Calculate the discounted payback period for an investment with an initial cost of $120,000, annual cash inflows of $30,000, and a discount rate of 10%.

Steps to Solve

1. Identify the variables

   • Initial investment cost, $C = 120,000$
   • Annual cash inflows, $CF = 30,000$
   • Discount rate, $r = 10% = 0.10$

2. Calculate the present value of cash inflows

   • The present value (PV) of cash inflows for each year must be calculated using the formula: $$ PV = \frac{CF}{(1 + r)^n} $$ where $n$ is the year.

3. Calculate cumulative discounted cash flows

   • Calculate the present value for each year until the cumulative present value equals or exceeds the initial investment.

   • For Year 1: $$ PV_1 = \frac{30,000}{(1 + 0.10)^1} = \frac{30,000}{1.10} \approx 27,273 $$

   • For Year 2: $$ PV_2 = \frac{30,000}{(1 + 0.10)^2} = \frac{30,000}{1.21} \approx 24,793 $$

   • For Year 3: $$ PV_3 = \frac{30,000}{(1 + 0.10)^3} = \frac{30,000}{1.331} \approx 22,533 $$

   • For Year 4: $$ PV_4 = \frac{30,000}{(1 + 0.10)^4} = \frac{30,000}{1.4641} \approx 20,485 $$

   • For Year 5: $$ PV_5 = \frac{30,000}{(1 + 0.10)^5} = \frac{30,000}{1.61051} \approx 18,637 $$

4. Sum the discounted cash flows

   • Cumulative cash flows:
     • After Year 1: $27,273$
     • After Year 2: $27,273 + 24,793 = 52,066$
     • After Year 3: $52,066 + 22,533 = 74,599$
     • After Year 4: $74,599 + 20,485 = 95,084$
     • After Year 5: $95,084 + 18,637 = 113,721$

5. Determine the year of payback

   • The cumulative cash flow exceeds the initial investment of $120,000$ in Year 5.
   • Calculate the exact time for payback in Year 5: $$ \text{Remaining amount} = 120,000 - 95,084 = 24,916 $$ $$ \text{Fraction of Year 5 to recover} = \frac{24,916}{18,637} \approx 1.34 \text{ years} $$
   • Therefore, discounted payback period is: $$ 4 + 1.34 = 5.34 \text{ years} $$