Mathematics of Finance

The Andreotti family—comprising Mrs. Andreotti, aged 40, Mr. Andreotti, aged 38, and their three young children—relocated to Barcelonain January 2020 when Mrs. Andreotti received a job offer from a leading investment banking giant. They rented a three-bedroom condominium in Barcelona for 2.000€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€ the first month, and the rent price will be growing bya 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.

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Option B: consisted in acquiring the property with a mortgage scheme for 40 years. The total price of the apartment is 800,000€. The family can pay an initial down payment of 200.000€and the rest (600k€) to be paid in constant monthly payments with an annual interest rate of a 2.4% compounded monthly.Mrs. Andreotti is fixing the maximum amount they can pay monthly in 2.000€.
1) In case of taking option A, what is the amount of the monthly payment the Andreotti family should pay in 40 years?(only the amount to be paid that month) Show the calculations and explain why. (10points)
2) In case of taking option A, how much money will have the Andreotti family paid in total after 40 years? (10points)
3) If the Andreotti family decides to leave Barcelona in 10 years, back to Italy, what is the present value of the rental contract offered by the owner as option A?(10points)
4) If Mrs. Andreotti decides to buy the apartment, and accepts Option B, what will be the amount of each monthly payment to be done during the next 40 years? (10points)
5) Mrs. Andreotti believes that, if she takes option B and acquires the ownership of the flat,she might be interested in selling the apartment in 40 years’ time, that is to say, when she has already paid it all.If she wants to recover absolutely all the money she’s spent in it (initial payments plus all monthly payments done), what will be the price she will ask for that apartment in that moment?(10points)
6) Mrs. Andreotti is very happy for knowing how to calculate future values and present values, because this helps her in taking this type of decisions. Having said that,she wonders what the future value of the flat will be in 40 years, if the interest rate for this type of operations is an annual 1.5% (comp. monthly). Can you help her?
Explain your answer and show your calculations.(10points)

7)The family is still thinking that the monthly payments they’ll have to afford during the next forty years are too much, and they believe they could convince the seller of making payments only once a year, at the end of each year. The interest rate would still be the same 2.4%, How much money will they save with this action?
a) What is the amount of the yearly payment to be done? (10 points)
b) What is the total amount they’ll have paid in total after 40 years? (10 points)
c) How much has the family saved (if any) by paying it yearly instead of monthly? (10 points)

8)In case that the Andreotti family pays the pending amount in yearly payments, the owner can only grant them a 2.4% during the first 10 years. There is the possibility that, after the first 10 years the interest rate increases to a 3.0% for the remaining 30 years.How much should the Andreotti family pay per year from year 11 onwards if this occurs?(10points)

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