Question

The Marconi family—comprising Mrs. Marconi, aged 40, Mr. Marconi, aged 38, and their three young children— relocated to Barcelona in January 2020 when Mrs. Marconi 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 Marconi 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 had been promoted by a state-owned construction company and was offering two alternatives:
Option A: renting the apartment with a perpetual contract, meaning forever. The Marconi 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 EUR the first month, and the rent price will be growing by a 0.1% monthly.

At the same time, they were not forced to ask for a loan, which represented a heavy burden off the Marconi’s.

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 EUR and the rest (600,000 EUR) to be paid in constant monthly payments with an annual interest rate of a 2.4% compounded monthly.

Mrs. Marconi establishes the maximum amount they can pay monthly as 2.000€.

• - In case of taking option A, what is the amount of the monthly payment the Marconi family should pay in 40 years (in month 480)? (only the amount to be paid that month) Show the calculations and explain why.

• - In case of taking option A, how much money will have the Marconi family paid in total after 40 years?

• - If the Marconi 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? (take the 2.4% compounded monthly as the interest/discount rate)

• - If Mrs. Marconi 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?

• - Mrs. Marconi 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 invested (initial payments plus all monthly payments done), what will be the price she will ask for that apartment at that moment? (don’t use the concept of Time Value of Money here, it is just about finding how much has Mrs. Marconi paid in total)

• - Mrs. Marconi 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). Find the Future Value of that apartment in 40 years. Explain your answer and show your calculations.

• - The family is still thinking that the monthly payments they’ll have to afford during the next forty years (we are still in option B) are too much, and they believe they could convince the seller of making constant payments only once per year, at the end of each year. The interest rate would still be the same 2.4% (but now that would be compounded yearly instead of monthly). What is the amount of the yearly payment to be done?

• - In this case (yearly payments) what is the total amount they’ll have paid in total after 40 years? (again, just find how much has Mrs. Marconi paid in total)

• - In this case (yearly payments), how much has the family saved (if any) by paying it yearly instead of monthly?

• - In case that the Marconi 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 Marconi family pay per year from year 11 onwards if this occurs?

Answer 1

Marconi family has an explicit rule of not exceeding monthly rental payments of 2000 €.

Option A is a perpetual contract, so it means that they can pay the rental for the selected house for ever ideally, taking into account the monthly incremental rate.

Option B is an ownership contract wherein you pay off the mortgage for the loan period after a particular downpayment. The repayment of the loan has nothing to do with the original price of the house.

• In case of taking option A, what is the amount of the monthly payment the Marconi family should pay in 40 years (in month 480)? (only the amount to be paid that month) Show the calculations and explain why

  This is a future value calculation, with the rate of increment of 0.1% monthly,hence the monthly payment in month 480 would be
  • €

• - In case of taking option A, how much money will have the Marconi family paid in total after 40 years?

The money Marconi family would have paid in a course of 40 years is the sum of all the FVs.This can be thought of as a sum of a geometric progression with constant multiple as 1.001(0.1% monthly) for a period of 480 months

The formula for sum in GP is

where a=1600, r=1.001 and n=480

so, total money to be paid is

€

• - If the Marconi 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? (take the 2.4% compounded monthly as the interest/discount rate)

  A)It can be calculated as
  • 
  • which comes out to 181033.6 €.

• - If Mrs. Marconi 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?

  This is a case of EMI calculation, where EMI is equated monthly installments
  • The formula for EMI is
    • where P=600,000€ as downpayment is deducted from the initial price of the house. r is 2.4/12=0.2% p.m and n is 480 months
    • Thus EMI is 1939.67€

• - Mrs. Marconi 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 invested (initial payments plus all monthly payments done), what will be the price she will ask for that apartment at that moment? (don’t use the concept of Time Value of Money here, it is just about finding how much has Mrs. Marconi paid in total)

• A)The recovered amount would be the initial downpayment and all the monthly payments, which can be calculated as

  200,000€ + 1939.67€*480 = 1,131,041.6€

• - Mrs. Marconi 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). Find the Future Value of that apartment in 40 years. Explain your answer and show your calculations.

  Mrs. Marconi can find the FV of the house by simply multiplying the initial price of the house by a compounding factor compounded for the entire period
  • The FV is = 1,457,148.96€

• - The family is still thinking that the monthly payments they’ll have to afford during the next forty years (we are still in option B) are too much, and they believe they could convince the seller of making constant payments only once per year, at the end of each year. The interest rate would still be the same 2.4% (but now that would be compounded yearly instead of monthly). What is the amount of the yearly payment to be done?

  This particular problem is similar to calculating EMI but with the caveat of each period being that of a whole year, we can name this EAI(Equated annual installment)
• where P is 600,000 €, r is 2.4% and n is 40 years
• So, EAI =23,500.96 €

• - In this case (yearly payments) what is the total amount they’ll have paid in total after 40 years? (again, just find how much has Mrs. Marconi paid in total)

  The total payment to be made after 40 years is EAI*40 which is computed as 940,038.58€

• - In this case (yearly payments), how much has the family saved (if any) by paying it yearly instead of monthly?

  The yearly payments save the family an amount of (1,131,041.6€ -  940,038.58€) = 191,003.02€

• - In case that the Marconi 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 Marconi family pay per year from year 11 onwards if this occurs?

  We need to draw up a repayment schedule of principal and interest for the first 10 years,
• The pending principal at the end of 10 years is 498505.1 euros. This is the initial principal for the subsequent EAI calculation, where n is 30 years and r is 3%
• EAI modified = = 25433.36€