# MPT and the connection to asset prices / initial capital

I have some question about MPT. Suppose we want to build a portfolio given $N$ assets: $A_1,\dots,A_N$. At time $t$ we build the portfolio using MPT, which yields some weight vector $w_t=(\lambda_1,\dots,\lambda_N)$, with $\sum_i\lambda_i = 1$. At time $t$, we have an initial capital $K_t$ and the asset prices are given by $S^1_t,\dots,S^N_t$.

My question is, given the weights $w_t$ which fraction of asset $i$ do I have to buy such that the portfolio is correctly built due to the MPT? In a first step, and for simplicity, we assume one can buy any fraction of an asset. Therefore I would build the portfolio as:

$\frac{S^1_t}{\lambda_1K}$ of asset $A_1$, $\frac{S^2_t}{\lambda_2K}$ of asset $A_2 \dots$ and $\frac{S^1_t}{\lambda_NK}$ of asset $A_N$. Is that correct? How are these things usually done in reality?

Moreover, assuming we have the classical optimization problem

$$\min\{w^T\Sigma w-qR^Tw\}$$

where $R$ is the expected return of the assets and the $\Sigma$ the covariance matrix given the constraint

$$\sum w_i = 1$$

It seems to me that the initial capital does not matter? Or how can the initial capital be linked to into the constraints?

Initial capital is not a real constraint in theoretical analysis, but might be a practical constraint in reality. The objective function you gave defines the efficient frontier corresponding to a given risk tolerance $q \in [0, \infty]$: $$\min\{w^T\Sigma w-qR^Tw\}$$
This criterion is among the other popular optimization criteria, such as minimum variance, maximum Sharpe ratio, etc. By solving these functions numerically, we can obtain the optimal weights, $W$, subject to the unity constraint $\sum w_i =1$. In a frictionless theoretical world, assuming total capital of $K$, one will thus allocate capital $K * w_i$ to asset i, which will translates to the $q_i$ shares to purchase in order to achieve MPT portfolio: $$q_i = \frac{K * w_i }{S_i}$$