Solution:  Evaluation of Investments with Different Lives

For this problem, we present a standard solution that we use for all investment decisions that involve different lives. The goal is to normalize the costs or cash flows in terms of the time period. In this particular example, the obvious thing to do, is to express the three options on a 'per day' basis. This way, we can immediately tell what the 'equivalent daily cost' is for the daily bus ticket: $2.00

For the other two options, the weekly and monthly pass, we calculate the Present Value of the cash flows first:

PVweekly pass = $5.00 + PV of an Annuity with t = 7, r = 0.01, and CF = $1.20  => PVweekly pass = $13.07
 

PVmonthly pass = $25.00 + PV of an Annuity with t = 30, r = 0.01, and CF = $1.10  => PVmonthly pass = $53.39
 

Find the Equivalent Daily Cost (EDCw and EDCm) by using the PV of an annuity formula, setting the annuity equal to respectively PVw and PVm:
 

For the weekly pass: $13.07 = [EDCw / 0.01] * [ 1 - (1 / 1.017)], hence  EDCw =  $1.94

For the monthly pass: $53.39 = [EDCm / 0.01] * [ 1 - (1 / 1.0130)], hence EDCm =  $2.07

Compare this to a daily cost of $2.00 if no pass is bought and we can conclude that we should buy weekly passes!

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