Question

You own a house on a plot of land. The land has a value of $150,000. Use of the house has a value to you. You believe that the first years benefit to you amounts to $8000. Assume that this is as though you get $8000 at the end of the first year. Thereafter the annual benefits diminish at 4% per year. Furthermore, assume that the house will have to be demolished in exactly 22 years. Assume that rates are 8% p.a. Assume the land value remains unchanged over time.

a. What is the total value of the house and land?

b. You enter into a contract to sell the house. The purchaser will take possession in exactly 6 years. The purchaser uses the same method of valuation as you have. However, you want payment today. What amount will the purchaser be willing to give you today?

Answer 1

Land Value = 150,000 | Benefit for first year = $8000 | Growth in Benefit = -4% | Time = 22 years

Rate = 8%

a) Since the benefit will be end of year and grows at -4% rate for 21 years, hence, it is a growing annuity.

Present value of Growing Annuity formula = (CF/(R-G))*(1-((1+G)/(1+R))^T)

G = -4% | R = 8% | T = 22 | CF = 8000

In this case, (1+G) will be compounded for 21 years as 8000 itself is the first year benefit, however, discount for (1+R) will be for 22 years to find the Value of house at Year 0.

PV of Annuity = (8000 / (8%+4%))*(1-((1-4%)²¹/(1+8%)²²))

Solving the above equation, we will get the Present value of the benefits for 22 years, which will be the Value of House.

Value of House = $ 61,463.33

Total Value of House and Land = Value of House + Value of Land

Total Value of House and Land = 61,463.33 + 150,000 = $ 211,463.33

b) Purchaser takes possession in = 6 years | Time remaining before demolition = 22 - 6 = 16 years

We will use same method to value the house for 16 years, as did in part (a). However, cashflow will change.

Cashflow at Year 6 = CF * (1+G)⁵ = 8000 * (1-4%)⁵ = 6,522.98 (Growth is for 5 years since 8000 itself is 1st year benefit)

We will use present value of growing perpetuity with below formula:

Present value of Growing Annuity formula = (CF/(R-G))*(1-((1+G)/(1+R))^T)

(1+G) will be compounded for 15 years, however, (1+R) discounting will be for 16 years.

Value of the house after 6 Years = (6522.98 / (8%+4%))*(1-((1-4%)^15)/((1+8%)^16))

Solving the above equation, we will get the value of the house after 6 years.

Value of the house after 6 years = $ 45,757.09

As value of land doesn't change, hence, value of land and house will be as given below:

Total Value of House and Land = 45,757.09 + 150,000 = $ 195,757.09