# How to derive Fisher's equation?

Guys help me derive Fisher's equation or suggest references that have the derivation.

I am looking for this relation: (1+i)=(1+r)(1+π)

You can find the derivation in Macroeconomics by Olivier Blanchard

We want to prove the relation, $$(1+i)=(1+r)(1+\pi)$$ where $$i$$ means the nominal interest rate,

$$r$$ stands for the real interest rate and

$$\pi$$ for the expected inflation

We also know that from the definition of real interest rate, $$r=i-\pi$$ $$\Rightarrow i=r+\pi$$

If we expand the RHS of the above equation, we get $$(1+r)(1+\pi)=1+r+\pi+(r\pi)$$

Observe that as both $$r \in (0,1), \pi \in (0,1)$$ and are usually small numbers, we can approximate their product to zero ($$r\pi \approx 0$$)

Therefore, we get $$(1+r)(1+\pi)=1+r+\pi+(r\pi)=1+r+\pi=1+i$$

Hence this proves the claim.

The inflation represents the loss of purchasing power of the monetary unit, terefore, the expected inflation rate will have to be subtracted as a first approximation from the yield on government bonds which is normally expressed in nominal terms.

Let

$$C$$ : capital

$$\ i_{r}$$: real interest rate

$$\ i_{n}$$: nominal interest rate

$$\ \pi_{e}$$: expected inflation

We can write

$$C \cdot\ \left (1 + i_{r} \right )= C \cdot\ \left(1 + i_{n} \right ) \cdot\ \left(1 - \pi_{e} \right )$$

Eliminating the factor $$C$$ from both members and carrying out the products we obtain $$1 + i_{r} = 1 - \pi_{e} + i_{n} - i_{n}\cdot\ \pi_{e}$$ that is $$\ i_{r} = i_{n} - \pi_{e} - i_{n}\cdot\ \pi_{e}$$

Generally, we have $$\ i_{n} \gg \ i_{n}\cdot\ \pi_{e}$$, so the term $$\ i_{n}\cdot\ \pi_{e}$$ is negligible in first approximation. As a results we get

(*) $$\ i_{r} \simeq \ i_{n} - \pi_{e}$$

In conclusion, by subtracting only the contribution of the expected inflation rate from the nominal rate as indicated by the formula (*), we will commit a $$\varepsilon$$ error.

Some details on the estimation and calculation errors of the real discount rate are provided below. As is known, interest rates are numerical values truncated/rounded to a certain decimal place or are known with a given uncertainty (think of the variance of expected inflation); the error in estimating $$i_{r}$$ in approximating formula will be due to two contributions: the truncation/rounding error $$\varepsilon_{1}$$ and $$\varepsilon_{2}$$ approximation error introduced in the formula obtained above. Let denote by $$\ \varepsilon_{n}$$ the estimate of the rounding error in the nominal interest rate and by $$\ \varepsilon_{\pi}$$ the estimate of the rounding error in the expected inflation rate. We will better represent the rate values as follows: $$i_{n} \pm \varepsilon_{n}$$ and $$\pi_{e} \pm \ \varepsilon_{\pi}$$. If the starting rates are truncated to the k+1-th decimal place and rounded to the k-th decimal place it means that the starting interest rates are known to exactly k-2 decimal places and we will write: $$\ i_{n } \pm 0.5\cdot\ 10^{-k}$$ and $$\ \pi_{e} \pm 0.5\cdot\ 10 ^{-k}$$. Using the general formula of error propagation, it will be possible to evaluate the number of exact digits with which $$\ i_{r}$$ is knowable and therefore ultimately the actualized capital. It is cleare that if the expected inflation rate is estimated with an error $$\ \varepsilon_{\pi}$$ greater than $$\varepsilon_{n}$$ the estimate of the rate in a more accurate form is useless since the precision of the result is determined by the quantity with the lowest number of exact digits. For completeness, the evaluation of errors is reported. The error resulting from the exact formula is due only to the propagation of rounding errors and is quantifiable in $$\varepsilon_{r}$$=$$\sqrt{\varepsilon_n^2 + \varepsilon_{\pi} ^2 + \left( - \pi_{e} \cdot \ \varepsilon_{n}\right )^2 + \left( -i_n \cdot \ \varepsilon_{\pi}\right )^2 }$$ , while the error resulting from the approximate formula can be quantified in $$\varepsilon_{r}^\prime$$=$$\sqrt{\varepsilon_n^2 + \varepsilon_{\pi}^2 + \left( i_{n} \cdot \ \pi_{e}\right )^2 }$$. Now assume that $$\ \varepsilon_{\pi} \gg \ \varepsilon_{n}$$ i.e. $$\ \varepsilon_{\pi }$$ is larger by two orders of magnitude ( i.e. larger by a factor of 100) than $$\ \varepsilon_{n}$$. We say that $$\ \varepsilon_{\pi}$$ is to be considered an infinitesimal of lower order than $$\ \varepsilon_{n}$$.

Reasoning in infinitesimal terms and taking into account that in a sum of infinitesimals the lower order infinitesimal prevails, we can neglect the terms that converge to zero faster and obtain: $$\varepsilon_{r} \sim \ \varepsilon_{\pi}$$ and $$\varepsilon_{r}^\prime \sim \ \varepsilon_{\pi}$$ where the symbol $$\sim \$$ denotes the asymptotic estimate.

Finally, it turns out that $$\varepsilon_{r}^\prime \simeq \ \varepsilon_{r}$$ or the values of the real interest rate can be known with equal uncertainty: what has just been described justifies the extensive use of the formula (*) .