This question is loosely related to finance. Basically, I was working on a model that involves finding out a value for a variable such that certain profit returns are met. First some definitions:
In $\text{year } i$, we have a portfolio return rate of $p_i$ and a market return rate of $m_i$, with a excess return rate $\alpha_i = p_i - m_i$. Here $p_i$ and $m_i$ are percentages in the interval $[0,1]$.
Suppose I define the target return $t_n$ at the end of $n$ years as such:
$$ t_n = \prod_{i=1}^{n}(1+p_i)^{\frac{1}{n}} - \prod_{i=1}^{n}(1 + m_i)^{\frac{1}{n}} $$
Note: Essentially this is the difference of the geometric means.
My question is,suppose $p_i$ and $m_i$ are known for $i \in [1,n-1]$ (that is, for every year except the last one). Can you find an expression for $\alpha_n$ (recall $\alpha_n = p_n - m_n$) such that $t_n \geq k$ for some $k \in \mathbb R^+$.
An example is as follows. Note that $p_1, m_2, k$ are known:
$$ \begin{array}{|c|c|} \hline \bf\text{Year 1} & p_1 & m_1 \\ \hline \bf\text{Year 2} & x & y \\ \hline \end{array} \\ $$
$$ t_2 = \left((1+p_1)(1+x)\right)^{\frac{1}{2}} - \left((1+m_1)(1+y)\right)^{\frac{1}{2}} \geq k $$
Can we find a way to express this equation so that there is no $x$ and $y$ but rather $\alpha = x-y$ so that we can find requirements for $\alpha$ such that $t \geq k$? I have a feeling this isn't possible without making additional assumptions about the market (i.e. guessing $p_n $ or $m_n$) but you never know.
Let me know if I was unclear anywhere.