for instance 52 (one year), will tend to go to 1, so that the hyperbolic discounting of a week in the far future is virtually zero, while the exponential discount factor is still 1/2, so there is still substantial discounting in the far future. Quasi-hyperbolic approximation The "quasi-hyperbolic" discount function (sometimes called "beta-delta discounting"), proposed by Laibson (1997), approximates the hyperbolic discount function above in discrete time by where β and δ are constants between 0 and 1; and D is the delay in the reward, but now it takes only integer values. The condition f(0) = 1 states that rewards taken at the present time are not discounted. Quasi-hyperbolic discounting retain much of the analytical tractability of exponential discounting while capturing the key qualitative feature of hyperbolic discounting. Explanations Uncertain risks 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). Uncertainty of this type can be quantified with Bayesian analysis. For example, suppose that the probability for the reward to be available after time t is, for known hazard rate λ, but the rate is unknown to the decision maker. If the prior probability distribution of λ is then the decision maker will expect that the probability of the reward after time t is which is exactly the hyperbolic discount rate. Similar conclusions can be obtained from other plausible distributions for λ. Applications More recently these observations about discount functions have been used to study saving for retirement, personal income to drug addiction., borrowing on credit cards, and procrastination. It has frequently been used to explain addiction. Hyperbolic discounting has also been offered as an explanation of the divergence between privacy attitudes and behaviour. Present values of annuities Present value of a standard annuity The present value of a series of equal annual cash flows in arrears discounted