Finance Preliminaries¶
Introduction¶
Our objective is to learn the theory and application of time series methods.
- We will focus on financial time series applications.
- The methods in this course are broadly applicable to any type of time series.
- We will use the R programming environment to work with financial time series data.
- The quantmod library will be especially useful:
> install.packages("quantmod")
Time Series Example¶
Let’s plot the historical prices for Facebook (FB).
> library(quantmod)
> getSymbols("FB",src="google", from="2012-01-01", to="2014-12-31")
> png(filename="fb.png")
> chartSeries(FB)
> dev.off()
Plot of Facebook Price¶
Facebook Returns¶
To plot just the closing prices:
> chartSeries(Cl(FB))
> chartSeries(FB$FB.Close)
Or daily returns:
> chartSeries(dailyReturn(Cl(FB)))
Plot of Facebook Returns¶
One-Period Return¶
Let \(P_t\) be the price of an asset at time \(t\).
- The gross return of the asset between dates \(t-1\) and \(t\) is:
\[\smash{\begin{align*}
R_t & = \frac{P_t}{P_{t-1}}
\,\,\,\, \text{or} \,\,\,\, P_t = P_{t-1} R_t.
\end{align*}}\]
- The net return is:
\[\smash{\begin{align*}
r_t & = R_t - 1 = \frac{P_t}{P_{t-1}} - 1 = \frac{P_t -
P_{t-1}}{P_{t-1}}.
\end{align*}}\]
- Note that the return can be computed between any two dates (i.e. daily, weekly, monthly, etc).
Multi-Period Return¶
The \(k\) -period gross return between dates \(t-k\) and \(t\) is:
\[\begin{split}\begin{align*}
R_t(k) & =
\frac{P_t}{P_{t-k}} = \frac{P_t}{P_{t-1}} \times
\frac{P_{t-1}}{P_{t-2}} \times
\cdots \times \frac{P_{t-k+1}}{P_{t-k}} \\
& = R_t R_{t-1} \cdots R_{t-k+1} \\
& = \prod_{j=0}^{k-1} R_{t-j}.
\end{align*}\end{split}\]
- The \(k\) -period net return is:
\[\smash{\begin{align*}
r_t(k) & = \frac{P_t - P_{t-k}}{P_{t-k}}.
\end{align*}}\]
Logarithmic Approximation¶
In general, for any small value \(\smash{\varepsilon > 0}\):
\[\begin{align*}
\ln(1+\varepsilon) & \approx \varepsilon.
\end{align*}\]
Thus,
\[\smash{\begin{align*}
\ln(R_t) & = \ln(1+r_t) \approx r_t.
\end{align*}}\]
Furthermore, by the definition of gross returns,
\[\smash{\begin{align*}
r_t & \approx \ln(R_t) = \ln(P_t/P_{t-1}) = \ln(P_t) - \ln(P_{t-1}).
\end{align*}}\]
Approximation for Multiperiod Returns¶
A similar relationship holds for the \(k\) -period net return:
\[\smash{\begin{align*}
r_t(k) & \approx \ln(P_t) - \ln(P_{t-k}).
\end{align*}}\]
Time Intervals¶
The interval of time for returns is of vital importance for understanding the data.
- Daily returns are very different from weekly, monthly, annual, etc. returns.
- Intra-day returns at various time scales (millisecond, second, minute) are very different from each other.
Aggregating Trading Intervals¶
When aggregating returns, we consider the following.
- There are approximately 250 trading days in a year.
- There are approximately 22 trading days in a month.
- There are 5 trading days in a week.
- U.S. equities markets are open from 9:30 am to 4:00 pm Eastern time - 6.5 hours each day.
- Thus there are approximately 6.5 hours, or 390 minutes or 23,400 seconds or 23,400,000 milliseconds in a trading day.
- Similarly, there are approximately 1625 trading hours, 97,500 trading minutes, 5,850,000 trading seconds and 5,850,000,000 trading milliseconds in a year.
Aggregating Returns¶
To aggregate net returns, we simply add them:
\[\begin{split}\begin{align*}
r_t(k) & = \ln(P_t) - \ln(P_{t-k}) \\
& = \ln(P_t) - \ln(P_{t-1}) +
\ln(P_{t-1}) - \ln(P_{t-2}) + \ln(P_{t-2}) \\
& \hspace{2in} - \ldots -
\ln(P_{t-k+1}) + \ln(P_{t-k+1}) - \ln(P_{t-k}) \\
& = r_t + r_{t-1} + r_{t-2} + \ldots + r_{t-k+1} \\
& = \sum_{j=0}^{k-1} r_{t-j}.
\end{align*}\end{split}\]
For example, to annualize daily returns,
\[\begin{align*}
r_t(250) & = \sum_{j=0}^{250} r_j.
\end{align*}\]
Example of Aggregating Returns¶
Get Exxon Mobile equities data for the week of March 23rd, 2015.
> getSymbols("XOM", from="2015-03-23", to="2015-03-27")
[1] "XOM"
> XOM
XOM.Open XOM.High XOM.Low XOM.Close XOM.Volume XOM.Adjusted
2015-03-23 85.02 85.78 85.01 85.43 17163200 85.43
2015-03-24 85.30 85.78 84.50 84.52 10099500 84.52
2015-03-25 85.05 85.57 84.77 84.86 11816000 84.86
2015-03-26 85.30 85.57 84.09 84.32 14388500 84.32
2015-03-27 84.04 84.05 83.33 83.58 11094600 83.58
- What are the daily returns?
- What is the weekly return?
Asset Classes¶
There are several broad classes of assets traded in financial markets.
- Equities.
- Futures.
- Options.
- Bonds.
- Currencies.
Indices¶
Indices are synthetic portfolios of assets that are not typically traded.
- The S&P 500 index is a portfolio of 500 equities and is not traded.
- To hold the S&P 500 index, one can:
- Purchase the 500 component equities in the correct proportions.
- Purchase shares in a mutual fund that tracks the index.
- Purchase shares of the SPY exchange traded fund (ETF).
- Purchase futures contracts on SPX.
Important Indices¶
- S&P 500 (SPX).
- VIX - portfolio of S&P 500 options which represents the expected value of a one-standard deviation move in the S&P 500 index over the next month (in annual terms).
- On March 30th, 2015, the closing value for VIX was 14.51 and the closing value for SPX 2086.24.
- Hence, the market expects the standard deviation of the SPX to be \(14.51/\sqrt{12} = 4.19\) percent or \(\smash{0.0419\times 2086.24 = 87.39}\) index points.
Important Assets¶
- SPY - SPX ETF.
- E-mini - Futures contract on the SPX.
- SPX Options.
- SPY Options.
- VIX Options.
- VIX Futures.
Near-Month VX Futures¶
> install.packages("Quandl")
> library(Quandl)
> VX1 = Quandl("OFDP/FUTURE_VX1",type="xts")
> chartSeries(VX1)
E-mini Near-Month Returns¶
> ES1 = Quandl("OFDP/FUTURE_ES1",start_date="2007-01-01",end_date="2015-03-27",type="xts")
> chartSeries(dailyReturn(ES1$Open))
Important Features of Returns¶
What do you notice about the E-mini returns?