Markowitz expected return time

Question

This is perhaps a rather silly question for the more experienced people in the community but it has been puzzling my mind for a while.

Let's say we have a portfolio of 10.000 dollar.

We will apply mean-variance model of Markowitz:

$$E[R_P] =w_1X_1 + w_2X_2$$

where $E[R_P]$ is the expected portfolio return and $w_i$ the weight for the return for a given asset $X_i$. I assume an investment horizon of one year.

So let's say the portfolio consists of stocks($X_1$) and bonds($X_2$) and it is an equal weighted portfolio (50/50).

The return will be 4% for bonds and 6% for stocks during the year. The return of the portfolio for one year is then

$$E[R] = (0.5 \cdot 0.06)+(0.5 \cdot 0.04) = 0.05$$

Now let's say the stock market had a really good run and in 6 months time, the target return of 6% was reached.

Now there are two possibilities according to me:

1. You can continue the equity position and hope for an anything above 6% return.
2. You close your equity positions and now you have 51.4% (6% return on 5.000 dollar) of your portfolio in cash that can either be put back in stocks or bonds.

My questions are as following:

1. Are my assumptions about a certain time frame correct?
2. Is there a method that can solve based on the standard deviation of bonds and equities to see where the cash should go?

I don't know if I have gone completely off the deep or not, any feedback would be more than welcome!

Answer 1

Time horizon

The mean-variance framework is based on a single time period of investment, i.e. you assume the same investment horizon $T$ for all investors (which can be one year, or any other period of time). Be aware of the resulting biases, if the "true" investment horizon of an investor differs from your assumptions (see here).

Optimum portfolio allocation

The optimum portfolio allocation is based on the co-variance of your assets. The variance of your portfolio's return is

$$\sigma_P^2 = \sum_{j=1}^N{\left( W_j^2\sigma_j^2 \right)} + \sum_{j=1}^N{\sum_{\substack{k=1 \\ k\neq j}}^N{\left( W_j W_k \sigma_{jk} \right)}}$$

with $\sigma_{ij}$ as the co-variance of asset $i$ and $j$. $W_i$ denotes the weight of asset $i$.

If you are interested in the portfolio with minimum variance of portfolio return (i.e. the portfolio with minimum risk) and assume two investment opportunities $X_S$ (stocks) and $X_B$ (bonds), than you have to calculate $\frac{\partial \sigma_P}{\partial W_S} =0$, with the constraint $W_S + W_B = 1$. This results in $W_S = \frac{\sigma_S^2}{\sigma_S^2 + \sigma_B^2}$.

However, the collection of all possible portfolios (based on all investment opportunities) defines a whole region in the mean-variance space. Assuming rational investors, you get a hyperbola of efficient portfolios called the efficient frontier. The decision to invest in a certain efficient portfolio is based on the investors risk-tolerance $q$. The optimum portfolio then is calculated by minimizing $$w^T \Sigma w - q \cdot r^T w$$ with the vector $w$ of weighting each asset, $\Sigma$ as the covariance-matrix of asset returns (which is the vector $r$).

I recommend you to read the wikipedia article as an introduction or chapter 5 and 6 in Elton et al. (2014) for a more formal and precise description (which also provides the case of a riskless investment opportunity).