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# Portfolio weightlifting (II)

### Enrique Millán

#### 07/02/2019

In a previous post, we took a look at the computation of a portfolio’s exposure to its allocations. Then, to show the effects of active management, we compared the return made by two portfolios. But there is so much more to look inside the financial time series.

Since we left a couple of cliffhangers, let’s jump into them now.

## Risk metrics

First of all, let’s begin with those hidden dangers we mentioned in the impressive performance accomplished by Free.  Recall that it generated way more profit than our Rebalanced portfolio. That could lead you to decide that it’s better to leave your money to itself rather than to pay someone to manage it. However, when we take a glance at their associated risk metrics, we find out why taking some care of where your money goes is worth it.

Our two indicators are going to be the annual volatility of the returns and drawdown.

### Volatility

Volatility is an indicator of the expected displacement of the return around its current value. The classic formula to compute it for a period T is



$Vol_t = \sqrt{T} \sigma\left(\ln 1 + r_j \right), \quad j = t-(T+1), \ldots, t-1$

(Remember, you have to compute it with the logarithmic returns! If you feel like reading in Spanish about it, check it at Quantdare).

Putting together both portfolio’s annual volatility, we see that our simple, yet effective, monthly rebalance was able to reduce the metric’s value by 21 % on average. And if you observe the plots carefully, you will see how when peaks of volatility take place on Free, on the Rebalanced curve they are damped or do not exist at all.

What are the implications of this reduction? When you are investing for the long term, you naturally want to create value, but mostly to preserve capital. Thus, what you want is not necessarily the highest return, but the most stable one. Indeed, a famous financial empirical fact is the low volatility anomaly.

### Drawdown

It’s time now for our fright measure. The drawdown reflects how much the accumulated return has dropped in for a given period since the last highest high. The more negative it is, the more the value of your portfolio dropped.

This could induce a feeling of insecurity, and perhaps make the investor think about moving his/her positions when the value of the portfolio drops too much (and as a consequence, overreact to the market. Remember that on the long term both portfolio’s fared really well).

Again, we see that the rebalancing has rewarding effects for most of the investing period. The Rebalanced portfolio has had less steep drops during stress periods, or even a null one, pumping up the feeling of calm for such volatile assets, we are dealing with a very volatile sector after all!

It is inevitably to fall when the market does so too, you just want to fall as less as possible to start from higher up when the rebound takes place.

So, with these two simple metrics, we have seen the effect of a very simple risk control strategy: to keep your portfolio equally weighted in as much as possible. Could you guess why I did not suggest to rebalance every day or every week?

## The number of shares

The problem with the weight formulation is that it leads to the computation of shares with decimal points. Naturally, the maths do not suffer from this problem, but you cannot go to your broker and ask for 17.345 shares of Apple on your next move.

To compute the initial shares you need to buy of a certain stock to reach a determined exposure level, you determine the ratio between the Assets Under Management you want to expose and the stock price,

$S_0 = \frac{I_0 w_0}{P_0}$

and take the closest smallest integer number (known as the floor operation in mathematics).

Since now you work with an integer number, you will always have some leftover cash that is not invested. You want that number to be as small as possible, but at the same time have enough left so that you can rebalance your exposure without problems.

Summing up, there is more to portfolio valuation than simply the return. Certainly, you are not going far if you do not capture the market growth when it takes place, but you want to make sure you are not boarding an airplane without a pilot, no matter how high it goes.

Once again, thank you for reading.