In the past weeks we discussed about returns and logarithmic returns, another important quantity is price volatility. Volatility is a key factor to take into account when building a portfolio in order to qualify if an asset is more or less risky (the higher the volatility, the higher the risk). Volatility appears in modern portfolio theory (more on that soon) and has gained a wide acceptance across the financial industry.

### Key takeaways:

• Volatility refers to the amount of uncertainty related to the size of changes in a security's value
• The volatility is the standard deviation of the returns
• Volatility is often associated with risk: the higher the volatility, the riskier the asset
• The higher the volatility the harder emotionally it is for an investor to not worry

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# What is volatility

Volatility is a statistical measure of the dispersion of returns for a given security or market index. — Investopedia

Volatility is the degree of variation of a trading price series over time, in particular a market (or an asset) is called volatile when there are big swings in price in either directions. In other words, volatility refers to the amount of uncertainty related to the size of changes in a security's value:

• A higher volatility means that a security's price is more likely to spread out over a large range of values
• A lower volatility means a security's price tends to be more steady

Investors care about volatility for many reasons:

1. The wider the swings in an investment's price, the harder emotionally it is to not worry;
2. Price volatility of a trading instrument can define position sizing in a portfolio (more on that soon);
3. When certain cash flows from selling a security are needed at a specific future date, higher volatility means a greater chance of a shortfall;
4. Higher volatility of returns while saving for retirement results in a wider distribution of possible final portfolio values;
5. Price volatility presents opportunities to buy assets cheaply and sell when overpriced;

Volatility is often associated with risk: the higher the volatility, the riskier the asset (This assumption is not entirely true, we will nuance it in another article).

# How to measure volatility

In order to measure volatility we need to quantify the "upness" and/or "downness" of the prices from one day to the other. How to measure up/down-ness? Returns!

Great! We already know how to do the first step of the computation: getting the returns. Or... Do we? Yes! But we will take something slightly different than returns. You guessed it: logarithmic returns!

Last week's article on logarithmic returns will now make full sense. In particular, the raw-log equality and time-additivity properties of log-returns. So, first step, the definition of logarithmic returns:

$r_{t}=\ln(\frac{p_{t}}{p_{t-1}}) \tag{1}$

The second (and last) step? Very simple: The volatility is just the standard deviation of the returns. The standard deviation is a measure of the amount of variation of a quantity, here the returns. It reflects the average amount a stock's price has differed from the mean over a period of time. It is calculated by determining the mean price for the established period and then subtracting this figure from each price point. The differences are then squared, summed, and averaged to produce the variance.

The definition of volatility becomes:

$\sigma_{t}^{2}=\frac{\sum_{i=1}^{n}(r_{t-i}-\bar{r})^{2}}{n-1} \tag{2}$

Where $$\sigma_{t}^{2}$$ is the volatility and $$n$$ the number of time intervals (i.e. the number of days if we consider daily returns).

Volatility is without a unit and is expressed as a percentage. While variance captures the dispersion of returns around the mean of an asset in general, volatility is a measure of that variance bounded by a specific period of time.

A critical decision in measuring the volatility is choosing a lookback window (i.e. the value of $$n$$). The longer the window, the more information we have for our estimate. However the shorter, the more quickly our estimate will respond to new information (I will write about this soon).

Thus, we can report daily volatility, weekly, monthly, or annualized volatility.

# Volatility in practice

Now that we know the theory, we can apply our knowledge to real life: What is the volatility of Google? Amazon? Apple? Netflix?

I computed the annualized volatility of different U.S. stocks using data from Yahoo Finance and the formula (2):

The annualized volatility of these stocks is in the range of 20-40% with an outlier: Netflix (NFLX) at 52%. Also, UPS and IBM have the lowest volatility of the group.

If what I wrote until now is correct (I hope!), the difference in volatility between NFLX and UPS/IBM should be noticeable with naked eye on the time series of prices. Remember, the volatility is a proxy of the dispersion of the distribution of returns. A wider distribution means a more "unstable" price.

So if we plot the prices of NFLX, UPS and IBM as a function of time we should see a lot more variations on NFLX than on UPS and IBM:

Indeed we see that NFLX price has a lot more "ups and downs" than IBM and UPS. The overall performance of Netflix is very good, so volatility is not a bad thing. However holding this stock would have been quite emotional, especially in the past 2-3 years.

Volatility is not only useful to characterize stocks, it is also used for everything that has a price. We've talked in the past about asset classes, and we sorted them into three risk profiles: Growth (high risk), Moderate (moderate risk) and Defensive (low risk).

Let's look at the annualized volatility of different assets belonging to these three groups:

The risk profiles that we defined seem to make sense overall.

The Growth Assets (green):

• Commodities, represented by Gold (GLD)
• Equities, represented by the S&P500 (SPY) (note: see how its volatility is lower than all the stocks presented before)
• Emerging Markets, represented by EEM
• Cryptocurrencies, represented by Bitcoin (BTC)
• Property / Real Estate, represented by REIT
• High-Yield Bonds, represented by JNK

They all tend to have a higher volatility than other asset classes.

The Moderate Assets (blue) are represented by AGG, a mix of bonds, which has a low volatility.

And finally the Defensive Assets (red) are represented by BSV, a mix of short-dated government bonds. It is indeed one of the lowest volatility you can find on the market.

That's all for today.

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