# 0% rate of return + inflation differs from inflation alone

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%.

• Percentages aren't additive. 1.05/1.04 - 1 isn't 0.01, either: it's 0.009615.... Jun 3, 2021 at 18:04
• Maybe I'm not understading it corretly but the thing is, I'm not changing anything when I put my money on a 'investment' that has no nomial rate, by being very loose about the term investment, the money will devalue by the inflation rate, but when I compute it I get a different result, as if by 'investing it' in a zero nominal rate I changed the real rate of return to be less than the inflation rate. Jun 3, 2021 at 18:59
• I think the discrepancy has something to do with the exact Fisher equation vs its approximation.
– Flux
Jun 3, 2021 at 20:44

## 2 Answers

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.

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.