Practice Problem – Payback & Discounted Payback

Practice Question

Intro to Finance
Capital Budgeting
Payback Period Rule
Short Answer

Company XYZ is considering Project A. Project A requires an initial investment of $75,000. It generates $35,000 each year for the coming 3 years. What is the payback period and the discounted payback period for this project if the proper discount rate is 18%?

Answer +
2.14 years and 2.94 years
Explanation +

To calculate the payback period and the discounted payback period for Project A, we will follow these steps:

Step 1: Define the Cash Flows
  • Initial Investment: $75,000
  • Annual Cash Inflows: $35,000 (for 3 years)
  • Discount Rate: 18%
Step 2: Calculate the Payback Period

The payback period is the time it takes for the cumulative cash flows to equal the initial investment.

  • Year 1: $35,000
  • Year 2: $35,000 + $35,000 = $70,000
  • Still short by $5,000 after Year 2
\[ \text{Fraction of Year 3} = \frac{5{,}000}{35{,}000} \approx 0.1429 \] \[ \text{Payback Period} = 2 + 0.1429 \approx 2.14 \text{ years} \]
Step 3: Calculate the Discounted Payback Period

We now discount each cash inflow to its present value using the formula:

\[ PV = \frac{C}{(1 + r)^t} \]
  • Year 1: \(\frac{35{,}000}{1.18} \approx 29{,}661.02\)
  • Year 2: \(\frac{35{,}000}{1.3924} \approx 25{,}185.23\)
  • Year 3: \(\frac{35{,}000}{1.6436} \approx 21{,}319.59\)

Cumulative Present Values:

  • End of Year 1: $29,661.02
  • End of Year 2: $54,846.25
  • Shortfall after Year 2: $75,000 - $54,846.25 = $20,153.75
\[ \text{Fraction of Year 3} = \frac{20{,}153.75}{21{,}319.59} \approx 0.944 \] \[ \text{Discounted Payback Period} = 2 + 0.944 = 2.94 \text{ years} \]
Final Answer

Payback Period: 2.14 years
Discounted Payback Period: 2.94 years