Question

The Andreotti family—comprising Mr. Andreotti, aged 40, Mrs. Andreotti, aged 38, and their three young children— relocated to Barcelona in 2020 when Mr. 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 Mr. Andreotti´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.

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

Mr. Andreotti is fixing the maximum amount he can pay monthly in 2.000€.

1) What is the maximum amount that Mr. Andreotti should pay? Show the calculations and explain why. (15 points)

2) What is the total amount that Mr. Andreotti will pay in total after 40 years? (15 points)

3) What is the present value of the rental contract offered by the owner as option A? (15 points)

4) Mr. Andreotti believes that he might be interested in selling the apartment in 40 years’ time, this is when he is planning to retire.

If the interest rates remain at the existing level, what will be the price of the apartment in that moment? (15 points)

5) Mr. Andreotti is very happy for knowing how to calculate future values and present values, because this helps him in taking this type of decisions. Having said that he wonders what the future value of the rental contract could be. Can you help him? Explain your answer and show your calculations. (10 points)

Answer 1

NOTE- There is a slight error in the problem statement. The initial down payment has to be 100,000 and not 1,000,000.

1) EMI = [P x R x (1+R)^N]/[(1+R)^N-1]

where P stands for the loan amount or principal,

R is the interest rate per month [if the interest rate per annum is 2.40%, then the rate of interest will be 2.40/(12 x 100)],

and N is the number of monthly instalments

Given,

Down payment	100000
Interest rate	2.40%
Compounding frequency	12
EMI	2000
Mortgage Tenure (years)	40

Here, N = 40*12 = 480

R = 2.40%/12 = 0.20%

EMI = 2000

Substituting these values in the formula above, we calculate the principal amount of mortgage.

Mortgage amount

616740

Hence, the maximum purchase price of the property can be (mortgage principal amount + down payment) = 716,740.

2) Total amount paid = EMI*N + Down payment = 2000*480 + 100000 = 1,060,000

3) Present Value (PV) of a growing perpetuity = D1 / (r-g)

where D1 = Monthly rent at the end of 1st month = 1600

r = interest rate per month = 0.20%

g = rental growth rate = 0.10%

PV = 1600 / (0.2% - 0.1%) = 1,600,000

4) Present purchase price of the property = 716,740

Interest rate = 2.40% compounded monthly

After 40 years, FV = PV*(1+r)^n = 716740*(1+0.20%)^480 = 1,870,113

5) FV of rental contract = PV*(1+r)^n = 1,600,000*(1+0.20%)*480 = 4,174,710