in year two results in a cumulative growth of only 30%, despite the fact that the average growth is still 15 percent! (In the previous example, a 15% average growth led to a 32% cumulative growth.) Therefore, to get an exact answer, it is necessary to correct for inflation in each year of growth BEFORE computing the average inflation-adjusted annual growth rate.

The last thing to note is that subtracting the inflation rate from the growth rate gives real growth only approximately. Suppose a price increases by 49% while the CPI increases by 30% in a given period. Then the inflation-adjusted growth factor for that period is 1.49/1.30, or about 1.146, from which you can conclude that the price grew by about 14.6%. Notice this is significantly different than saying 49%-30%=19%. But for small growth rates, in the 0 to 10% range, this error becomes very small.

Dear Mr. Freakwenter:

Since 1983, Lancaster house prices have grown at an average annual rate of 4.9%, according to the US Office of Federal Housing Enterprise Oversight's House Price Index. [Click here for a nifty chart]

Inflation as defined by Wikipedia is "is a rise in the general level of prices of goods and services in an economy over a period of time."

Is it fair to say that the inflation-adjusted annual growth of house prices in Lancaster is close to 1.9% (assuming that the CPI has been averaging 3% per year)?

[real person]

Dear real person,

Subtracting inflation from growth to get real growth isn't exactly right, but it's usually better than nothing.

One source of confusion on this issue is that the averaging of growth rates can be done in two ways. Notice that a 10% growth in year 1 followed by a 20% growth in year 2 results in a cumulative 32% growth (1.1 times 1.2 = 1.32). You might think that this implies an "average annual growth" of 16%. But I'm pretty sure that the standard averaging method is to just average the individual annual growth numbers, for an average of 15% growth. (Investopedia agrees.)

A bizarre feature of this standard definition of "average annual growth rate" is that a given average growth rate does not uniquely determine the cumulative growth. A 0% growth in year 1 followed by a 30% growth in year two results in a cumulative growth of only 30%, despite the fact that the average growth is still 15 percent! (In the previous example, a 15% average growth led to a 32% cumulative growth.) Therefore, to get an exact answer, it is necessary to correct for inflation in each year of growth BEFORE computing the average inflation-adjusted annual growth rate.

The last thing to note is that subtracting the inflation rate from the growth rate gives real growth only approximately. Suppose a price increases by 49% while the CPI increases by 30% in a given period. Then the inflation-adjusted growth factor for that period is 1.49/1.30, or about 1.146, from which you can conclude that the price grew by about 14.6%. Notice this is significantly different than saying 49%-30%=19%. But for small growth rates, in the 0 to 10% range, this error becomes very small.

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