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In the Wikipedia article on annuity, different formulas are given, and their difference is due to whether the first payment is made in the first period or not, whether we compute the present or future value, whether the annuity is a perpetuity or not, etc.

In macroeconomics, e.g. page 8 here, I often recognize something of the following terminology: "If the interest rate is $r$, then the annuity of $x$ (which may be initial wealth) is $\frac{r}{1+r}x$." Now, the problem is that I do not understand why we would say that the annuity of $x$ is $\frac{r}{1+r}x$ in any sense given in the article on Wikipedia. What am I missing here?

Let me mention some examples.

  1. If the interest rate is $r$, then the present value of an annuity-immediate for $n$ periods with "rent" $x$ is $x\frac{1-(1+r)^{-n}}{r}$. If $n=1$ here, then we have $x\frac{1}{1+r}$ (I pick $n=1$ to somehow get the expression $\frac{r}{1+r}x$).

  2. For the future value way of writing an annuity-immediate, we have $x\frac{(1+r)^{n}-1}{r}$ which is $x$ for $n=1$.

  3. For an annuity-due, we have the former expressions in 1 and 2 multiplied by $(1+r)$.

  4. For a perpetuity I get $x\frac{1}{r}$ for first payment in second period.

  5. For a perpetuity I get $x\frac{1+r}{r}$ for first payment in first period.

So, in no case do I get $\frac{r}{1+r}x$. I feel there is something I am missing in my understanding of the concept of an annuity.

Looking at 5, I think that they mean by the "annuity value of $x$" the annuity value of some present value $x$. So in 5, the present value is $x=y\frac{1+r}{r}$ for some other rent $y$, and thus the annuity value of $x$ is $y=\frac{r}{1+r}x$?

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