Question

A transit district has asked for assistance in determining the proper fare for its bus system. An effective annual interest rate of 7% is to be used. The following additional information was compiled.

Cost per bus

Bus life

Salvage Value

Miles driven per year 37,440

Number of passengers80,000

Operating cost $1.00 per mile in the first year, increasing $0.10 per mile

after each year thereafter

$60,000 20 years $10,000

8. If the fare is to remain constant for the next 20 years, the break-even fare per passenger is most nearly:

(A) $0.51 per passenger (B) $0.61 per passenger (C) $0.84 per passenger (D) $0.88 per passenger

9. If the transit district decides to set the per-passenger fare at $0.35 for the first year, approximately how much should the passenger fare go up each year thereafter such that the district can break even in 20 years?

(A) $0.022 increase per year (B) $0.036 increase per year (C) $0.067 increase per year (D) $0.072 increase per year

Answer 1

solution.

Initial cost ( cost per bus) = $60000

Salvage Value = $10000

Operating cost year 1 = 37440*1=$37440

Addition in operating cost form year 2= 0.1*37440=$3744

Let the Constant fare be f, Total Revenue = 80000*f=$80000f

PV of operating cost will be PV of $37440 for 20 years and PV of gradient of $3744 for 20 years

PV of $37440 for 20 years=A*(1-(1+r)^-n)/r

=37440*(1-(1+7%)^-20)/7%

=37440*(1-1.07^-20)/0.07

=37440*(1-0.2584)/0.07

=37440*0.7416/0.07

=$396639.89

PV of gradient of $3744 for 20 years=A*((1+r)^n-r*n-1)/((1+r)^n*r^2)

=3744*((1+7%)^20-7%*20-1)/((1+7%)^20*7%^2)

=3744*(1.07^20-20*0.07-1)/(1.07^20*0.07^2)

=3744*(3.8697-1.4-1)/.8697*0.0049)

=3744*(1.4697)/0.0190

=$290196.99

Total PV of operating cost = 396639.89+290196.99=$686836.88

PV of Salvage Value = 10000/(1+7%)^20 =10000/1.07^20=10000/3.8697=$2584.19

Total PV of Cost = 60000+686836.88-2584.19=$744252.69

PV of Revenue =A*(1-(1+r)^-n)/r

=80000f*(1-(1+7%)^-20)/7%

=80000f*(1-1.07^-20)/0.07

=80000f*(1-0.2584)/0.07

=80000f*0.7416/0.07

=$847521.14f

At break even Total cost=Total revenue

or, 847521.14f=744252.69

or, f = 744252.69/847521.14=$0.88

Hence fare should be $0.88 per passenger

If constant fare is $0.35, then PV of constant fare =847521.14*0.35 = $296632.40

Hence Remaining cost to be recovered = 744252.69-296632.40=$447620.29

Hence PV of gradient fare shall be =$447620.29

or, 447620.29=A*((1+7%)^20-7%*20-1)/((1+7%)^20*7%^2)

or, 447620.29=A*(1.07^20-20*0.07-1)/(1.07^20*0.07^2)

or, 447620.29=A*(3.8697-1.4-1)/.8697*0.0049)

or, 447620.29=A*(1.4697)/0.0190

or, A = 447620.29*0.0190/1.4697

or, A = 5786.75

Hence Fare per passenger = 5786.75/80000=$0.07

PV of constant Fare =847521.14*0.45=$381384.51

Gradient Fare= 80000*0.05 = 4000

PV of Gradient Fare = 4000*((1+7%)^20-7%*20-1)/((1+7%)^20*7%^2)

=4000*(1.07^20-20*0.07-1)/(1.07^20*0.07^2)

=4000*(3.8697-1.4-1)/.8697*0.0049)

=4000*(1.4697)/0.0190

=$309410.53

Total PV of Revenue =381384.51+ 309410.52=$690795.03

Remaining cost for break even =744252.69- 690795.03=$53457.66

Hence Subsidy per passenger = 53457.66/80000=$0.67.

ANSWER. OPTION (C) $0.067 increase per year