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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?

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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: $$q_i = \frac{K * w_i }{S_i} $$

However, in reality, this number is rarely round number. Since floating number of shares cannot be purchased from standard market, one must apply truncation or rounding operations to the optimization results.

There are certain situations when initial capital does matter. In these cases, one should then pull the capital constraints in to the optimization procedure:

1) It matters when initial capital is too small, such that individual allocation is significantly impacted by rounding and truncation. For portfolio with infinite capital, these rounding effect can be neglected. However, for a small portfolio, truncation operations may lead to substantial impact. For example, for a futures contract, the minimum contract value is usually in the range of 100K dollars. In this case, a small portfolio with total capital of 200K may not be able to allocate capital efficiently without deviating from the optimization results.

2) When initial capital is too large, such that individual allocation can substantially move the market and cause liquidity concerns. In this case, one should supply a maximum capital allocation constraints to selected assets.

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