0% rate of return + inflation differs from inflation alone

Question

Given that

Real Rate of Return = (1 + nominal rate) / (1 + inflation rate) -1

If my nominal rate equals the inflation rate I get 0, meaning that my money is worth the same no matter the inflation rate.
If my nominal rate is greater than the inflation I get positive values, meaning my money is worth more the longer I leave in this investment. And vice-versa.
But something weird happens when the nominal rate is 0%, such as when I leave my money in my wallet. In that case the real rate will be

RR = (1 + 0%)/(1 + 5%) -1 = -4.76%

Am  I missing something? How do I interpret that? Shouldn't it be -5%?
EDIT:
Also, is there anything wrong with the following formula?

Fake Real Rate of Return = ((1 + inflation rate) / (1 + nominal rate) -1 ) * - 1

In that case when the nominal rate is 0% we get the inflation rate, and when the inflation rate equals the nominal rate we get 0%.

glibdud · Answer

If inflation was 100%, would you expect your money to lose 100% of its value?
Say you have $1000 cash. Widgets cost $1. You can currently buy 1000 widgets with your cash.
After 5% inflation, widgets now cost $1.05, but you still have the same $1000 cash. You can now only buy $1000/$1.05 = 952.38 widgets. Your money only has 952.38/1000 = 95.238% of the buying power it had previously, a loss of 4.762%. This is consistent with the value calculated by the formula.

Brady Gilg · Answer

Percentages are multiplicative, not additive.
If you make a 5% gain, and then have a 5% loss, you are not back to where you started, because 1.05 * 0.95 = 0.9975. The loss that balances out a 5% gain is a (1-1/1.05) ~= 4.76% loss, because 1.05*0.9524 ~= 1.

0% rate of return + inflation differs from inflation alone

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