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The equation $$ P = \frac{Cn}{y}\left(1-\frac{1}{(1+\frac{y}{n})^{nT}}\right)+\frac{M}{(1+\frac{y}{n})^{nT}} $$

represents the present value (price $P$) of a government bond which pays $C$ dollars $n$ times a year plus a $M$ value at the end of the $T$ years.

I'm trying to get the yield $y$ by solving this equation for $y$.

Wolfram Alpha cannot help me (time exceeded), nor any algebraic online solver I could find.

Is it even possible to solve this equation for $y$?

I can use excel solver to find y numerically, but I would like to get an equation for it.

Update:

Found online an approximate formula:

$$ y \approx \frac{C+\frac{M-P}{nT}}{\frac{M+P}{n}} $$

Update 2:

The Bond Price Formula (as it is called) is derived from this:

$$ P = \sum_{m=1}^{nT}\frac{C}{(1+\frac{y}{n})^m}+\frac{M}{(1+\frac{y}{n})^{nT}} $$

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  • $\begingroup$ It depends on what you allow in the solution. The general quintic, for instance, cannot be solved exactly by radicals, but it can be solved by elliptic functions. Are you asking for a Galois-theoretic proof of the unsolvability by radicals of the specific family of equations you wrote? $\endgroup$
    – user41593
    Commented Feb 28, 2017 at 8:02
  • $\begingroup$ Thanks for you comments, @EmanueleTron , but I simply didn't understand anything you said ;D I included an approximate formulate I found online. $y$ is called yield-to-maturity(YTM), in case you are curious. Do the approximate formula helps you understand what I need? $\endgroup$
    – motobói
    Commented Feb 28, 2017 at 16:06

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Let $z=1+y/n$, then your equation is equivalent to $$ Pz^{nT}=C\sum_{k=0}^{nT-1}z^k + M. $$ So you're basically asking,

is a polynomial equation $\sum_{k=0}^N c_kx^k=0$ solvable if you know that $c_1=c_2=\dots= c_{N-1}$?

This can be thought of as Galois theory. I don't know a proof that it's impossible under these conditions, but Wolfram Alpha can't solve e.g. $$9x^5+x^4+x^3+x^2+x+13=0$$ by radicals.

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