Question 2: Optimal Replacement (Lecture 7)

Setting up the problem

I am using the paper chiefly as a source of real-world data. There are a number of methodological issues with the paper, and I am not using all of the information in the paper, so you should not expect your results to match those of the paper.

The costs to King County Metro associated with a newly purchased hybrid bus in Washington State are as follows:

• The bus is purchased in Year 0 for $958,000
• The salvage/resale value of the bus is $1,000, no matter when it is re-sold or salvaged.[1]
• Operating and Maintenance costs are $48,179.61 in Year 0 and go up by $2,184.27 per year thereafter.[2]
• King County Metro's MARR is derived from an APR of 7%, compounded monthly. (You'll have to turn this APR into an interest rate before you can use it as the MARR.)[3]
• The "planning horizon" is 100 years. (You don't need to consider economic lifetimes greater than 100 years.)

2.a. Writean equation for Equivalent Annual Costs (EAC)

Use your knowledge of discounted cash flow analysis to write an equation for the Equivalent Annual Costs (EAC) of the bus as a function of K, the number of years the bus is kept before being sold and replaced with a new bus.Please write your equation using the appropriate discount factor notation (e.g. (P/F,6%,K)). Do NOT try to expand out the factors further(e.g. 1/(1+6%)^K), as that just makes your reasoning more difficult to follow.

Write your equation below

EAC(K) =

[1] From the paper: "The salvage values for the two buses are assumed to be $1,000 regardless of bus type or bus age according to King County Metro's request".

[2] The paper had the following equation for per-mile operating & maintenance costs: 1.458+0.661t, where t is the age of the bus. The baseline scenario assumed the bus would be driven for 33,045 miles per year. 33,045 x 1.458 = 48,179.61, and 0.661 x 33,045 = 2,184.27.

[3] From the paper: "A 7.0% annual discount rate (APR) [...] [is] assumed to be constant throughout the planning time horizon according to King County Metro's request". The authors of the paper don't appear to know the difference between an APR and a discount rate, so didn't list a compounding period. I picked a monthly compounding period for the purposes of our problem, as that's fairly common.