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{{Refimprove|date=June 2010}}
Given two similar rewards, humans show a preference for one that arrives sooner rather than later. Humans are said to ''discount'' the value of the later reward, by a factor that increases with the length of the delay. In [[behavioral economics]], '''hyperbolic discounting''' is a particular mathematical model thought to approximate this discounting process; that is, it models how humans actually make such valuations. Hyperbolic(contracted; show full)

The degree of discounting is vitally important in describing hyperbolic discounting, especially in the discounting of specific rewards such as money. The discounting of monetary rewards varies across age groups due to the varying discount rate.<ref name="Green et al"/> The rate depends on a variety of factors, including the species being observed, age, experience, and the amount of time needed to consume the reward.<ref>Loewenstein, G. and 
[[Drazen Prelec|Prelec, D.]] (1992). ''Choices Over Time'' New York, Russell Sage Foundation</ref><ref>Raineri, A., and Rachlin, H. (1993). The effect of temporal constraints on the value of money and other commodities. ''Journal of Behavioral Decision-Making, 6,'' 77-94.</ref>

==Mathematical model==
(contracted; show full)
:<math>f_{QH}(0)=1\,</math>, and
:<math>f_{QH}(D)=\beta \times \delta^D\,</math>,

where β and δ are constants between 0 and 1; and again ''D'' is the delay in the reward, and ''f''(''D'') is the discount factor. The condition ''f''(0) = 1 is stating that rewards taken at the present time are not discounted.

Quasi-hyperbolic time preferences are also referred to as 
"[[present-biased preferences|present-biased]]" or "beta-delta" preferences. They retain much of the analytical tractability of [[exponential discounting]] while capturing the key qualitative feature of discounting with true hyperbolas.

==Explanations==

===Uncertain risks===
(contracted; show full)
* Ainslie, G. (2001) ''Breakdown of Will'' Cambridge, Cambridge University Press, ISBN 978-0521596947
* Musau, A. (2009): Modeling Alternatives to Exponential Discounting, MPRA Paper 16416, University Library of Munich, Germany.
* Rachlin, H. (2000). ''The Science of Self-Control'' Cambridge;London: Harvard University Press

[[Category:Cognitive biases]]
[[Category:Behavioral finance]]


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