Question

The Andreotti family—comprising Mrs. Andreotti, aged 40, Mr. Andreotti, aged 38, and their three young children— relocated to Barcelona in 2020 when Mrs. Andreotti received a job offer from a leading investment banking giant. For the next six years, they rented a three-bedroom condominium for 2.000€ in Barcelona per month, which included parking and condominium fees.

While renting made life easy, the Andreotti family began weighing the pros and cons of purchasing a flat, in the same building, that became available in June 2020. In the past three years, the real estate market had softened somewhat, and the cost of the flats were stable. The idea of home ownership as a form of pension investment appealed to the couple. The monthly rents could be used for mortgage payments instead.

While searching for the right property they found a nice apartment with 200 square meters, very close to Diagonal-Numancia, one of the best locations of the city.

The apartment was owned and been promoted by a state-owned construction company and was offering to type of alternatives:

Option A: renting the apartment with a perpetual contract, meaning for ever and ever. The Andreotti family thought that could be a good solution for them.

The family was very happy living in that area, and they had the chance to live there forever at an offered price of 1.600€ per month. The contract contained a clause stating that the rent price will be growing at a 0.1% monthly.

At the same time, they were not forced to ask for a loan, which represented a heavy weight in the Andreotti family’s shoulders.

Option B: consisted in acquiring the property with a mortgage scheme for 40 years. The ownership was demanding an initial down payment of 1.000.000€. The total price of the apartment was still not clear, it seems there was some space for negotiation.

Mrs. Andreotti knew that the interest applicable rates were very attractive, around an annual 2.4% compounded monthly, this is supposed to be the market rate for this type of activities.

Mrs. Andreotti is fixing the maximum amount they can pay monthly in 2.000€.

THE QUESTION

What is the present value of the rental contract offered by the owner as option A?

Answer 1

Rent per month = €1,600

Number of periods = forever

Monthly growth of rental price = 0.1%

Rate of interest which Mrs.Andreotti can borrow = 2.4% per annum or 0.2% per month

Since the rental per month of €1,600 is offered forever, this becomes a perpetual cash flow.

Thus, present value of a growing perpetuity = PMT / (r-g)

where PMT = periodical payment = €1,600

r = interest rate (discount rate) = 0.2%

g = growth rate = 0.1%

Present value of the rental contract offered by the owner as option A = €1,600 / (0.2%-0.1%) = €1,600/0.1% = €1,600,000