Question

Gen. Robert E. Lee's Boyhood Home is for Sale

The historic Virginia home that Confederate Gen. Robert E. Lee grew up in hit the market (in April 2018) for $8.5 million. (Trapasso, C.)

Robert E. Lee's father Henry rented the home in 1812, according to The Washington Post. The family lived there for over 80 years, including Robert E. Lee from age five to when he went to West Point in 1825. He again visited five years after the Civil War ended, The Post reported. (Leayman, E.)

The home's other claim to fame is that President George Washington also dined and lodged there before the Lee family moved in. (Trapasso, C.)

Built in 1795, the brick house was listed on the National Register of Historic Places in 1986. (Trapasso, C.)

The home had been used as a residence until 1966. The Stonewall Jackson Memorial Foundation purchased the home and opened it to the public. Unable to make ends meet, the foundation sold the home in 2000 to Mark and Ann Kington for $2.5 million. (Trapasso, C.)

The boyhood home of Robert E. Lee in Alexandria was listed on the market with a significant price drop. Previously priced at $8.5 million, the six bedroom is available for $6.2 million (March 2019). (Leayman, E.)

1. Calculate the annual compound growth rate of the house price since the house was sold to Mark and Ann Kington (since 2000) until the house was listed for sale at a reduced price in 2019. (Round the number of years to the whole number). Please show your work.
2. Assume that the growth rate you calculated in question #1 remains the same for the next 30 years. Calculate the price of the house in 30 years after it was listed at a reduced price in 2019. Please show your work.
3. Assume that the growth rate you calculated in question #1 remains the same since Robert E. Lee's father Henry rented the home in 1812. Calculate the price of the house in 1812. (Round the number of years to the whole number). Please show your work.
4. Assume the growth rate that you calculated in #1 prevailed since 1795. Calculate the price of the house in 1795. (TIP: To get the answer correctly you need to use the price of the house in your calculations in dollars with all zeros). Please show your work.
5. You were using the time value of money concept to answer question #4. Think about the time line for that problem. What is the time point 0 in that problem? Please explain your answer.
6. In April 2018, the listed price of the house was $8.5 million. Calculate the annual compound growth rate of the house price since the house was sold to Mark and Ann Kington (since 2000) until the house was listed for sale in 2018. Compare with your answer to the question #1.
7. Using the growth rate from question #6, calculate the price of the house in 1812. (Round the number of years to the whole number). Please show your work. Compare with your answer to the question #3.

Answer 1

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).