# Portfolio weightlifting (I)

### Enrique Millán

#### 30/01/2019

Both portfolio valuation and management have many facets, but one they have in common is attribution, that is, how much each asset or collection of them is contributing to the return.  Thus, in these series of posts, we are going to wax technical about the computation of asset exposure. We will show how an initially balanced allocation, left to itself navigate the markets, can create a risky situation if you don’t intervene.

When we talk about exposure, we are referring to the weight each asset has in our portfolio. The higher we are exposed to an asset or a sector, the more our portfolio will rise when the given line goes up. However, it will capture the downs in the same proportion.

## The FAANG

For the sake of example, we are going to set up a portfolio with the five most important technology companies in the States. They are known as the FAANG: Facebook – Amazon – Apple – Netflix and Google. In the following chart, we can see how all their prices have skyrocketed since 2012.

However, prices are not often the best metric to determine value. Could you tell which is the company that has grown the most? You might be tempted to say Google, or Amazon, right? Let’s take a look at each company’s cumulative return to get a better picture.

Surprise! Netflix is actually the company that has grown the most in the past years. In fact, if you had invested 100 USD by the time it started selling on the market, today you would have shares worth 3200 USD. Alas, how many of us knew of the existence of Netflix back then?

## A portfolio’s performance

Let’s do some backtesting (simulation and analysis of past financial performance) with these five giants to learn how to correctly compute a portfolio’s performance through a weight formulation. This mathematical dressing, the weight, is very appealing from an analytical point of view. However, when put into practice, it reveals a fundamental fragility due to the integer nature of shares (I will let you ponder that one).

We set up two portfolios, Free and Rebalanced. The difference lies in the fact that for Rebalanced we will sell and buy every first business day of the month such that it becomes again an equally distributed portfolio. Free, on the other hand, will be left alone without intervention. In the financial jargon you would say Rebalanced is actively managed.

Now the technical stuff. All of the following seems trivial when explained out loud, but there are subtle nuances to bear in mind. For each day, we have compute the portfolio’s performance and what proportion of the cake each asset has in our allocation. To this proportion we call the asset weight.

For each day t, the return for each asset i is

$r_{i,t} = \frac{P_{i,t}}{P_{i,t-1}}-1, \quad i = 1, \ldots, N$

where the prices are taken at the close of each session, and adjusted for possible splits and dividend payments. With these returns and the asset weights, the portfolio’s evolution that day is given by

$r_{P,t} = \sum_{i=1}^{N} \omega_{i, t-1}r_{i,t}$

Aha! The return on day t of the portfolio is given by the sum of yesterday’s weights times today’s returns. Why? Well, simply because we are computing the returns with the prices at the session close. Therefore, we will capture the evolution of day t with whatever we had in our portfolio the day before when the markets closed. This formulation forces you to work changing your allocations on day t-1, just before the markets closed, if you want to capture the effect on day t.

Indeed, on day the weights, i.e. the exposure, cannot be known a priori without the returns. To compute each asset weight on day t, you need to put into context how much the assets worth has changed with respect to the portfolio. But you need to do it in absolute terms, not in relative ones, otherwise you will run into trouble with the sign!

$\omega_{i,t} = \omega_{i,t-1} \left (\frac{1+r_{i,t}}{1+\sum_{i=1}^{N}\omega_{i,t-1}r_{i,t}} \right )$

We have applied these ideas and equations to the period of time for which we have shown the price evolution, and we find something you might have not expected: the assets which grow the most eat up those that don’t.

This has serious implications. In our backtest, Netflix represents most of the portfolio’s allocation by 2019, which means that if Netflix drops tomorrow more than you can afford, most of your money will follow (which is actually something that happened during the second half of the year in 2018).

## Comparison with the technological index

Since during the overall period Netflix has grown so much, when we compare the cumulative return of both portfolios, we are not surprised to see Free winning over Rebalanced.

Free has outperformed the index associated with the technological companies, the Nasdaq Composite 100, by almost 800%, while our Rebalanced portfolio has “roughly” made it 400% better than the index.

Next week we will determine the dangers hidden behind the extra return made by Free and take a look at how these weights can be used to understand better where your profit and losses coming from.