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Our professor calculated the present value of a bond with $T=10$ years, $FV=10,000$€, $C=700$€ p.a. and an expected rate of return $r$. He wrote $$\begin{align}PV&=C\cdot\sum_{n=1}^{10}\frac{1}{(1+r)^n}+\frac{FV}{(1+r)^n}\\&=700\cdot\color{red}{\frac{1-\frac{1}{(1+r)^n}}{(1+r)^n}}+\frac{FV}{(1+r)^n}\end{align}$$ I don't understand the red step (I guess he transformed the sum).

In our lectures we have to different formulas for calculating the present value (with cash flows $C_n$): $$PV=\sum_{n=1}^T\frac{C_n}{(1+r)^n} \tag{1}$$ and annuity (value of $C$ received each year for $T$ years): $$PV = \frac{C}{r}\left(1-\frac{1}{(1+r)^T}\right).\tag{2}$$ I understand that if in $(1)$ $C_n=C\ \forall n=1,...,T:$ $$PV=C\cdot\sum_{n=1}^T\left(\frac{1}{(1+r)}\right)^n $$ Maybe it's connected to the geometric sum? Thanks for every help!

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  • $\begingroup$ Professor may have made a mistake in the denominator of the red portion. Need $r$ there and not $(1+r)^n$ $\endgroup$ Commented Jan 21, 2024 at 18:14
  • $\begingroup$ May be. Then it would look more like $(1)$ and $(2)$. But how do I simplify the sum to the red fraction? $\endgroup$ Commented Jan 21, 2024 at 18:16

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Not sure what the issue is here, but $$S_n=\frac{a(x^n-1)}{x-1}$$ for the sum of the first $n$ terms of a series with first term $a$ and common ratio $x$. In your case $a=x=\frac{1}{1+r}$ so $$\sum_{n=1}^{10} \frac{1}{(1+r)^n}=\frac{1}{r}\left( 1-\frac{1}{(1+r)^{10}}\right)$$ which as @nbbo2 has pointed out is your red term with the denominator corrected to be $r$ instead of $(1+r)^n$.

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