Question

Ruth Gehrig has received contract offers from three professional aquatic badminton teams. The details

are below. Which contract is most valuable?

a.Salary of $500,000 per year for 7 years, starting next year.

b. Salary of $333,333 per year for 7 years, starting next

year, plus a $1 million signing bonus, received immediately.

c. $350,000 per year for 3 years, starting next year. In years 4 - 7, the salary increases to $700,000 per year.

The interest rate is 2% per year.

Please use the PV formulas and not excel. I am trying to learn the formulas first :)

Answer 1

Answer:

a: Salary of $500,000 per year for 7 years, starting next year.

We will use PV of annuity due formula to get PV as of starting next year and then discount the same to get PV as of today:

PV of annuity due = A + A (1- (1 + r) ^{- (n - 1)}) /r

PV as of starting next year = 500000 +500000 * (1 - (1 + 2%) ^{- (7 - 1)} ) / 2%

= $3,300,715.4453

PV as of now =PV as of starting next year / (1 + 2%) = $3,300,715.4453 / (1 + 2%) = $3,235,995.53

PV = $3,235,995.53

(b) Salary of $333,333 per year for 7 years, starting next year, plus a $1 million signing bonus, received immediately.

PV as of starting next year = 333333 +333333 * (1 - (1 + 2%) ^{- (7 - 1)} ) / 2% = 2200474.7631

PV as of now = PV as of starting next year /(1 + 2%) + $1,000,000 = $2200474.7631 /(1 + 2%) +1000000 = $3,157,328.20

PV = $3,157,328.20

(c) $350,000 per year for 3 years, starting next year. In years 4 - 7, the salary increases to $700,000 per year.

PV at the start of year 4 = 700000 + 700000 * (1 - (1 + 2%) ^{- (4 - 1)} ) / 2% = 2718718.29085

PV (as of now) of years 4 - 7 salary = 2718718.29085 / (1 + 2%) ⁴ =2,511,675.45766

PV at the start of next year of salary from year 1 to year 3 = 350000 + 350000 * (1 - (1 + 2%) ^{- (3 - 1)} ) / 2% = 1,029,546.32834

PV (as of now) of salary from year 1 to year 3 = 1,029,546.32834 / (1+ 2%) = 1,009,359.14543

PV = PV of salary from year 1 to year 3 + PV of years 4 - 7 salary = 1,009,359.14543 + 2,511,675.45766 = $3,521,034.60

PV = $3,521,034.60

Since PV of contract 3 is highest, Contract 3 is most valuable.