How to Calculate Hyperbolic Discounting
Hyperbolic discounting values a delayed reward at PV = V ÷ (1 + k × t), where k measures impatience and t is the wait, so each extra period of delay costs less than the one before it.
The Hyperbolic Discounting formula
Calculator
Enter a delayed reward, an impatience parameter and the wait to get its hyperbolic present value.
The payoff that arrives after the wait, not what it is worth today.
How fast value falls with delay. At k = 1 a one year wait halves the reward.
Measured in the same periods as k.
Waiting out the delay leaves the reward worth $75 to this person today.
- Hyperbolic discount factor
- 0.5
- Cost of the first year of waiting
- $50
- Cost of one more year at this delay
- $15
- Constant rate with the same present value
- 41.42%
- Constant-rate cost of that extra year
- $21.97
The wait leaves 0.5 of every dollar, which is 1 ÷ (1 + k × t).
Moving the reward from today to one year out costs $50, the steepest stretch of the curve.
Adding one more year on top of the delay you entered costs only $15, so patience gets cheaper the further out you already are.
An exponential discounter would need 41.42% a year to arrive at the same value over this delay.
That constant-rate discounter gives up $21.97 for the same extra year, against $15 here, which is the gap the two shapes create.
How to calculate Hyperbolic Discounting, step by step
- 1Write down the reward and the delay. V is the size of the future payoff and t is how long you have to wait for it, measured in the same periods as k.
- 2Choose the impatience parameter k. k scales how fast value falls with delay. At k = 1 a one year wait halves the reward, and at k = 0.5 the same halving takes two years.
- 3Build the discount factor. Multiply k by t, add 1, then divide 1 by the result. That factor is the share of each dollar the wait leaves behind.
- 4Divide the reward by (1 + k × t). PV = V ÷ (1 + k × t) turns the delayed reward into what it is worth to the person today.
- 5Repeat one period later and subtract. Recompute at t + 1. The value given up by waiting one more period keeps shrinking as t grows, and that shrinking is what makes the curve hyperbolic instead of exponential.
Worked example: Hyperbolic Discounting
A reward of $150 arrives in 2 years and impatience is set at k = 0.5 per year. The discount factor is 1 ÷ (1 + 0.5 × 2) = 0.5, so PV = 150 × 0.5 = $75. The first year of waiting is the expensive one: it takes the reward from $150 down to 150 ÷ 1.5 = $100, a cost of $50. A third year of waiting costs much less, moving the value from $75 to 150 ÷ 2.5 = $60, a cost of $15. A constant-rate discounter who also valued this reward at $75 would be charging 41.42% a year, and for that person the third year of waiting would cost $21.97 instead of $15.
Hyperbolic Discounting questions
What does k mean in the hyperbolic discounting formula?
k sets how impatient the person is and it is measured per period, so k = 0.5 per year means one year of delay adds 0.5 to the denominator. A larger k pulls the present value down faster. When k is 0 the formula returns the full reward, because waiting costs nothing.
How is hyperbolic discounting different from exponential discounting?
Exponential discounting multiplies by the same factor every period, so the percentage lost to waiting is identical wherever the wait starts. Hyperbolic discounting divides by 1 + k × t, so the percentage lost to one more period of waiting shrinks the further out the reward already sits. The first shape keeps a person's ranking of two rewards stable over time and the second does not.
Why does hyperbolic discounting produce preference reversals?
The curve is steep near the present and flat far out, so a short extra wait costs a lot when it starts today and very little when it starts years from now. A person can therefore prefer the larger, later reward while both options are distant and switch to the smaller, sooner one once the sooner reward becomes available immediately. Nothing about the rewards changed, only where the pair sits on the curve.
What happens to the present value when k rises?
A higher k makes the denominator larger, so the present value falls and the person leans harder toward the sooner payoff. Doubling k does not halve the present value, because k sits in the denominator next to the 1, so the relationship between impatience and value is not proportional.
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