### Investment Recovery and Discounted Payback Period Calculation**Problem Statement:**If a project costs $90,000 and is expected to return $24,500 annually, how long does it take to recover the initial investment? What would be the discounted payback period at $i = 15\%$? Assume that the cash flows occur continuously throughout the year.---**Calculation Requirements:**1. **Determine Payback Period:** - **Formula:** The payback period is calculated by dividing the initial investment by the annual return. - **Input:** - Initial investment: $90,000 - Annual return: $24,500 - **Output:** - Payback period in years (rounded to one decimal place)2. **Calculate Discounted Payback Period:** - **Formula:** The discounted payback period considers the present value of future cash flows. Each yearly return must be discounted at the rate of 15%. - **Assumptions:** Cash flows occur continuously throughout the year. - **Input:** - Annual return: $24,500 - Discount rate: 15% - **Output:** - Discounted payback period in years (rounded to one decimal place)---**Solution Process:***Firstly, compute the regular payback period:*\[ \text{Payback Period} = \frac{\text{Initial Investment}}{\text{Annual Return}} \]\[ \text{Payback Period} = \frac{90,000}{24,500} \]\[ \text{Payback Period} \approx 3.7 \text{ years} \]*Next, compute the discounted payback period:*Since cash flows are continuous, we will integrate the present value of returns until the total equals the initial investment:\[ \int_0^T 24,500 e^{-0.15t} \, dt = 90,000 \]Solving this integral-based equation will provide the discounted payback period:\[ \int_0^T 24500 e^{-0.15t} \, dt = \left[ \frac{24,500}{-0.15} e^{-0.15t} \right]_0^T = -163,333.33 (e^{-0.15T} - 1) \]Set and solve it for $T$:\[

The payback period time frame is the time allotment it takes to recuperate the expense of a…

Answered: payback period is years. (Round to one…

payback period is years. (Round to one decimal place.)

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

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