Exponential growth of a capital is given by the expression $$C(t)=C_0(1+r)^t$$
A typical shape of investments is initially decreasing and gradually becoming exponential later on. This behavior can be modeled by generalizing the hyperbolic cosinus
$$\frac{e^t+e^{-t}}{2}$$
into
$$C(t) = C_1 (1+r_1)^t + C_2 (1-r_2)^t$$
with positive $r_1, r_2$ and $r_2<1$.
At time $t=0$, the initial capital is the sum of the growing component $C_1$ and the decaying one $C_2$, $C(0)=C_1+C_2$
When the minimum of the function is at $t_{min}>0$, we can estimate the time when the capital is recovered, some $t>t_{min}$ where $C(t)=C(0)$.
Does this two component function have a name and derivation in finance?
The function $$C(t) = C_1 (1+r_1)^t + C_2 (1-r_2)^t$$ is minimum at $$t_*=-\frac{\log \left(-\frac{C_1 \log (1+r_1)}{C_2 \log (1-r_2)}\right)}{\log (1+r_1)-\log (1-r_2)}$$
Now, if you want to approximate $t$ such that $C(t)=\text{something}$, you can use a series expansion around $t_*$ which will be $$C(t)=C(t_*)+ \frac 12 C''(t_*) (t-t_*)^2 + O\left((t-t_*)^3 \right)$$ with $$C''(t)=C_1 \log^2 (1+r_1)\,(1+r_1)^t+C_2 \log^2 (1-r_2)\,(1-r_2)^t \quad > ~~0$$ But all of that requires that $C(t_*) <0$.