# annuities and perpetuities

Annuity & Perpetuity calculations – Solve the following mathematical problems within the topic of annuities and perpetuities.

Mrs. Johnson has just moved to London with her new job, in which she received a significant salary increase. Being new to the city, she is now trying to decide whether she should rent an apartment or buy one. She has spent the first weekend looking at different options and has shortlisted two options.

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• Option 1: Rent an apartment, which would cost her a total of 3,000£ per month including everything. Every year the rent is due to increase by 2.50%. The contract is perpetual meaning she could theoretically stay there forever if she wanted.

• Option 2: Buy an apartment, which costs 1,350,000£ with an initial down payment of 150,000£ and a 30-year loan for the remaining 1,200,000£. The loan would carry an interest of 1.5%. After 30 years she would own the apartment 100%

Assumptions

1. She will buy or rent the apartment on the 1st of January

2. Payments happen at the end of each period

Questions

1. In case Mrs. Johnson chooses option

2, how much will she have to pay each month for her loan?

2. In option 1, Mrs. Johnson is unsure how much rent she will have to pay in the future. Show the monthly rent per year from year 1-10 3. How much rent would she have paid in total over the 10 years?

4. If she chooses option 1, she believes she can invest 150,000£ today and in addition, 900£ each month in a financial product that offers 6% return. Assuming she plans to retire in 30 years, how much will she have saved up?

5. Alternatively, if she chooses option 2, she will not be able to invest anything (other than what she invests in her apartment). She believes that the apartment will be worth approximately 1,800,000£ in 30 years at which point she will have no debt. Which options provides the largest savings after 30 years?

6. Upon retirement she expects to have a yearly cost of living of 37,500£ in year one, which will increase each year by 3.5%. She believes she can continue to grow her investments at an annual return of 5.5%. As she does not know how old she will grow, she would like to assume that she needs payments in perpetuity to never run out of money. Will the two options allow her to live this way?

7. Mrs. Johnson is somewhat uncertain how long she will stay in London. She is considering moving move outside of the city in 10 years.

a. Calculate how much her investment would have grown to if she chooses option 1

b. If she chooses option 2, she believes that she would be able to sell the apartment after 10 years and have 375,000£ left after paying the rest of her loan back

c. Which option is best with 10-year horizon? Submission file format: Word document with all the answers, clearly identifying all steps, results, and including comments for each item of the case

Submission file format: Word document with all the answers, clearly identifying all steps, results, and including comments for each item of the case.