**What is Risk or Volatility?**

In finance, risk and return go always together. You cannot really separate return from risk. These are like two sides of the same coin. You can only interpret properly a given amount of return only if you are thinking of the return that, of the risk that investors are exposed to when they expect or they get those returns.

In finance, we may think about two or three ways of summarizing mean returns and again those different ways of summarizing mean returns. They rather designed to answer different questions and they give you obviously different answers, numerical and, in terms of interpretation and exactly the same thing, is about risk, with one difference.

There are many more ways to access the risk of an asset than they are to access the mean return of an asset. And the reason for this is if I ask ten people to evaluate the performance of an asset over ten years all of them will actually give me the same number. They will look at the value of the asset at the beginning and at the end. They will look at the cash flows that they got in between and then they would calculate either the mean return or the return between the beginning and the end of the period. But there is no controversy there. Ten people looking at the same data will come up with exactly the same return obtained by being invested in the asset over that ten-year period. If I were to ask those ten people to give me an idea of the risk of the asset then I might get ten completely different answers. Some people might focus on variability; some people might focus on losses. Some people might focus on how large those losses were. Some other people would focus on how frequently those losses happen, and so, you know, we open a little bit of Pandora’s box. We do not really know what is going to come out and that is important that you know from the beginning that when we assess the risk of an asset there are multiple ways of doing so. Of all those ways, we are going to focus on the two that are what we typically call the standard of modern corporate finance ways of assessing risk. And that is, just so that you get the names, are what we call **volatility**, sometimes also called the **standard deviation of returns** and **beta**.

Before we actually define and calculate the standard deviation of returns or volatility and beta, keep in mind that we could actually assess the risk of the four assets we have been working on, or any other asset in many different ways. Now, **modern portfolio theory** rather goes back to the early 50s. And in the early 50s a guy by the name of **Harry Markowitz** that eventually won the Nobel Prize in economics precisely for his contributions to the risk of individual assets and to the risk of the portfolio. What Markowitz actually proposed, is that one way we can think of the risk of an individual asset is the variability of the asset. And that is, technically speaking, what we call, the standard deviation of returns. So, you may or may not know about, if I give you a very long series of returns, I can actually look at the distribution of returns. I can calculate the mean of that distribution, and that would be the **arithmetic mean** that we already talked about. And I can calculate the standard deviation, which basically gives you an idea of dispersion around that arithmetic mean return. So, you know, I could look at some asset, like for example, a one-year treasury bill. In which, if we look at all the historical returns, we calculate the mean return and we look at the dispersion around those mean returns. Well, that dispersion is not going to be very large. And the reason it’s not going to be very large because, is simply because, you didn’t get very high returns, and you didn’t get very low returns. You get returns that are more or less clustered closely around that mean return. Now, let’s suppose that I give you instead of the distribution of returns of an **emerging market**, like Russia. Well, you know, you calculate the arithmetic mean return, but then you get returns that are very far away on the positive side, and very far away from the negative side from that mean return. That means that you get a lot more dispersion, you get a lot more variability.

Why is that important?

Well because one way you are thinking about this volatility or standard deviation of returns is simply as a measure of variability or uncertainty. You know, in the case of the one-year treasury bill, history tells me that I’m not going to make a lot of money and I’m not going to lose a lot of money is that, you know, I have returns that are more or less predictable within a fairly narrow range. But if I look at the history of the Russian market, that actually will tell me that I have a huge uncertainty. Because there are periods in which I could have more than doubled my money and there are periods in which I could actually lose more than half of my money and that uncertainty is precisely what the standard deviation tends to capture that volatility, that uncertainty, that variability is something that in finance we do not spend a lot of time trying to think of the actual meaning of a number. And another way to put that is to say that typically we look, we use volatility in relative terms.

What does that mean?

Well, let’s go back to our data set that’s the calculation of the volatility numbers. And let me remind you of one thing. Let me remind you that the numbers that we have here are total returns.

Year |
USA |
Spain |
Egypt |
World |

2004 |
10.7% | 29.6% | 126.2% | 15.8% |

2005 |
5.7% | 4.9% | 161.6% | 11.4% |

2006 |
15.3% | 50.2% | 17.1% | 21.5% |

2007 |
6.0% | 24.7% | 58.4% | 12.2% |

2008 |
-37.1% | -40.1% | -52.4% | -41.8% |

2009 |
27.1% | 45.1% | 39.7% | 35.4% |

2010 |
15.4% | -21.1% | 12.4% | 13.2% |

2011 |
2.0% | -11.2% | -46.9% | -6.9% |

2012 |
16.1% | 4.7% | 47.1% | 16.8% |

2013 |
32.6% | 32.3% | 8.2% | 23.4% |

AM |
9.4% |
11.9% |
37.2% |
10.1% |

GM |
7.6% |
7.9% |
21.4% |
7.7% |

SD |
17.9% |
28.1% |
64.0% |
20.1% |

Beta W |
0.9 |
1.1 |
1.5 |
1.0 |

The numbers that we have here are dollar returns. We have the arithmetic mean returns already calculated. And what these numbers basically mean, and remember we’re not doing any formulas here. You have the formulas in the technical note that compliments this particular session. But, you know, if you look at the number for the US, it’s 17.9%, 28% for Spain, 64% in Egypt and about 20% of the world market. And what I was meaning before, when I said that we usually use this variable in relative terms, is that we compare the 18% for the US with the 28% with Spain. And we say, well, returns in Spain tend to fluctuate more and they’re more uncertain than they are in the US. We compare the 28% of Spain with 64% of Egypt and we say, well, you know, look at, just look at the returns. Returns in Egypt tend to be far more variable than they are in Spain and the US. In other words, we never tried and, you know, as much as, for example, for the geometric mean return. We gave it very clear and precise definition of what they, each of those numbers meant. So, for example, the 7.7% for the world market was the mean annual compound rate at which a capital invested evolved time. We give a very precise definition. We are not going to do that for the standard deviation. And more often than not, that’s the way we used it. The higher the number, the more uncertainty, the more variability in the data.

What does 28% of one mean? Well, if you actually look at the formula that we use to calculate a standard deviation, strictly speaking, what that is, and you hold on to your seat here, this is the square root of the average quadratic deviation, with respect to the arithmetic mean return. Now, you can as well, forget that. You are not going to use the definition in that way. What matters to you, is that 28 is higher than 18, 64 is higher than 28 and 17, and 20 is lower than 64. And that gives you an idea of **relative variability**, relative volatility, of the markets that we are actually discussing. So, that is as far as the interpretation of the volatility or standard deviation goes. The higher this number, the more dispersion around the mean, the more variability in the data, the more uncertainty we have about the returns that we observed, but obviously, the more uncertainty we have about the returns that we expect from this particular from this particular asset. And again, we’re not really going to go much further, in terms of trying to explain what we mean by volatility.

We are going to stay with the fact that the higher this number is, then the more uncertainty you’re going to have about the expected returns that that asset might actually give you, particularly from the point of view of your pocket. More uncertainty means more uncertainty in terms of what capital you are going to have, at the end of any given period. And how that capital, is going to be fluctuating over time. Now, one more thing and then we move on to the next measure of risk which is **beta**. And another thing with volatility is you know, the reason that sometimes we use it in relative terms is that if you work in finance you would have some numbers in the back of your head. So, for example, historical volatility, in annual terms of the US equity market, is between 17 and 20%, depending on the periods that you actually look at. But if you have that number in the back of your head and someone shows you an equity market with a volatility of 60%, well, you know, that that market is actually very volatile. You’re going to have a lot of uncertainty, but if someone actually brings you an asset with a volatility, annual volatility of 5%, then you know that this is a very stable asset, at least compared to the US market. So, the way that we typically use volatility is, again, not only in relative terms but also in relative terms after having a few numbers in the back of our head. So again, two numbers that you may want to put in the back of your head is historical volatility of the US equity market between 17 and 20%. Historical bola, volatility of the US bond market between eight and 11%, again, depending on the periods that you look at.