Question 1.
As a financial advisor, you are consulting a client who has won the lottery. The prize for the competition can be paid in 4 different ways described below. The annual discount rate is 11%.
(1) $1,140,000 now
(2) A one-time payment of $1,000,000 in 5 years and an additional one-time payment of $1,100,000 in 10 years
(3) 5 consecutive annual payments of $400,000, with the first payment made at the end of year 3 (that is, at t=3) and the remaining payments made at the end of each of the four following years
(4) 25 consecutive annual payments, with the first payment of $270,000 made at the end of year 2 and the remaining payments (which occur at year end) declining by 10 % per year.
Questions:
(1a) What is the present value of each alternative and which option would you recommend?
(1b) Suppose that your client would like to replace alternative (1) with a perpetual stream of consecutive annual payments (made at year-end), which starts at the end of year 5 and grows by 6% per year forever. What should be the amount of the first payment at the end of year 5 so that you are indifferent between these two alternatives?
(1c) You are comparing alternative (2) to a stream of annual inflows of $160,000 each (payments occur at year-end), with the first payment made at the end of year 1. What is the minimum whole number of payments in this new income stream that would make it more valuable than alternative (2) in present value terms?
Question 2.
Payday lenders are firms that make short-term (often one- to two-week) loans to consumers. The intent is to provide households with some extra cash in advance of the next paycheck. According to the loan terms, if you obtain a two-week loan of $500, in two weeks (i.e., 14 days), you have to repay $565.45.
Questions:
(2a) What is your two-week interest charge expressed as a percentage of the loan received? (i.e. your periodic interest rate over the 14-day period)?
(2b) If the interest is compounded every 14 days, how many compounding periods do you have in one year (assuming 365 days in a year)?[1] Using this number of compounding periods and the periodic rate for the 14-day loan you found in (2a), calculate the APR of the payday lenders’ loan.
(2c) Using the number of compounding periods and the APR that you found in (2b), find the Effective Annual Rate (EAR) on this loan. As in (2b), interest is compounded every 14 days and there are 365 days in a year.
(2d) Suppose the payday lender charges the same APR as it does right now for a $500 loan, but changes its compounding frequency to daily. Assuming daily compounding (365 days in a year) and the APR found in (2b), what would be the EAR on this loan?
Question 3.
You are evaluating a project to produce a movie and are wondering how much you can afford to pay the actors.
Actors’ Compensation
The shooting of the movie will commence in month 5 and will take 12 months to complete. Actors are paid on a monthly basis, and their salary grows by 1% each month. The first salary payment to actors will occur at the end of month 5 and the remaining 11 payments will occur at the end of each of the next eleven months.
The Movie
After the movie is released, it will be shown in theaters for 5 months, producing cash inflows at the end of each month. The first cash inflow, to be received at the end of month 18, is estimated to be $25 million. For the remaining four months, the inflow will decrease at a monthly rate of 10%, as the viewership gradually declines and the picture moves to budget theaters.
The Decision
You are now negotiating with several famous actors and wonder about the following question: what is the maximum amount of the first monthly payment to the actors so that the movie creates at least $20 million in present value terms? Assume that the actors’ salary is the only expense in the movie production, that there are no taxes in this particular example, and that the annual discount rate is 24% APR with monthly compounding. This discount rate reflects the high risk and uncertainty associated with producing a movie.
Please do not round the number of periods to an integer; retain two digits after the decimal point.