The Stack has just written and recorded the single greatest rock song ever made. The boys in the band believe that the royalties from this song will pay the band a handsome $200,000 every year forever. The record studio is also convinced that the song will be a smash hit and the royalty estimate is accurate. The record studio wants to pay the band up front and not make any more payments for the song. What should the record studio offer the band if it uses a 5% discount rate, a 7.5% discount rate, or a 10% discount rate?

The band from the previous problem agrees to the one-time payment at a 5% discount rate, but it wants to figure the royalty payments from the beginning of the year, not the end of the year. How much more will the band receive?

One time royalty should be $4,000,000 (r=5%), $2,666,666 (r=7.5%) or 2,000,000 (r=10%).

The band receive $200,000 more if it wants to figure the royalty payments from the beginning of the year

Explanation:

One time royalty payment = a

Interest rate = r

Current Value of

first yearly royalty = 200,000 * 1 / (1+r) second yearly royalty = 200,000 * 1 / (1+r) ^2 ... and so on indefinitely

It's a indefinite geometric series where the total value is given by

a = 200,000 * [1 / (1+r) ] / [1 - 1 / (1+r) ]

r = 0.05 we have a = 200,000 * (1/1.05) / [1 - (1/1.05) ] = $4,000,000

r = 0.075 we have a = 200,000 * (1/1.075) / [1 - (1/1.075) ] = $2,666,666

r = 0.1 we have a = 200,000 * (1/1.1) / [1 - (1/1.1) ] = 2,000,000

If the yearly royalty is supposed to come at the beginning of the year, value of each term in the series will increase by (1+r)

Then a becomes: a = 200,000 [1 - 1 / (1+r) ] = 200,000/[1 - (1/1.05) ] = $4,200,000

The band receive an addtional amount of $4,200,000 - $4,000,000 = $200,000