To find the discounted payback period, we need to calculate the present value of each cash flow and then add them up until the sum equals the initial investment.
Using a discount rate of 12%, we can calculate the present value of each cash flow as follows:
Year 1: PV = $75 / (1 + 0.12)^1 = $66.96
Year 2: PV = $105 / (1 + 0.12)^2 = $83.34
Year 3: PV = $100 / (1 + 0.12)^3 = $68.45
Year 4: PV = $50 / (1 + 0.12)^4 = $28.64
Now, we can add up the present values of the cash flows until we reach the initial investment of $260:
Year 0: -$260
Year 1: -$260 + $66.96 = -$193.04
Year 2: -$193.04 + $83.34 = -$109.70
Year 3: -$109.70 + $68.45 = -$41.25
Year 4: -$41.25 + $28.64 = -$12.61
The discounted payback period is the point at which the cumulative present value of the cash flows equals the initial investment. From the calculations above, we can see that the discounted payback period is between Year 3 and Year 4, since the cumulative present value of the cash flows is still negative at the end of Year 3 but becomes positive by the end of Year 4.
To estimate the discounted payback period, we can use linear interpolation to estimate the fraction of the final year that is needed to reach the initial investment.
Fraction of Year 4 needed = $12.61 / $28.64 = 0.44
Therefore, the discounted payback period is approximately 3.44 years.