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We have trouble understanding the time value of money

In my recent article on superannuation tax concessions, I focused on discounting for illiquidity. And it got me thinking about a close conceptual cousin, ‘hyperbolic discounting’.

People can make irrational decisions when it comes to money. For example, behavioural experiments that reveal subjects who prefer to have say, $50 today over $100 in six months’ time. Reflecting a huge annual discount rate. This type of behaviour known as ‘hyperbolic discounting’ places a clear preference on immediate consumption over deferred consumption.

Curiously, those same individuals would often prefer $100 in nine months’ time over $50 in three months. A small time shift removes the immediacy bias. These choices are time-inconsistent which if plotted, follow a hyperbolic pattern. Hence the term ‘hyperbolic discounting’.

Financial considerations like investment returns aside, both liquidity and hyperbolic discounting reflect a time preference, placing a higher value on money today than in the future. Both think that future money isn’t worth full-face value today, but for different reasons.

With liquidity discounting, there is a cost recognition of not being able to use tied up funds today, there is an inaccessibility factor. Whereas with hyperbolic discounting, the cost is purely psychological where a person simply prefers current consumption over it being delayed.

Both exhibit steep discounting that flattens over time, whether with respect to access or reward. And hyperbolic discounting can be heavier, earlier, with bigger individual variation driven by personality and context. Liquidity discount rates reflect more objective constraints.

I have already covered liquidity discounting in that previous article, so focusing more on hyperbolic discounting, a classic real-life example exists with so-called ‘payday loans’. This is where borrowers pay exorbitant interest rates on short duration loans that see them through to ‘payday’.

Payday loan lenders exploit hyperbolic discounting by targeting borrowers who have immediate cash needs for things like rent and groceries, and for possibly short-term rewards. People accept irrationally high interest rates to satisfy immediate needs and wants. A small time shift comes at a high cost. This is hyperbolic discounting in action, with people acting against their long-term interest.

And as noted prior, immediacy bias reduces with delayed access, so that placing a small waiting period on access to payday funds, might reduce the number of people willing to take up such loans.

Yet it is not just low-income earners on the edge that can succumb to hyperbolic discounting, as the phenomenon can present under different guises and affect many.

If hyperbolic discounting could be defined as the favouring of short-term benefits at the expense of long-term financial wellbeing, then other examples of it might include:

  • Not saving enough for retirement. This might occur if for example, voluntary contributions are not made when there is capacity to do so. While retirement funds could be substantially boosted, the benefits are a long time away, and the cost is immediate with less money to spend now. The long-term payoff is heavily discounted, reflecting in this instance both a hyperbolic immediacy cost and an illiquidity cost for the inaccessible nature of super.
  • Early super withdrawals. Like those during the pandemic made by many people even when not necessarily needed. They valued the extra liquidity far more than future income in retirement.
  • Not maximising mortgage repayments. Even knowing that total costs will be higher over the term of the loan, it is far more gratifying to have the extra funds available for discretionary spending today.

Though not extreme and irrational examples, compared to the classic ‘$50 now, $100 later’ case, these milder forms of hyperbolic discounting still highlight the immediacy bias that exists in the lives of many of us.

Indeed, hyperbolic discounting might be thought of as ‘behavioural discounting’. It’s probably a more apt description for discounting future outcomes not just for financial considerations, but also for psychological biases. A process that can place weight on emotion and impatience as much as, or even more than inflation, expected returns, and real constraints like lack of access.

Again, these cognitive biases result in immediate rewards being given disproportionate weight, with preferences being time-inconsistent. Certainty versus uncertainty also comes into it, where the distant future aligns with an uncertain outcome, so we prefer the certainty of immediate reward even if it is less valuable.

And such biases can lead to poor decision making. Which is why it is important to understand that hyperbolic discounting exists so that we can make informed and better decisions in the future.

 

Tony Dillon is a freelance writer and former actuary.

 

7 Comments
Tony Dillon
July 26, 2025

Love your work Dudley.

The hyperbolic discount factor formula is typically not exponential, and of the form 1 / (1 + kt) over time t, curve parameter k.

Classic finance discounting is exponential, and of the form 1 / ((1 + r)^t), time t, discount rate r.

In my discounting for illiquidity article (https://www.firstlinks.com.au/the-rubbery-numbers-behind-super-tax-concessions), I used a declining concave function for the time t discount rate: r(t) = r(max) x SQRT((T - t)/T), T = time at access. That is, r is not constant. And so the discount factor at time t becomes: PRODUCT(1 / (1 + r(i))), i = t to T.
r(max) = 3.73%

Dudley
July 27, 2025

"r(max) = 3.73%":
That is about the real interest rate on mortgages.

The rate of change of:
= (1 / (EXP(1) ^ (t / k)) - 1) + 1
k ~= mort-gage term, 25 y.
is constant at close to 4% across all values of t, including negative.

Having difficulty articulate the meaning of negative time.
In the context of mortgages, it is something like borrowing k future payments shifts the origin t' by -k units.

The expression:
= 1 / ((1 + r)^t)
is essentially similar to:
= (1 / (EXP(1) ^ (t / k)) - 1) + 1

Tony Dillon
July 27, 2025

Dudley, you have happened on a key relationship here in compound interest mathematics, with your inquisitiveness.

Your expression, (1 / (EXP(1) ^ (t / k)) - 1) + 1, reduces to 1 / (EXP(0.04 x t)), when k = 25.

And you are saying, 1 / (EXP(0.04 x t)) “is essentially similar to” 1 / ((1 + r)^t) at “close to 4%”, i.e: 1 / ((1 + 0.04)^t)

That is, EXP (0.04 x t) “is essentially similar to” (1 + 0.04) ^ t

That is, EXP (0.04) “is essentially similar to” (1 + 0.04)

That is, EXP (i) “is essentially similar to” (1 + i), for any interest rate i

To prove this, consider a typical bank account. Banks quote interest rates that are in practice applied monthly. So one-twelfth of the quoted rate compounds every month. For example, a quoted rate of 6% applies 0.5% every month to the account balance. But because it compounds monthly, the effective annual interest rate is 6.17%. It makes intuitive sense that the effective rate is more, because compounding occurs 12 times per year. Mathematically:

6.17% = (1 + 0.06/12)^12 - 1. Formularised: (1 + i) = (1 + i(p)/p)^p, where i(p) = the quoted bank interest rate, applied and compounded every 1/pth of the year.

Now, as p approaches infinity (that is, interest is applied and compounded continuously), then (1 + i(p)/p)^p approaches EXP(i(p)).

So we have (1 + i) = EXP(i(p)), at p = infinity. Note, when interest is compounded continuously, i(p) is known as the ‘force of interest’, and when it equals 6%, the effective annual rate (i) is 6.1837%. That is, i “is essentially similar to” i(p), which proves your point.

Note, I had r(max) = 3.73% in my article, which is the annual compound rate required for a value to triple over 30 years. That is, indifference to having say $100 today over $300 in 30 years, as per the article. Perhaps coincidental that you tied this to a mortgage rate of 4% over 25 years, with your observation resulting from 4% being, 1 divided by 25.

Keep up the good work.

David
July 25, 2025

The most famous quotation for short time being preferred over the longer time comes from Shakespeare's Richard III: "My kingdom for a horse".

Andrew Smith
July 24, 2025

Simple formula everyone should be aware of on the time value of money, property price should double every decade to reflect value at the 7% rate; only a minority of properties achieve this value or more ie. tread water

The EU is recommending financial literacy in curricula citing understanding the time value of money as an essential life literacy.

Dudley
July 24, 2025

Hyperbolic?

More like exponential function:
Y = (1 / k) * (1 / (EXP(1) ^ t) - 1) + 1
k ~= mort-gage term, 25 y.

The majority prefer money yesterday (mort-gage) to money tomorrow (savings / capital).

Which results in property price gains, reinforcing brought forward demand for money yesterday.

Until money rent becomes widely unaffordable, then the preference changes to preferring money now and later.

Dudley
July 24, 2025

Or is it more like:
= (1 / (EXP(1) ^ (t / k)) - 1) + 1
k ~= mort-gage term, 25 y.

 

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