1. Risk-factor returns
In QRM, the aim is to model the fluctuations in key risk factors which affect the value of a portfolio.
2. Risk-factor returns
These fluctuations are called the risk-factor changes or risk-factor returns or simply returns.
However, there are a few different ways of defining returns.
Let (Z_t) be a time series containing the values of a risk factor at a time at a set of regularly-spaced times, which could represent days, weeks, months, etc. For illustration, let's suppose it is a daily series.
Here are three different ways of defining risk-factor returns (X_t):
In the first definition, the returns are the differences of the risk-factor values, known as simple returns. This is the simplest definition but not the most common. It tends to be used when the risk factors have very small values close to zero (like certain interest-rate series).
The second definition is the easiest to interpret. The relative returns are the differences divided by the initial values. If you multiply the relative returns by one hundred, you get percentage changes. So if a stock has a relative return of point zero two, it gains 2% in value; if it has a relative return of minus point zero three, it falls 3% in value.
In the third definition, the returns are the differences of the log-values of the risk factors; these are log-returns. This definition is, in fact, the most widely used. Here are a few of the reasons why log-returns are popular.
3. Properties of log-returns
If you build a model for the log-returns of a risk factor, you know that the risk-factor can never become negative. This is generally a desirable feature for risk factors that are prices and rates. However, some risk factors can become negative under unusual market conditions, an example being short-term interest rates.
Log returns are, in fact, very close to relative returns for typical values. It can be shown with some algebra that log of Z_t minus log of Z_(t-1) can be approximated by Z_t minus Z_(t-1) divided by Z_(t-1)
Finally, there are a couple of theoretical reasons for using log-returns:
It is particularly easy to aggregate them to calculate longer-interval returns, such as weekly or monthly returns, as we will see later in this chapter.
Also, there is a very famous model for prices known as the Black-Scholes-Merton model, or geometric Brownian motion (GBM). The model is widely used for pricing derivatives. If a price series follows GBM, then the log-returns will be normally distributed. That's all you really need to know about this model at this stage.
4. Log-returns in R
It's worth remarking, however, that you are going to find that log-returns generally aren't normally distributed in practice, at least for short time horizons.
To calculate the log-returns of the S&P 500 index, you apply the functions log and diff in succession. Diff calculates differences. Here is some code to illustrate.
Note that you get an NA in the first position when you apply these operations. It is best to remove the first value prior to working with the return series.
5. Log-returns in R (2)
Now let's plot the log-returns. You see, a fairly noisy series mostly oscillating between minus point zero five and plus point zero five, which can be approximately interpreted as 5% movements up and down. However, there are some very extreme values. The largest negative return corresponds to the famous stock market crash on Black Monday in 1987.
6. Let's practice!
Now it's your turn to calculate and plot some returns in the next exercises.