# An Application Of Modern Portfolio Theory In Practice

By: Christian Kjølhede

It is know to most financial professionals that an intelligent investor diversifies his portfolio to avoid unnecessary risk, which is the unsystematic risk of the individual asset. By doing so the investor enhance the risk/return ratio of the portfolio, which can then be geared to suit the risk preferences that the investor might have.

The Modern Portfolio Theory first introduced by Harry Markowitz in 1954 shows, how the optimal portfolio can be determined. This is based on the co-movement of the different risky asset portfolio candidates and the individual return premium of the risky asset over the risk free return (often determined as the return on a 3-month Treasury Bill). A method to calculate the optimal portfolio weight for assets will be determined by the following matrix calculation know as Black’s 72 method:

Where W is a vector with the weights of each risky asset, is the inverse covariance matrix of the risky assets and RP is a vector with the risk premium of each asset. The premiums and covariances can be based on either historical data or forecasts.

This might seem as a very straightforward way to get an optimal portfolio, but the problem with this approach is that it has some assumption, which makes huge limitations to the use of it practically.

Firstly, there is assumed to be no transaction costs for trading, which is clearly not the case as you must at least pay for the liquidity as a price taker through the bid-ask spread and might very well also be charged commission when trading as an individual investor. More stocks included at even small weights might incur higher transaction costs, which might exceed the marginal advantage of some of the weight allocations with Black’s 72 method.

However, the largest limitation of the model is due to short selling. The model simply assumes that the proceeds from a short sell can be invested in the another asset with a better risk diversification and return profile, so that short sales will even be profitable for short position on assets increasing in price, as long as the price increase on the long positions in the portfolio are higher.

In the real world regulation T enforced by the Federal Reserve Board in the US and similar regulation in other countries as well demands the proceeds from the short sale to be locked in an margin account as collateral. For this reason, the proceeds cannot be used to make long positions and the efficient portfolio suggested by Black’s 72 method. Because short positions do not generate free cash to invest in long position, which you assume to be better it is no longer a relative performance investment (borrowing money at a low asset return to invest in a high asset return), as Black’s 72 method assumes. Now the reason for short positions would solely be, if you expect the stock to have a negative price development. In that case, you must also consider, that there is an additional 50% of the short asset value required as initial margin by regulation T as well, and that brokers you use my demand an even higher margin. There is also a limited time horizon of the short position, before it must be closed or remade, which results in additional commission costs.

What would then be the practical way to make an efficient portfolio with respect to the limitations on short sales, which are not taken into account Black’s 72 method. The best way is to optimize the risk/return of your portfolio by using linear programming to iterate the weights, which give the best risk/return measured by the highest Sharpe ratio of the portfolio (Return premium of the portfolio divided by the portfolio volatility).

With this procedure, the weights of the portfolio can constrained to be only positive, so no short sales. You can also constrain the weights to certain sizes if you want to put a cap on the amount of each asset, asset class, industry etc. You can even make a negative weight if you want to sell short one of the assets, as you assume the price to go down. Then you can re-optimize the weights of you portfolio with constraints.

Of course, this gives a less efficient portfolio than the optimal by using Black’s 72 method. But this portfolio can actually be acquired in practice and even be modified to personal preferences that might trump the preference simply of absolute efficiency.