Annuities are streams of payments that the owner of an annuity receives for some specified period of time. The holder of an annuity can sell it to someone else who then becomes the recipient of the remaining stream of payments that are still owed.
A: Some people who retire and own their own home finance their retirement by selling their house for an annuity: The buyer agrees to pay $x per year for n years in exchange for becoming the owner of the house after n years.
(a) Suppose you have your eye on a house down the street owned by someone who recently retired.
You approach the owner and offer to pay her $100,000 each year (starting next year) for 5 years in exchange for getting the house in 5 years. What is the value of the annuity you are offering her assuming the interest rate is 10%?
(b)What if the interest rate is 5%?
(c) The house’s estimated current value is $400,000 (and your real estate agent assures you that homes are appreciating at the same rate as the interest rate.) Should the owner accept your deal if the interest rate is 10%? What if it is 5%?
(d) True/False: The value of an annuity increases as the interest rate increases.
(e) Suppose that, after making the second payment on the annuity, you fall in love with someone from a distant place and decide to move there. The house has appreciated in value (from its starting value of $400,000) by 10% each of the past two years. You no longer want the house and therefore would like to sell your right to the house in 3 years in exchange for having someone else make the last 3 annuity payments. How much will you be able to get paid to transfer this contract to someone else if the annual interest rate is always 10%?
B: In some countries, retirees are able to make contracts similar to those in part A except that they are entitled to annuity payments until they die and the house only transfers to the new owner after the retiree dies.
(a) Suppose you offer someone whose house is valued at $400,000 an annual annuity (beginning next year) of $50,000. Suppose the interest rate is 10% and housing appreciates in value at the interest rate. This will turn from a good deal to a bad deal for you when the person lives n number of years. What’s n? (This might be easiest to answer if you open a spreadsheet and you program it to calculate the value of annuity payments into the future.)
(b) Recalling that the sum of the infinite series 1/(1+x)+1/(1+x)2 ++1/(1+x)3+... is 1/x, what is the most you would be willing to pay in an annual annuity if you want to be absolutely certain that you are not making a bad deal?