# decoding this formula about nominal and real return [closed]

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

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

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

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

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?

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## closed as off-topic by Joshua Ulrich, olaker♦Jan 2 '14 at 19:39

• This question does not appear to be about quantitative finance within the scope defined in the help center.
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This question appears to be off-topic because interpreting badly-written lecture notes is unlikely to be useful to future visitors. – Joshua Ulrich Jan 2 '14 at 17:32

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.

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