The Payback Gap Is Pure Capital Cost
The Payback Gap Is Pure Capital Cost
Ada re-derives this chapter’s own numbers step by step, at full precision
ADA · CALCULATION AUDIT
The Payback Gap Is Pure Capital Cost
A 12-building campus with 2,400 rooms and 480 Wi-Fi APs already logs about 18,000 daily connected devices, so the chapter repurposes that Wi-Fi data (plus 50 supplementary PIR sensors) to cut a $2.8 million/year HVAC bill by 22%. The hybrid approach costs $33,500 to implement and claims a payback near 20 days, against $480,000 and roughly 284 days for a dedicated PIR deployment. This audit asks the question that payback gap invites: is the hybrid approach really paying back faster because it saves more, or purely because it costs less to install?
Companion to the chapter Sensing with Existing Infrastructure — every number here comes from that chapter.
See the relationship before changing it
The figure reads from left to right. The blue card is hybrid installation cost. The middle card applies this page's rule. The green card is simple payback. Walk the arrows once: set the input, apply the rule, then read the result with its unit.
The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only hybrid installation cost, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 33.5 thousand USD.
- 2
Name the relationship. payback = capital / (616 thousand USD / 365 days)
- 3
Substitute with units. 33.5 / 616 x 365 = 19.8 days
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change hybrid installation cost
Try Predict the direction of payback = capital / (616 thousand USD / 365 days). Test another hybrid installation cost, then compare simple payback.
Observe With identical savings, installation capital alone scales the payback time. Reset hybrid installation cost to 33.5 and compare simple payback.
Explain With identical savings, installation capital alone scales the payback time.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Use annual energy cost $2800000, saving 22%, hybrid capital $33500, and dedicated-PIR capital $480000 to recompute both payback periods.
Both options save $1687.67/day, while payback changes from 19.9 to 284.4 days; the 14.33× time ratio exactly matches the capital-cost ratio.
Because the model assigns identical savings to both designs, that term cancels in their payback ratio and installation capital alone creates the apparent speed advantage.
Ada: The worked example says the hybrid approach pays back in about 20 days and dedicated PIR in about 284 days. Those numbers are correct, but the reason the gap is so wide is worth pinning down, because it decides what the comparison is really testing.
Both scenarios claim the same annual saving, so the daily saving is shared:
- Annual:
$2,800,000 x 22% = $616,000; per day$616,000 / 365 = $1,687.67 - Hybrid payback:
$33,500 / $1,687.67 = 19.9 days - Dedicated payback:
$480,000 / $1,687.67 = 284.4 days
Now take the ratio of those two payback times:
- Payback ratio:
284.4 / 19.9 = 14.33 - Capital-cost ratio:
$480,000 / $33,500 = 14.33
They are identical to three digits, and they must be, because the savings term cancels when you divide one payback by the other. The hybrid does not pay back 14x faster because it saves more; the model gives it zero savings advantage. The entire speed-up is capital cost, 93% lower (1 - 33,500 / 480,000 = 0.930). That reframes the decision honestly: the infrastructure approach wins purely on what it costs to install, so the only question that actually matters is whether its +/-10% zone accuracy is good enough for HVAC scheduling. The accuracy trade, not the ROI, is where the real risk sits.
Every number above is taken from the chapter’s own material and re-derived step by step.
Technical boundaries: This simple payback comparison omits discount rate, maintenance and replacement, commissioning downtime, occupancy-estimation error, energy-price changes, tax, residual value, and unequal realised savings.