This is a follow up question to this
I found two different ways to "adjust for inflation" on a daily basis given an annual rate (without doing daily compounding). Both yield the same result at the end of the timeframe however I can't determine if there is any error in the interpolated result.
$$r: \text{annual rate}$$ $$t: \text{time in days}$$ $$years: \text{total time in years, constant}$$ $$PV: \text{Present Value, constant}$$ $$FV(t): \text{Future Value for a given time } t \text{ (in days)}$$
For example, if $years = 2$ then $t \in [0, 730]$
Plots below are generated using $r=0.50$ and $PV=1000$
- The first approach is using Giskard's proposal (see related anwer) and let the rate fixed and let the time vary:
$$x = \sqrt[365]{1 + r} - 1$$ $$FV(t) = PV(1 + x)^t$$
This produced the following plot for 1 year:
- The second solution fixes the time and changes the interest rate based on time:
$$x(t) = \frac{r}{365} * \frac{t}{365}$$ $$FV(t) = PV(1 + x(t))^{years}$$
This produced the following plot for 1 year:
And this for 2 years:
I think the second approach is incorrect because when computing 2 years, the value for one year changed from $666.67$ to $\approx 640$ whereas in the first method the original value is kept but I cannot justify why this happens.