What it is (short definition)
– Discounted payback period (DPBP): the number of years required for the present value of a project’s future cash inflows to equal the project’s initial cash outlay. It uses discounted cash flows, so it accounts for the time value of money (the idea that a dollar today is worth more than a dollar in the future).
Why it’s used
– Managers and investors use DPBP as a simple way to measure how quickly a project returns its invested capital after adjusting future receipts for the appropriate discount rate. It helps compare projects when a quick recovery of capital is important.
Key terms (defined)
– Payback period: the time it takes for cumulative nominal (undiscounted) cash inflows to equal the initial investment.
– Discount rate: the rate used to convert future cash flows into present value; reflects required return, cost of capital, or inflation + risk premium.
– Present value (PV): current worth of a future cash flow, calculated as PV = CF / (1 + r)^t, where CF = cash flow, r = discount rate, and t = time period.
Step-by-step method to calculate DPBP
1. Estimate the initial cash outlay (year 0) and the expected cash inflows for each future period.
2. Select an appropriate discount rate (r).
3. Discount each future cash inflow to present value using PV = CF / (1 + r)^t.
4. Make a cumulative running sum of the discounted inflows starting from year 1.
5. Identify the first period in which cumulative discounted inflows are equal to or exceed the initial outlay.
6. If the cumulative sum exceeds the initial outlay within a period, compute the fractional year:
fractional year = (remaining amount to recover at start of period) / (discounted cash flow in that period).
7. DPBP = number of full years before recovery + fractional year (if needed).
Decision rule (how managers typically use it)
– Set a maximum acceptable payback window (a cutoff). If DPBP is shorter than the cutoff, the project passes this screening test; if not, it fails. Note: DPBP is a screening tool, not a complete measure of project profitability.
Main advantages
– Incorporates the time value of money (unlike the simple pay
back) — so cash flows received later are worth less when computing recovery time.
Main advantages
– Accounts for the time value of money: discounted cash flows (future receipts adjusted for discounting) are used instead of raw nominal receipts.
– Simple and intuitive: expresses recovery time in years, which managers find easy to interpret for liquidity and risk screening.
– Useful for liquidity-focused screening: helps exclude projects that tie up capital longer than management’s allowable window.
Main limitations (what DPBP does not do)
– Ignores cash flows after the payback point. A project can have a short discounted payback but a negative net present value (NPV) thereafter (or vice versa).
– Requires selecting a discount rate; results are sensitive to that choice. The discount rate often reflects cost of capital, but using an inappropriate rate misstates recovery time.
– Does not measure overall profitability or value creation (unlike NPV). It cannot rank mutually exclusive projects reliably.
– Uses a cutoff (maximum acceptable payback) that is arbitrary and may bias against long-term value-creating investments.
Key formulas (definitions)
– Present value of a single cash flow: PV_t = CF_t / (1 + r)^t
where CF_t is the cash flow at time t, r is the discount rate, t is years.
– Cumulative discounted cash flow at end of year t: Cum_t = sum_{i=1..t} PV_i.
– Fractional year when recovery happens mid-period: fractional year = (remaining amount to recover at start of period) / (discounted cash flow in that period).
– Discounted payback period (DPBP) = number of full years before recovery + fractional year (if needed).
Worked numeric example (step-by-step)
Assume: initial investment = $10,000 today; expected nominal cash inflows = $3,000 at the end of each year for 5 years; discount rate r = 10% (0.10).
Step 1 — discount each year’s cash flow:
– Year 1 PV = 3,000 / 1.10 = 2,727.27
– Year 2 PV = 3,000 / 1.10^2 = 2,479.34
– Year 3 PV = 3,000 / 1.10^3 = 2,253.94
– Year 4 PV = 3,000 / 1.10^4 = 2,048.14
– Year 5 PV = 3,000 / 1.10^5 = 1,862.86
Step 2 — compute cumulative discounted inflows:
– End of Year 1 cumulative = 2,727.27
– End of Year 2 cumulative = 2,727.27 + 2,479.34 = 5,206.61
– End of Year 3 cumulative = 5,206.61 + 2,253.94 = 7,460.55
– End of Year 4 cumulative = 7,460.55 + 2,048.14 = 9,508.69
– End of Year 5 cumulative = 9,508.69 + 1,862.86 = 11,371.55
Step 3 — locate recovery point:
– Initial outlay $10,000 is not recovered by end of Year 4 (cumulative 9,508.69) but is recovered sometime in Year 5.
Step 4 — compute fractional year:
– Remaining to recover at start of Year 5 = 10,000 − 9,508.69 = 491.31
– Discounted cash flow in Year 5 = 1,862.86
– Fractional year = 491.31 / 1,862.86 = 0.2637
Step 5 — DPBP:
– DPBP = 4 full years + 0.