en.wikipedia.org/wiki/Hyperbolic_discounting

For annual cash flows in arrears, present value is computed by discounting each payment back to the present and summing them.

A peer-reviewed paper on discounting formulas states that the present value of a stream of payments is the sum of discounted payments, and for Mazur hyperbolic discounting gives:

[
PV = \sum_{t=0}^{T-1} A \cdot \frac{1}{1+\omega (t + D)}
]

That is the discrete-payment formula. Separately, a Cambridge paper defines the hyperbolic discount factor for a single delayed reward as (1/(1+kd)), so one payment received after delay (d) has present value (A/(1+kd)).

Putting those together, an annuity with annual payments in arrears should be valued as:

[
PV = \frac{P}{1+k} + \frac{P}{1+2k} + \cdots + \frac{P}{1+kD}
]

—not (P\ln(1+kD)/k).

The article’s formula fails even in the simplest case. If there is one annual payment (so (D=1)), hyperbolic discounting implies the present value must be:

[
PV = \frac{P}{1+k}
]

But the article’s formula gives:

[
PV = P\frac{\ln(1+k)}{k}
]

These are not the same. For example, if (k=1), the correct value is (P/2 = 0.5P), while the article’s formula gives (P\ln 2 \approx 0.693P).

So the quoted formula is not correct for a series of equal annual cash flows in arrears. It corresponds to integrating a continuous payment stream, not valuing a discrete annual annuity.

The expression

[
\frac{1}{1+kt}
]

is the hyperbolic discount factor/function, not the discount rate.

A Cambridge paper defines the hyperbolic discount function as (F(d)=1/(1+kd)) and then explicitly gives the corresponding hyperbolic discount rate as (r(d)=k/(1+kd)). A Springer paper similarly distinguishes the hyperbolic discount formula from the instantaneous discount rate, stating that for the Mazur hyperbolic formula the rate is (\omega/(1+\omega t)).

So the article’s statement is mathematically wrong: the Bayesian derivation yields the discount factor (1/(1+kt)), while the associated discount rate is

[
\frac{k}{1+kt}.
]