Payback Period

The payback period is a method used to estimate the time required to recover the initial investment in an asset or project. It is often used as an initial screening tool to determine if a project is worth further consideration.

• The payback period (N, P) is the estimated time it takes for revenues and economic benefits to recover the initial investment (P) and a specified rate of return (i%).
• Net Cash Flow (NCF): The estimated net cash flow for each year t, where NCF = receipts – disbursements.
• Two forms of payback analysis:
  • No-Return Payback (i = 0%): Also called simple payback, it focuses on recovering the initial investment.
  • Discounted Payback (i > 0%): Considers the time value of money by including a specific rate of return.

Calculations:

1. Without Interest (i = 0%)

(a) Varying Net Cash Flow (NCF)

$$0 = -P + \sum_{t=1}^{n_p} NCF_t$$

• $P$ = Initial Investment
• $NCF_t$ = Net Cash Flow in year $t$
• $n_p$ = Payback period (years)
• The sum of cash flows over time must equal the initial investment.

(b) Uniform Net Cash Flow (NCF is constant each year)

$$n_p = \frac{P}{NCF}$$

• The investment is recovered in a fixed number of years based on uniform cash inflows.

2. With Interest (Discounted Payback Period, i > 0%)

(a) Varying Net Cash Flow (NCF changes each year)

$$0 = -P + \sum_{t=1}^{n_p} NCF_t \cdot (P/F, i, t)$$

• The future cash flows are discounted using the Present Worth Factor $(P/F, i, t)$: $$(P/F, i, t) = \frac{1}{(1 + i)^t}$$
• Each year’s cash flow is converted to its present value and summed until it equals $P$.

(b) Uniform Net Cash Flow (NCF is constant each year)

$$0 = -P + NCF \cdot (P/A, i, n_p)$$

• The sum of discounted cash flows over $n_p$ years equals the initial investment.
• Present Worth of an Annuity Factor $(P/A, i, n_p)$ is given by: $$(P/A, i, n_p) = \frac{(1 – (1 + i)^{-n_p})}{i}$$

Benefits of Payback Analysis:

• Easy to understand and apply .
• Helps to quickly identify potentially acceptable projects.
• Reduces information search by focusing on the time to recover the initial investment.
• Discounted payback can provide a sense of the risk involved in undertaking an alternative.
• Provides additional insights when used with other evaluation methods like PW, AW, ROR, or B/C.

Drawbacks of Payback Analysis:

• Does not measure the overall earning power of a project .
• Fails to account for the present and future value of money for simple payback.
• Neglects all cash flows occurring after the payback period, which can unfairly bias alternative selection.
• Can favor short-lived assets over longer-lived assets with higher returns due to the neglect of cash flows after ( n_p ).

Cautions

• Payback analysis should not be used as the primary measure of worth to select an alternative because it can unfairly bias alternative selection.
• When evaluating multiple alternatives, neglecting cash flows after ( n_p ) may lead to economically incorrect decisions .
• PW or AW analysis should always be the primary selection method when comparing alternatives .

In summary, while the payback period offers simplicity and ease of understanding, its limitations make it more suitable as a supplemental tool for initial project screening rather than a definitive measure for economic decision-making.