Question

1. Annual compound growth rate of the house price from 2000 to 2019 listing:

Price in 2000 P0 = 2.5 million; price in 2019 P1 = 6.2 million

Number of years n = 2019 - 2000 = 19

Total growth rate G = P1/P0 -1 = 6.2/2.5 -1 = 1.48 or 148%

Annual compounding rate g: (1+g)^n = (1+G)

(1+g)^19 = (1+148%)

g = (1+148%)^(1/19) -1 = 4.90%

2. Listing price P1 = 6.2 million; n = 30 years; g = 4.90% per year

Price after 30 years P2 = P1*(1+g)^n

= 6.2*(1+4.90%)^30 = 26.01 million

3. Number of years from 1812 to 2000 n = 2,000 - 1,812 = 188

Price in 1812 P3 = P0/(1+g)^n = 2,500,000/(1+4.90%)^188 = $312.57

4. Number of years from 1795 to 2000 n = 2,000 - 1,795 = 205

Price in 1795 P4 = P0/(1+g)^n = 2,500,000/(1+4.90%)^205 = $138.68

5. The time point 0 in part(4) is 1795. Taking an interval of one year (since annual growth rate is being applied), time point 205 is 2000 (calculated as 1,795 + 205 =2,000). So, to calculate the price at t = 0, we need to discount the given price at t = 205.

6. P0 = 2.5 million in year 2000

Listed price L1 = 8.5 million in year 2018

n = 2,018 - 2,000 = 18

Total growth rate G1 = (L1/P0) - 1 = (8.5/2.5) -1 = 240%

Annual growth rate g1: (1+g1)^n = (1+G1)

g1 = (1+G1)^(1/n) -1 = (1+240%)^(1/18) -1 = 7.04%

In comparison to the growth rate calculated in 2018 (g = 4.90%), this is an increase of 7.04% - 4.90% = 2.14%

7. L1 = 8.5 million; n = 2,018 - 1,812 = 206

Price in 1812 P5 = L1/(1+g1)^n

= (8,500,000)/(1+7.04%)^206 = $7.03

This represents a lower price of 312.57 - 7.03 = 305.54, compared to the price of 312.57 in part (3).

Based on these answers, explain the changes in the value of this house?

Answer 1

The difference in the value of the house is because of using different growth rates in the two cases.

In the first case, we assumed the compounded growth rate as 4.90% and calculated the House price in the year 1812 = $312.57

In the second case, we assumed the compounded growth rate as 7.03% and calculated the House price in the year 1812 = $7.03

In case 1 the price in 2019 is $6.2 mn but in case 2 the price in 2018 is $8.5 mn. So we can see that the house price decreased from 2018 to 2019. This made the growth Rate also decrease from 7.03% to 4.90%.

In case we considered only the list prices in 2018 and 2019. Growth rate = (6.2/8.5) - 1 = -27.06%    

Using this growth rate of -27.06%, if we calculated the price of the house in 1812.

P0 = 8.5 million in year 2018

Number of years from 1812 to 2018 = 206

Price in 1812 = p0/(1+g)^n = 8,500,000/(1-27.06%)^205 = 1,435,587,229,094,060,000,000,000,000,000,000,000.00

We can see that it is an abnormally huge amount.

Hence extrapolating the growth rates is not a fair way to calculate the historic and future prices of properties because there will always be periods of economic booms and downturns. Using such growth rates will lead to abnormally high or low valuations for historical or future prices.

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