Question

An All-Pro offensive lineman is in contract negotiations. The team has offered the following salary structure (note: today is Year=0). The appropriate discount rate is 12% per year, compounded monthly.

Year	Salary ($)
1	$3,500,000
2	$4,000,000
3	$4,500,000
4	$5,000,000

a) The lineman wants the timing of the payments modified. (He is happy with the overall value of the contract – so no increase in overall value, measured in PV terms -- is needed.)

Instead of the initial terms offered, he would like to receive a $3,000,000 signing bonus, today. In addition, he wants the remaining overall value of the contract paid out in equal quarterly installments (beginning one quarter from today) over the next four years (last payment at t=4. So, there are 16 quarterly payments). What payment amount should he receive each quarter (for the next four years)?

b) The lineman has changed his mind. He now wants the overall value of the contract increased by $2,000,000. In addition, he wants to receive quarterly payments (over the next four years) that grow by 2% each quarter. If the first quarterly payment is received one quarter from today and the last is received at t=4, what amount should he receive in the first quarterly payment (received 3 months from today)?

Please answer both parts of the question and please show your work

Answer 1

discount rate , i = 12% = 0.12

discount rate per month = m = i/12 = 12/12 = 1% = 0.01

effective annual rate , r = (1+(i/12))¹² -1 = (1+(0.12/12))¹² -1 = 0.126825 or 12.6825%

overall contract value, c = [ salary in year1/(1+r)] + [ salary in year2/(1+r)²] + [ salary in year3/(1+r)³] + [ salary in year4/(1+r)⁴]

= [ 3,500,000/(1.126825)] + [ 4,000,000/(1.126825)²] + [4,500,000/(1.126825)³] + [ 5,000,000/(1.126825)⁴]

= 3106072.2884 + 3150264.5097 + 3145162.2733 + 3101302.0253

=$ 12,502,801.0968

a)

signing bonus , s= $3,000,000

remaining contract value, C = c - s = 12,502,801.0968 - 3,000,000 =$ 9,502,801.0968

interest rate per quarter , x = (1+r)^(1/4) -1 = (1.126825)^(1/4) -1 = 0.030301 =3.0301%

no. of quarters in 4 years , n = 4*4 = 16

let the equal quarterly amount = A

C = A*PVIFA(16, 3.0301%)

where PVIFA = present value interest rate factor of annuity

PVIFA(16, 3.0301%) =[ (1+x)ⁿ -1]/((1+x)ⁿ *x) =

[ (1.030301)¹⁶ -1]/((1.030301)¹⁶ *0.030301) = 12.53224629

A = C/PVIFA(16, 3.0301%) = 9,502,801.0968/12.53224629 =$ 758,267.9812 or $758,267.98 ( after rounding off)

b)

new overall contract value, N = c +2,000,000 = 12,502,801.0968 + 2,000,000 = 14,502,801.0968

growth rate , g = 2% = 0.02

no. of quarters in 4 years , n = 4*4 = 16

let the first quarterly payment = A

N = A*[(1-((1+g)/(1+x))ⁿ )/(x-g)] = A*[(1-((1.02)/(1.030301))¹⁶ )/(0.030301 - 0.02)]

N = A*14.41756941

A = 14,502,801.0968/14.41756941 = $1,005,911.654 or $1,005,911.65 ( after rounding off)