Question

The Marco family—comprising Mrs. Marco aged 40, Mr. Marco, aged 39, and their three young children— relocated to Barcelona in January 2020 when Mrs. Marco received a job offer from an international firm. They rented a three-bedroom condominium in Barcelona for 2.100€ per month, which included parking and fees.

While renting made life easy, the Marco family began weighing the pros and cons of purchasing a flat, in the same building, that became available in June 2020. The idea of home ownership as a form of long-term investment appealed to the couple. The preliminary rental payments could be used for mortgage payments instead.

While searching for the right property they found a nice apartment at 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 I: renting the apartment with a perpetual contract, meaning forever.

The family was very happy living in that area, and they had the chance to live there forever at an offered price of 1,650 EUR the first month, and the rent price will be growing by a 0.125% monthly. This option would prevent the Marco family from applying for a loan, which represented a heavy burden off the family’ budget.

Option II: consisted in acquiring the property with a mortgage scheme for 35 years. The total price of the apartment is 875.000€. The family can pay an initial down payment of 275,000 EUR and the rest (600,000 EUR) to be paid in constant monthly payments with an annual interest rate of a 2.75% compounded monthly.

Mrs. Marc establishes the maximum amount they can pay monthly as 2.250€.

1) In case of taking option I, what is the amount of the monthly payment the Marco family should pay in 35 years (in month 420)? (only the amount to be paid that month, show the calculations)

2) In case of taking option I, how much money will have the Marco family paid in total after 35 years?

3) If the Marcon family decides to leave Barcelona in 10 years, to attend a better offer elsewhere, what is the present value of the rental contract offered by the owner as option I? (consider 2.75% compounded monthly as the interest/discount rate)

4) If Mrs. Marco decides to buy the apartment, and accepts Option II, what will be the amount of each monthly payment to be done during the next 35 years?

Answer 1

Answer to Question 1:
The question requires what would be the payment to be made after 35 years for the 420th month
According to Option I, First rental Payment to be paid is EUR 1650
There after the rental payment to be paid would be growing 0.125% monthly
Accordingly I am providing the formula using below table for calculation of monthly Rent
First month Payment =	EUR 1650*(1)		1650
Second month Payment =	EUR 1650*(1+0.125%)	1652.0625
Third month Payment =	EUR 1650*[(1+0.125%)]^2	1654.127578
Fourth month Payment =	EUR 1650*[(1+0.125%)]^3	1656.195238
Nth month Payment =	EUR 1650*[(1+0.125%)]^(N-1)
480th month Payment =	EUR 1650*[(1+0.125%)]^(420-1)	2784.86171
Answer to Question 2:
The question requires sum of total monthly rental payments made in the first 35 years(i.e., 420 months payments) under Option I
The monthly payments are in a Geometric Progression Growing at the common ratio(r) of (1+0.125%) i.e., r=1.00125
First No. in the series of Geometric Progression (a) is 1650
Sum of first n terms in a geometric progression is = a*([1-(r^n)]/[1-r])
Sum of first 420 monthly payments in a geometric progression is = 1650*([1-(1.00125^420)]/[1-1.00125])
Which equals to	910674.2294
Answer to Question 3:
The Question requires to calculate present value of rental contract of 10 years under Option I
Given discount rate to be taken is 2.75% componded monthly
I would show show the discounting of first four payments look firstly
First month Payment =	[EUR 1650*(1)]
Second month Payment =	[EUR 1650*(1+0.125%)]/(1+0.0275)
Third month Payment =	[EUR 1650*[(1+0.125%)]^2]/(1+0.0275)^2
Fourth month Payment =	[EUR 1650*[(1+0.125%)]^3]/(1+0.0275)^3
Nth month Payment =	[EUR 1650*[(1+0.125%)]^(N-1)]/(1+0.0275)^(N-1)
The Discounting of each months payment is in a Geometric progression Growing at the common ratio (r) of (1+0.125%)/(1+0.0275) i.e., r=0.9745
First No. in the series of Geometric Progression (a) is 1650
as already provided in Answer 2 Sum of first n terms in a geometric progression is = a*([1-(r^n)]/[1-r])
Therefore sum of Discounted 10 years (i.e., 120 monthly payments) in the geometric progression is = 1650*([1-(0.9745^120)]/[1-0.9745])
Which equals to	61790.0337
Answer to Question 4:
The question requires If Option B is accepted what would be the monthly payments to be made
Given Total Cost of the Apartment = EUR 875,000
Initial downpayment that the family are willing to make is = EUR 275,000
Remaining Amount of EUR 600,000 is to be paid in 35 years (i.e., monthly) in equal installments
Equated Installment formula is	{P*[i*((1+i)^n)]}/((1+i)^n)-1
Where P equals to Principal amount to be collected
And I is the discount rate , n is no. of installments
in our Question P= EUR 600,000; i = 2.75%; n= 420
Therefore Equated Annual installment is	(600000*(0.0275*((1+0.0275)^420)))/((1+0.0275)^420)-1
Which equals to	16500.18583