Quantile with periodic investing

Question

Short Version

Can I get a quantile of such an expression? \begin{equation} \sum_{k=1}^{n} A_k\exp(\mathcal{N}(t_k\mu-\sigma\sqrt{t_k}/2,\sigma))) \end{equation}

I know I can do it for one part of the summation as stated here, however I would like to know if that can be also the case for the summation.

Long Version

Let's say I have a periodic investment, in a really simplified case, we have the user deposits $A_1$, $A_2$ and $A_3$ as in the following image.

The deposits $A_1$, $A_2$ and $A_3$ happen at $t_1$, $t_2$ and $t_3$ respectively, being the period of time between each transaction $t$. The return of the period of time $t$ is $\mu$ and the volatility is $\sigma$.I want to know how much I have at time $t_3$ with a probability of $90\%$.

My variables are normally random distributed. This means that for example in order to project $A_1$ to the times $t_1$, $t_2$ and $t_3$ we will have the following equations:

\begin{equation} X_{A_1,t_1}\sim A_1\exp(\mathcal{N}(0t\mu-\sigma\sqrt{0t}/2,0\sigma))=A_1\\ X_{A_1,t_2}\sim A_1\exp(\mathcal{N}(t\mu-\sigma\sqrt{1t}/2,\sigma\sqrt{t}))\\ X_{A_1,t_3}\sim A_1\exp(\mathcal{N}(2t\mu-\sigma\sqrt{2t}/2,\sigma\sqrt{2t})) \end{equation}

In the first equation you have exactly the same money because your deposit was it $t_1$, the same time that we analyze. Now, we know that we can calculate a quantile for a normal distribution is $\Phi^{-1}_P=\mu+\sqrt{2}\sigma \operatorname {erf} ^{-1}(2P-1)$ where $P \in [0,1]$, in our case $.9$ because we want to verify $90\%$ and $\operatorname {erf}$ is the error function. Thus, we can calculate the quantile of $A_1$ in $t_3$ as follows:

\begin{equation} \Phi^{-1}_{A_1,t_3|P}= A_1\exp(2t\mu-\sigma\sqrt{2t}/2+\sqrt{2}\sigma\sqrt{2t} \operatorname {erf} ^{-1}(2P-1)) \end{equation}

Some reference to this equation here. My question is, can I sum all the values of the quantiles to $t_3$? or, how can I obtain the values of the quantiles at $t_3$?

I tried the following equation:

\begin{equation} \Phi^{-1}_{t_3|P}=\Phi^{-1}_{A_1,t_3|P}+\Phi^{-1}_{A_2,t_3|P}+\Phi^{-1}_{A_3,t_3|P} \end{equation}

As shown before, the last term of the quation is just $A_3$ but the others depend on the time and the quantile. This equation is not right, I have some dissimilarities with a montecarlo simulation I did to verify the results.

I'm almost certain I must not sum quantiles the way I did, but I cannot find the proper resource to verify the equations.

Answer 1

The random variable $$X = \sum_{k=1}^{n} A_k\exp(\mathcal{N}(t_k\mu-\sigma\sqrt{t_k}/2,\sigma)))$$ emerges as a weighted sum of individual random variables that are log-normally distributed.

Unfortunately, even if we assume that the individual r.v. involved in the sum are independent, a sum of log-normals (here $X$) possesses no analytically tractable probability density function.

You will therefore not be able to find a closed-form expression for the quantiles of $X$.

A practical solution however could be to use a "Monte Carlo" simulation in the loose sense: draw a (sufficient) number $M$ of random samples from $X$, $(x_i)_{i=1,...,M}$; sort them in ascending order and infer the desired quantile

These related questions (here and there) may also help I guess.