Observing Rates
The Difficulty of Observing Interest Rates
Present Value and Risk Adjusted Returns
Understanding risk management, expected values and even gambling are important to finance and economics. Much of conventional economic theory depends on a notion of an "equilibrium" rate of interest. To compute the present value of various assets, you need to discount "cash flows" up to some time horizon, according to some expected risk adjusted rate of return.
Importantly, the discount rate, used for computing expected values, is about the opportunity cost of alternative investments. The discount rate does not need to be the return of any asset in particular, and it does not need to be "risk free", it is merely risk adjusted.
In fact, a "risk free" asset does not need to exist, it is enough to assume that the level of risk is stable across time. If risk is stable across time, then an assets expected cash flow will follow a simple exponential function.
Infinite Sums, Convergent Series
The discount rate defines the boundary of convergence
Any cash flow with stable risk across time, whose return exceeds the risk, creates an infinite series of payments which diverges to infinity. However, because we define present value in terms of opportunities available, we adjust each term according to the discount rate. Any series of cash payments growing equal to or faster than the discount rate will diverge, and any series of cash payments growing less than the discount rate will converge to some finite value.
if an asset were to diverge, that would simply mean we need to change the discount rate, and then the series now converges.
For simplicity's sake it is often easier to assume that an asset totally fails, or not, but the calculation is similar regardless, you simply sum the probability of each outcome
Martingale Betting and The St. Petersburg Paradox
One interesting mathematical puzzle is the dilemma of martingale betting strategies and the st petersburg paradox.
Conclusion
It can take up to 7 years to observe a rate difference of 10%, and up to 70 years to observe a difference in rates of just 1%