Fridges and Freezers are the only appliances I'm aware of where it works out like that though.
Edit: Here's a case study for those that don't believe me.
http://www.t2c.org.uk/chilling/new-case-study-of-20-yr-old-f...
I wonder whether this calculation holds once you account for the energy used to develop, construct, market and transport the new fridges that replace the older-but-working fridges. It's a genuine question - I haven't seen such calculation done. My current intuition is that the extra marginal energy used in manufacturing a new fridge dwarfs the energy savings of replacing the old one before it dies.
This [1] gives the embodied energy for a fridge as 5900 Mj which equals 1600 kwh.
So broadly speaking you new fridge 'pays' for itself in 2 years. I suspect its worse that, but even if its 5 years, my point still stands.