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| In [[economics]], '''hyperbolic discounting''' is a time-''inconsistent'' model of [[discounting]].
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| The discounted utility approach: Intertemporal choices are no different from other choices, except that some consequences are delayed and hence must be anticipated and discounted (i.e. reweighted to take into account the delay).
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| 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. This process is traditionally modeled in form of [[exponential discounting]], a time-''consistent'' model of discounting. A large number of studies have since demonstrated that the constant discount rate assumed in exponential discounting is systematically being violated.<ref>{{cite journal |last=Frederick |first=Shane |first2=George |last2=Loewenstein |first3=Ted |last3=O'Donoghue |year=2002 |title=Time Discounting and Time Preference: A Critical Review |journal=[[Journal of Economic Literature]] |volume=40 |issue=2 |pages=351–401 |doi=10.1257/002205102320161311 }}</ref> Hyperbolic discounting is a particular mathematical model devised as an improvement over exponential discounting, in the sense that it better fits the experimental data about actual behavior. But note, the time inconsistency of this behavior has some quite perverse consequences. Hyperbolic discounting has been observed in humans and animals.
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| In hyperbolic discounting, valuations fall very rapidly for small delay periods, but then fall slowly for longer delay periods. This contrasts with exponential discounting, in which valuation falls by a constant factor per unit delay, regardless of the total length of the delay. The standard experiment used to reveal a test subject's hyperbolic discounting curve is to compare short-term preferences with long-term preferences. For instance: "Would you prefer a dollar today or three dollars tomorrow?" or "Would you prefer a dollar in one year or three dollars in one year and one day?" For certain range of offerings, a significant fraction of subjects will take the lesser amount today, but will gladly wait one extra day in a year in order to receive the higher amount instead.<ref>{{cite journal |last=Thaler |first=R. H. |year=1981 |title=Some Empirical Evidence on Dynamic Inconsistency |journal=Economic Letters |volume=8 |issue=3 |pages=201–207 |doi=10.1016/0165-1765(81)90067-7 }}</ref> Individuals with such preferences are described as "[[present-biased preferences|present-biased]]".
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| Individuals using hyperbolic discounting reveal a strong tendency to make choices that are inconsistent over time – they make choices today that their future self would prefer not to have made, despite using the same reasoning. This [[dynamic inconsistency]] happens because the value of future rewards is much lower under hyperbolic discounting than under exponential discounting.<ref name="Laibson1997QJE">{{cite journal |authorlink=David Laibson |last=Laibson |first=David |year=1997 |title=Golden Eggs and Hyperbolic Discounting |journal=[[Quarterly Journal of Economics]] |volume=112 |issue=2 |pages=443–477 |doi=10.1162/003355397555253 }}</ref>
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| ==Observations==
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| The phenomenon of hyperbolic discounting is implicit in [[Richard Herrnstein]]'s "[[matching law]]," which states that when dividing their time or effort between two non-exclusive, ongoing sources of reward, most subjects allocate in direct proportion to the rate and size of rewards from the two sources, and in inverse proportion to their delays. That is, subjects' choices "match" these parameters.
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| After the report of this effect in the case of delay,<ref>{{cite journal |last=Chung |first=S. H. |last2=Herrnstein |first2=R. J. |year=1967 |title=Choice and delay of Reinforcement |journal=Journal of the Experimental Analysis of Behavior |volume=10 |issue=1 |pages=67–74 |doi=10.1901/jeab.1967.10-67 }}</ref> [[George Ainslie (psychologist)|George Ainslie]] pointed out that in a single choice between a larger, later and a smaller, sooner reward, inverse proportionality to delay would be described by a plot of value by delay that had a [[hyperbolic function|hyperbolic shape]], and that when the larger, later reward is preferred, this preference can be reversed by reducing both rewards' delays by the same absolute amount. That is, for values of ''x'' for which under current conditions it would be obviously rational to prefer ''x'' dollars in (''n'' + 1) days over one dollar in ''n'' days (''e.g.'', ''x'' = 3), a large subset of the population would (rationally) prefer the former alternative given large values of ''n'', but even among this subset, a large (sub-)subset would (irrationally) prefer one dollar in ''n'' days when ''n'' = 0. Ainslie demonstrated the predicted reversal to occur among pigeons.{{how?|date=November 2012}}{{Vague|date=March 2009}}<ref name="Ainslie1974">{{cite journal |authorlink=George Ainslie (psychologist) |last=Ainslie |first=G. W. |year=1974 |title=Impulse control in pigeons |journal=Journal of the Experimental Analysis of Behavior |volume=21 |issue=3 |pages=485–489 |doi=10.1901/jeab.1974.21-485 }}</ref>
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| A large number of subsequent experiments have confirmed that spontaneous preferences by both human and nonhuman subjects follow a [[hyperbolic curve]] rather than the conventional, "[[exponential discounting|exponential]]" curve that would produce consistent choice over time.<ref name="Green et al">{{cite journal |last=Green |first=L. |last2=Fry |first2=A. F. |last3=Myerson |first3=J. |year=1994 |title=Discounting of delayed rewards: A life span comparison |journal=Psychological Science |volume=5 |issue=1 |pages=33–36 |doi=10.1111/j.1467-9280.1994.tb00610.x }}</ref><ref>{{cite journal |last=Kirby |first=K. N. |year=1997 |title=Bidding on the future: Evidence against normative discounting of delayed rewards |journal=Journal of Experimental Psychology: General |volume=126 |issue=1 |pages=54–70 |doi=10.1037/0096-3445.126.1.54 }}</ref> For instance, when offered the choice between $50 now and $100 a year from now, many people will choose the immediate $50. However, given the choice between $50 in five years or $100 in six years almost everyone will choose $100 in six years, even though that is the same choice seen at five years' greater distance.
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| Hyperbolic discounting has also been found to relate to real-world examples of self-control. Indeed, a variety of studies have used measures of hyperbolic discounting to find that drug-dependent individuals discount delayed consequences more than matched nondependent controls, suggesting that extreme delay discounting is a fundamental behavioral process in drug dependence.<ref>{{cite book |last=Bickel |first=W. K. |last2=Johnson |first2=M. W. |year=2003 |chapter=Delay discounting: A fundamental behavioral process of drug dependence |editor1-first=G. |editor1-last=Loewenstein |editor2-first=D. |editor2-last=Read |editor3-first=R. F. |editor3-last=Baumeister |title=Time and Decision |location=New York |publisher=Russell Sage Foundation |isbn=0-87154-549-7 }}</ref><ref>{{cite journal |last=Madden |first=G. J. |last2=Petry |first2=N. M. |last3=Bickel |first3=W. K. |last4=Badger |first4=G. J. |year=1997 |title=Impulsive and self-control choices in opiate-dependent patients and non-drug-using control participants: Drug and monetary rewards |journal=Experimental and Clinical Psychopharmacology |volume=5 |issue= |pages=256–262 |doi= |pmid=9260073 }}</ref><ref>{{cite journal |last=Vuchinich |first=R. E. |last2=Simpson |first2=C. A. |year=1998 |title=Hyperbolic temporal discounting in social drinkers and problem drinkers |journal=Experimental and Clinical Psychopharmacology |volume=6 |issue=3 |pages=292–305 |doi=10.1037/1064-1297.6.3.292 }}</ref> Some evidence suggests pathological gamblers also discount delayed outcomes at higher rates than matched controls.<ref>{{cite journal |last=Petry |first=N. M. |last2=Casarella |first2=T. |year=1999 |title=Excessive discounting of delayed rewards in substance abusers with gambling problems |journal=Drug and Alcohol Dependence |volume=56 |issue=1 |pages=25–32 |doi=10.1016/S0376-8716(99)00010-1 }}</ref> Whether high rates of hyperbolic discounting precede addictions or vice-versa is currently unknown, although some studies have reported that high-rate discounting rates are more likely to consume alcohol<ref>{{cite journal |last=Poulos |first=C. X. |last2=Le |first2=A. D. |last3=Parker |first3=J. L. |year=1995 |title=Impulsivity predicts individual susceptibility to high levels of alcohol self administration |journal=Behavioral Pharmacology |volume=6 |issue=8 |pages=810–814 |doi=10.1097/00008877-199512000-00006 }}</ref> and cocaine<ref>{{cite journal |last=Perry |first=J. L. |last2=Larson |first2=E. B. |last3=German |first3=J. P. |last4=Madden |first4=G. J. |last5=Carroll |first5=M. E. |year=2005 |title=Impulsivity (delay discounting) as a predictor of acquisition of i.v. cocaine self-administration in female rats |journal=Psychopharmacology |volume=178 |issue=2–3 |pages=193–201 |doi=10.1007/s00213-004-1994-4 |pmid=15338104}}</ref> than lower-rate discounters. Likewise, some have suggested that high-rate hyperbolic discounting makes unpredictable (gambling) outcomes more satisfying.<ref>{{cite journal |last=Madden |first=G. J. |last2=Ewan |first2=E. E. |last3=Lagorio |first3=C. H. |year=2007 |title=Toward an animal model of gambling: Delay discounting and the allure of unpredictable outcomes |journal=Journal of Gambling Studies |volume=23 |issue=1 |pages=63–83 |doi=10.1007/s10899-006-9041-5 }}</ref>
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| 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>{{cite book |last=Loewenstein |first=G. |authorlink2=Drazen Prelec |last2=Prelec |first2=D. |year=1992 |title=Choices Over Time |location=New York |publisher=Russell Sage Foundation |isbn=0-87154-558-6 }}</ref><ref>{{cite journal |last=Raineri |first=A. |last2=Rachlin |first2=H. |year=1993 |title=The effect of temporal constraints on the value of money and other commodities |journal=Journal of Behavioral Decision-Making |volume=6 |issue=2 |pages=77–94 |doi=10.1002/bdm.3960060202 }}</ref>
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| ==Criticism==
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| An article from 2003 noted that the evidence might be better explained by a [[similarity heuristic]] than by hyperbolic discounting.<ref>{{cite journal |last=Rubinstein |first=Ariel |year=2003 |title="Economics and Psychology"? The Case of Hyperbolic Discounting. |journal=International Economic Review |volume=44 |issue=4 |pages=1207–1216 }}</ref> Similarly, a 2011 paper criticized the existing studies for mostly using data collected from university students and being too quick to conclude that the hyperbolic model of discounting is correct.<ref>{{cite article |last1=Andersen |first1=Steffen |last2=Harrison |first2=Glenn W. |last3=Lau |first3=Morten |last4=Rutström |first4=E. Elisabet |year=2011 |title=Discounting Behavior: A Reconsideration }}</ref>
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| A study by Daniel Read introduces "subadditive discounting": the fact that discounting over a delay increases if the delay is divided into smaller intervals. This hypothesis may explain the main finding of many studies in support of hyperbolic discounting—the observation that impatience declines with time–while also accounting for observations not predicted by hyperbolic discounting.<ref>{{cite journal |last=Read |first=Daniel |year=2001 |title=Is time-discounting hyperbolic or subadditive? |journal=Journal of risk and uncertainty |volume=23 |issue=1 |pages=5–32 |doi=10.1023/A:1011198414683 }}</ref>
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| ==Mathematical model==
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| ===Step-by-step explanation===
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| Suppose that in a study, participants are offered the choice between taking ''x'' dollars immediately or taking ''y'' dollars ''n'' days later. Suppose further that one participant in that study employs exponential discounting and another employs hyperbolic discounting.
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| Each participant will realize that ''a'') they should take ''x'' dollars immediately if they can invest the dollar in a different venture that will yield more than ''y'' dollars ''n'' days later and ''b'') they will be indifferent between the choices (selecting one at random) if the best available alternative will likewise yield ''y'' dollars ''n'' days later. (Assume, for the sake of simplicity, that the values of all available investments are compounded daily.) Each participant correctly understands the fundamental question being asked: "For any given value of ''y'' dollars and ''n'' days, what is the minimum amount of money, ''i.e.'', the minimum value for ''x'' dollars, that I should be willing to accept? In other words, how many dollars would I need to invest today to get ''y'' dollars ''n'' days from now?" Each will take ''x'' dollars if ''x'' is greater than the answer that they calculate, and each will take ''y'' dollars ''n'' days from now if ''x'' is smaller than that answer. However, the methods that they use to calculate that amount and the answers that they get will be different, and only the exponential discounter will use the correct method and get a reliably correct result:
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| * The exponential discounter will think "The best alternative investment available (that is, the best investment available in the absence of this choice) gives me a return of ''r'' percent per day; in other words, once a day it adds to its value ''r'' percent of the value that it had the previous day. That is, every day it multiplies its value once by (100% + ''r''%). So if I hold the investment for ''n'' days, its value will have multiplied itself by this amount ''n'' times, making that value (100% + ''r''%)^''n'' of what it was at the start – that is, (1 + ''r''%)^''n'' times what it was at the start. So to figure out how much I would need to start with today to get ''y'' dollars ''n'' days from now, I need to divide ''y'' dollars by ([1 + ''r''%]^''n''). If my other choice of how much money to take is greater than this result, then I should take the other amount, invest it in the other venture that I have in mind, and get even more at the end. If this result is greater than my other choice, then I should take ''y'' dollars ''n'' days from now, because it turns out that by giving up the other choice I am essentially investing that smaller amount of money to get ''y'' dollars ''n'' days from now, meaning that I'm getting an even greater return by waiting ''n'' days for ''y'' dollars, making this my best available investment."
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| * The hyperbolic discounter, however, will think "If I want ''y'' dollars ''n'' days from now, then the amount that I need to invest today is ''y'' divided by ''n'', because that amount times ''n'' equals ''y'' dollars. [There lies the hyperbolic discounter's error.] If my other choice is greater than this result, I should take it instead because ''x'' times ''n'' will be greater than ''y'' times ''n''; if it is less than this result, then I should wait ''n'' days for ''y'' dollars."
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| Where the exponential discounter reasons correctly and the hyperbolic discounter goes wrong is that as ''n'' becomes very large, the value of ([1 + ''r''%]^''n'') becomes much larger than the value of ''n'', with the effect that the value of (''y''/[(1 + ''r''%)^''n'') becomes much smaller than the value of (''y''/''n''). Therefore, the minimum value of ''x'' (the number of dollars in the immediate choice) that suffices to be greater than that amount will be much smaller than the hyperbolic discounter thinks, with the result that they will perceive ''x''-values in the range from (''y''/[(1 + ''r''%)^''n'') to (''y''/''n'') (inclusive at the low end) as being too small and, as a result, irrationally turn those alternatives down when they are in fact the better investment.
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| ===Formal model===
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| [[File:Hyperbolic vs. exponential discount factors.svg|thumb|300px|right|Comparison of the discount factors of hyperbolic and exponential discounting. In both cases, <math>k=1</math>. Hyperbolic discounting is shown to value future assets higher than exponential discounting.]]
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| Hyperbolic discounting is mathematically described as:
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| :<math>f_H(D)=\frac{1}{1+kD}\,</math>
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| where ''f''(''D'') is the [[discount factor]] that multiplies the value of the reward, ''D'' is the delay in the reward, and ''k'' is a parameter governing the degree of discounting. This is compared with the formula for exponential discounting:
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| :<math>f_E(D)=e^{-kD}\,</math>
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| ====Simple derivation====
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| If <math>f(n)=2^{-n}\,</math> is an exponential discounting function and <math>g(n)=\frac{1}{1+n}\,</math> a hyperbolic function (with n the amount of weeks), then the exponential discounting a week later from "now" (n=0) is <math>\frac{f(1)}{f(0)}=\frac{1}{2}\,</math>, and the exponential discounting a week from week n is <math>\frac{f(n+1)}{f(n)}=\frac{1}{2}\,</math>, which means they are the same. For g(n), <math>\frac{g(1)}{g(0)}=\frac{1}{2}\,</math>, which is the same as for f, while <math>\frac{g(n+1)}{g(n)}=1-\frac{1}{n+2}\,</math>. From this one can see that the two types of discounting are the same "now", but when n is much greater than 1, for instance 52 (one year), <math>\frac{g(n+1)}{g(n)}\,</math> will tend to go to 1, so that the hyperbolic discounting of a week in the far future is virtually zero, while the exponential is still 1/2.
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| ===Quasi-hyperbolic approximation===
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| The "quasi-hyperbolic" discount function, proposed by Laibson (1997),<ref name="Laibson1997QJE"/> approximates the hyperbolic discount function above in [[discrete time]] by
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| :<math>f_{QH}(0)=1, \, </math>
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| and
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| :<math>f_{QH}(D)=\beta \times \delta^D,\,</math>
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| 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.
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| Quasi-hyperbolic time preferences are also referred to as "beta-delta" preferences. They retain much of the analytical tractability of [[exponential discounting]] while capturing the key qualitative feature of discounting with true hyperbolas.
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| ==Explanations==
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| ===Uncertain risks===
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| Notice that whether discounting future gains is rational or not—and at what rate such gains should be discounted—depends greatly on circumstances. Many examples exist in the financial world, for example, where it is reasonable to assume that there is an implicit risk that the reward will not be available at the future date, and furthermore that this risk increases with time. Consider: Paying $50 for dinner today or delaying payment for sixty years but paying $100,000. In this case, the restaurateur would be reasonable to discount the promised future value as there is significant risk that it might not be paid (e.g. due to the death of the restaurateur or the diner).
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| Uncertainty of this type can be quantified with [[Bayesian probability|Bayesian analysis]].<ref name="sozou1998">{{Cite doi|10.1098/rspb.1998.0534}}</ref> For example, suppose that the probability for the reward to be available after time ''t'' is, for known hazard rate λ
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| :<math>P(R_t|\lambda) = \exp(-\lambda t)\,</math>
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| but the rate is unknown to the decision maker. If the [[prior probability]] distribution of λ is
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| :<math>p(\lambda) = \exp(-\lambda/k)/k\,</math>
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| then, the decision maker will expect that the probability of the reward after time ''t'' is
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| :<math>P(R_t) = \int_0^\infty P(R_t|\lambda) p(\lambda) d\lambda = \frac{1}{1 + k t}\,</math>
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| which is exactly the hyperbolic discount rate. Similar conclusions can be obtained from other plausible distributions for λ.<ref name="sozou1998"/>
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| ==Applications==
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| More recently these observations about [[discount function]]s have been used to study [[Retirement#Saving_for_retirement|saving for retirement]], borrowing on [[credit card]]s, and [[procrastination]].
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| It has frequently been used to explain [[substance dependence|addiction]].<ref>{{cite journal |last=O'Donoghue |first=T. |first2=M. |last2=Rabin |year=1999 |title=Doing it now or later |journal=[[The American Economic Review]] |volume=89 |pages=103–124 }}</ref><ref>{{cite journal |last=O'Donoghue |first=T. |first2=M. |last2=Rabin |year=2000 |title=The economics of immediate gratification |journal=[[Journal of Behavioral Decision Making]] |volume=13 |pages=233–250}}</ref>
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| Hyperbolic discounting has also been offered as an explanation of the divergence between privacy attitudes and behaviour.<ref>{{cite book |last1=Acquisti |first1=Alessandro |last2=Grossklags |first2=Jens |editor1-first=J. |editor1-last=Camp |editor2-first=R. |editor2-last=Lewis |chapter=Losses, Gains, and Hyperbolic Discounting: Privacy Attitudes and Privacy Behavior |title=The Economics of Information Security|year=2004 |publisher=Kluwer |pages=179–186}}</ref>
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| ==Present Values of Annuities==
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| ===Present Value of an Standard Annuity===
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| The present value of a series of equal annual cash flows in arrears discounted hyperbolically:
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| :<math>V = P \frac{\ln(1+kD)}{k}\,</math>
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| Where V is the present value, P is the annual cash flow, D is the number of annual payments and k is the factor governing the discounting.
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| ==See also==
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| * [[Time value of money]]
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| * [[Time preference]]
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| * [[Intertemporal choice]]
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| * [[Deferred gratification]]
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| * [[Akrasia]]
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| * [[Temporal motivation theory]]
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| ==References==
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| {{Reflist|30em}}
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| Benartzi and Thaler (2004)
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| Bickel et al. (1999)
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| Hendrickx et al. (2001)
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| ==Further reading==
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| *{{cite journal |last=Ainslie |first=G. W. |year=1975 |title=Specious reward: A behavioral theory of impulsiveness and impulsive control |journal=Psychological Bulletin |volume=82 |issue=4 |pages=463–496 |doi=10.1037/h0076860 |pmid=1099599}}
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| *{{cite book |last=Ainslie |first=G. |year=1992 |title=Picoeconomics: The Strategic Interaction of Successive Motivational States Within the Person |location=Cambridge |publisher=Cambridge University Press |isbn= }}
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| *{{cite book |last=Ainslie |first=G. |year=2001 |title=Breakdown of Will |location=Cambridge |publisher=Cambridge University Press |isbn=978-0-521-59694-7 }}
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| *{{cite book |last=Rachlin |first=H. |year=2000 |title=The Science of Self-Control |location=Cambridge; London |publisher=Harvard University Press |isbn= }}
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| {{DEFAULTSORT:Hyperbolic Discounting}}
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| [[Category:Cognitive biases]]
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| [[Category:Behavioral finance]]
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If you do, then you'll be truly happy with Roulette Sniper. The only factor that I would suggest is make certain you really get your head around the software program (though it's simple to use) before you wager a lot of cash with it.
Many have misplaced their monetary trustworthiness and balance simply because of indiscriminate taking part in with borrowed cash, like credit score playing cards. That is a nicely-recognized path for numerous and they all end up in the same preposterous scenario. It is poor information to borrow money to play on-line.
Seeing SoCal through Harlan's eyes is really a present. In accordance to Harlan, a lot of Redondo Seaside ('RB') has remained the same, besides of course, for enhancements 'here and there' as RB by itself grew. His stories, about the individuals with whom he grew up, as nicely as our limitless selection of California beach towns (exactly where he went from boyishly cute to totally handsome) are endless.
You can neglect unguaranteed signature financial loans to get enjoyment methods! You can neglect journeys on the Internet Internet casino, betting about extraordinary the concept prosperous on the plastic card! Cease obtaining those lotto passes! Get yourself out from the bad monetary financial debt by no means-ending cycle.
Here's how likelihood is at work at casinos. Probability, in easier terms, is the possibility of getting the jackpot. You are usually utilizing likelihood every working day and all of us have a knack at selecting the right guess. Even though most of us consider it as luck, our right options are introduced about by our innate capability to figure out patterns. You simply have to sharpen this inborn mathematical ability to win large on online casino gambling.
you choose or whether or not you win the sport or not, and frequently you can rapidly bypass the game to go straight to the results. In most cases, it doesn't make a difference what sq. , or play some little sport. These can be enjoyable because you don't have to wait around for months to see whether or not a prize comes at your doorway or not. These instants frequently involve some kind of flash sport where you spin a digital Slot machine, pick a sq. Some sweepstakes will allow you to know immediately whether you gained or not.
If you don't perform with a awesome head then you might shed even much more. Always have a great technique and that can happen only if you are nicely knowledgeable. If you are dropping as well much then transfer on or quit. Once you have registered with an online casino for gambling remember that you are there to win and not to shed cash. Keep your feelings under verify at all times. Do not lose your awesome when you are dropping cash. Remember understanding is energy.
It is much better to stand if your playing cards complete to a difficult 17 or above. If the total of your playing cards arrive up to 17 and neither of the cards is an ace, this is called as a difficult total. Because the ace may be counted as one or 11, it tends to make a gentle total when added with an additional card.
Below are eight of the important steps to take if you think your cherished 1 has a gambling addiction. Even though this can be a extremely troubling issue, there are numerous issues that you can do to assist. You know if your cherished 1 has taken his love for poker or blackjack as well far if you discover that he's been borrowing money just to gamble, or if he has began to pawn his watches or jewellery just to get his kicks in the casino.
We must consequently trip the fluctuation of roulette completely in a way to balance the books of our bank stability. This assumption is cocktail for mistake, because when taking part in online roulette, the successful and losing end result is unpredictable.