Question

Assume that two athletes sign 10-year contracts that pay out a total of $100 million over the life of the contracts. One contract will pay the $100 million in equal installments over the 10 years. The other contract will pay the $100 million in installments, but the installments increase 5% per year. Which athlete received the better deal? You must show your work and explain your position in detail.

Answer 1

The athelete who receives the constant installments
will receive each year 100 million/10 = 10 million per year
The athelete who received the growing installment will
receive installments in the series as F*1.05^n, where F =
the first installment, F*1.05, the second installment,
F*1.05^2 the third installment and so on, 100 million
being the sum of the installments.
This is a geometric series
Sum of the first Sn terms of a geometric sequence is given by the formula
Sn=a1(1−r^n)/(1−r), r≠1
where n is the number of terms, a1 is the first term and r is the common ratio.
Substituting values we have
100 million = F*(1-1.05^10)/(1-1.05) =	12.577893
F (the first installment) = 100/12.577893 =	$     7.9505	millions
The installments would be	7.95
	8.35
	8.77
	9.20
	9.66
	10.15
	10.65
	11.19
	11.75
	12.33
	100.00
In contrast the equal installments would be 10 million each year.
The constant installments would be preferred as it is higher for the
first 5 years than the growing installments.
This would mean higher present value for the constant installments.
Using a discount rate of 10%
The PV of constant installments would be 10*(1.1^10-1)/(0.10*1.1^10) =		$               61.45	million
The PV of the growing installments would be
=(( P/(r-g))*[1-((1+g)/(1+r))^n
where P = first payment, r = rate of interest per period,
g = growth rate and n = number of periods.
Substituting values we have
PV of the growing installments = ((7.95/(0.10-0.05))*((1-((1.05/1.10)^10)))) =		$               59.15	million
As can be seen the PV of equal installments is higher as
the initial installments are more.