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Deriving Real Returns from Nominal Returns and Inflation

Article Quant Q&A · Author: Lost1
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Summary

The document interprets notation for nominal wealth, real wealth, and the price level, then derives the relationship between nominal return, inflation, and real return. Real wealth is nominal wealth divided by the price level. If nominal wealth grows by the nominal return and the price level grows by inflation, the real growth factor is the nominal growth factor divided by the inflation growth factor.

Subtracting one from that ratio gives the exact real return: the nominal return less inflation, divided by one plus inflation. The explanation also identifies a likely formatting error in the original first equation, where a division by the later price level appears disconnected. The derivation depends on the stated definitions and period alignment; it is an algebraic identity, not an empirical result or an approximation that simply subtracts inflation from nominal return.

Key ideas

  • Real wealth is nominal wealth divided by the price level.
  • Nominal wealth growth and price-level growth jointly determine real wealth growth.
  • The exact real return equals nominal return minus inflation, divided by one plus inflation.
  • The original equation appears to contain a formatting error in its first line.

Tags

StatisticsFixed income
Full text
# decoding this formula about nominal and real return


# decoding this formula about nominal and real return












I am sorry if the following question is not quantitative finance.

I am reading this thing badly written lecture notes, which says

$W^r_1 \equiv W_1/P_1^g = (W_0^rP_0^g)(1+R)=/P_1^g$

$(1+R^r)\equiv W_1^r/W_0^r = (1+R)(1+\pi)$

$R^r\equiv \Delta_1^r/W_0^r = (R-\pi)/(1+\pi)$

I am guessing $R^r$ is the rate of real return. $R$ is nominal rate of return and $\pi$ is inflation. but what the hell is $P_1^g$ and $W_1^r$ I don't really understand the first formula in particular.

I am guessing $W_1$ is the wealth at time 1?

## Answer by crcvd (score 3, accepted)

https://quant.stackexchange.com/a/9700

I'm guessing ${W_t^r}$ and ${W_t}$ correspond to real and nominal endowment at time $t$, respectively, and that ${P_t^g}$ is the price level at time $t$. In that case, $W_t^r \equiv W_t/P_t^g$ follows, and if endowment grows at a nominal interest rate $R_t$, then $W_t = W_{t-1}(1+R_t)$. We can write $W_{t-1}=(W_{t-1}^rP_{t-1}^g)$, so by substitution $W_t=(W_{t-1}^rP_{t-1}^g)(1+R_t)$.

I suspect the last part of the first line, $=/P_1^g$, contains a typo. There should be no space between this and the previous text: that way we can substitute $W_t$ back into the expression for $W_t^r$ and get to the adequate expression:

$W_t^r = (W_{t-1}^rP_{t-1}^g)(1+R_t)/P_t^g$.

What bugs me is the second line: we can define the inflation rate $\pi_t$ as $P_t^g=(1+\pi_t)P_{t-1}^g$. We are given $(1+R_t^r)\equiv W_t^r/W_{t-1}^r$, so the equation should resolve to:

$\dfrac{W_t^r}{W_{t-1}^r}=\dfrac{W_t/P_t^g}{W_{t-1}/P_{t-1}^g}=\dfrac{W_t}{W_{t-1}}\dfrac{P_{t-1}^g}{P_t^g}=(1+R_t)\dfrac{1}{1+\pi_t}=\dfrac{1+R_t}{1+\pi_t}$

With the above in hand, all we need to do is subtract one to arrive at your final expression:

$R_t^r=\dfrac{1+R_t}{1+\pi_t}-1=\dfrac{1+R_t-(1+\pi_t)}{1+\pi_t}=\dfrac{R_t-\pi_t}{1+\pi_t}$

EDIT: This, of course, is a restatement of the Fisher equation.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.