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

If P₀ is the price today, g is CAGR and P_n is the price n years later then

P_n = P₀ x (1 + g)ⁿ

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.

6.2 = 2.5 x (1 + g)^{(2019 - 2000)} = 2.5 x (1 + g)¹⁹

Hence, g = (6.2 / 2.5)^1/19 - 1 = 4.90%

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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.

P₃₀ = 6.2 x (1 + g)³⁰ = 6.2 x (1 + 4.90%)³⁰ =  $ 26.01 mn

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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.

P₂₀₁₉ = 6,200,000 = P₁₈₁₂ x (1 + g)^{(2019 - 1812)} = P₁₈₁₂ x (1 + 4.90%)²⁰⁷ = P₁₈₁₂ x  19,835.69

Hence, P₁₈₁₂ = 6,200,000 /  19,835.69 = $  312.57

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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.

P₂₀₁₉ = 6,200,000 = P₁₇₉₅ x (1 + g)^{(2019 - 1795)} = P₁₇₉₅ x (1 + 4.90%)²²⁴ = P₁₇₉₅ x 44,707.22

Hence, P₁₇₉₅ = 6,200,000 / 44,707.22 = $ 138.68

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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.

Year 1795 is the time 0 in the problem.

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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.

8.5 = 2.5 x (1 + g)^{(2018 - 2000)} = 2.5 x (1 + g)¹⁸

Hence, g = (8.5 / 2.5)^1/18 - 1 = 7.04%

This growth rate is better (higher) than the growth rate we calculated in part (1) earlier.

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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.

P₂₀₁₈ = 8,500,000 = P₁₈₁₂ x (1 + g)^{(2018 - 1812)} = P₁₈₁₂ x (1 + 7.04%)²⁰⁶ = P₁₈₁₂ x 1,209,152.11

Hence, P₁₈₁₂ = 8,500,000 / 1,209,152.11 = $ 7.03

This price is much lower than what we calculated earlier.