1. International equity portfolio
2. International equity portfolio
Suppose that a UK investor has invested 30% of her wealth in the FTSE index, 40% in the SP500 index, and 30% in the SMI index, that is, the Swiss market index.
This investor is subject to five risk factors:
The FTSE, S&P 500, and SMI indexes.
The GB pounds - US dollar exchange rate.
The GB pounds - Swiss Franc exchange rate.
The first thing to do is to examine the behavior of these risk factors.
The five daily series are merged into a multivariate time series with the merge command. The all = FALSE option selects only the days on which values for all series are available. The series is then truncated at the end of 2012 to give 13 years of data 2000 - 2012.
3. Displaying the risk factors
The risk factors are then plotted with plot (dot) zoo(). It is clear that there is a fall in all the major indexes around the 2008-09 crisis. The strengthening of the Swiss Franc against the UK pound is also throughout most of the period is also evident.
In the first exercise after this video, you will examine the log-returns of these risk factors in more detail and check they conform to the stylized facts.
You should find as usual that the risk factor returns are heavy-tailed and serially dependent.
4. Historical simulation
To estimate the portfolio loss distribution, you are going to use a simple technique called historical simulation that is widely used in the financial industry and which is based on the nonparametric idea of resampling past data.
The historical risk-factor changes are resampled and applied to the current portfolio in order to construct the series of losses and profits that would result if those risk-factor changes happened again.
In order to implement historical simulation, it is necessary to write a function called a loss operator that takes a vector of risk-factor changes or log-returns as input and returns the resulting loss or profit on the current portfolio as output.
The whole history of log-returns can then be passed through this function to get the corresponding time series of historically simulated losses and gains.
The distribution of these simulated data is taken as the model for the portfolio loss distribution.
Generally, the time series of historically simulated losses and gains also has the heavy tails and strong serial dependencies found in the underlying return series. This will be confirmed for the international equity portfolio example in one of the exercises.
A note on the loss operator function. This is something that has to be written specifically for the portfolio in question at the time point in question. It is not difficult, but it requires a bit of programming. In the exercises, the loss operator functions that you need will be provided to keep the scripts relatively simple.
5. Empirical estimates of VaR and ES
When the losses have been simulated, they can be used to get estimates of VaR and expected shortfall.
Since VaR is a quantile, it can be estimated by taking a sample quantile of the simulated losses.
The function that computes sample quantiles is called quantile(). The displayed code generates 100 normally distributed losses, sorts them from biggest to smallest and displays the top 8 losses.
The 95% sample quantile is an interpolated value between the 5th and 6th largest loss. The estimate one point five nine is not too far away from the true value one point six four for a normal.
6. Empirical estimates of VaR and ES
Now what about estimating expected shortfall? Since ES is the expected loss given the loss exceeds the quantile it can be estimated by taking the average of the simulated losses that exceed the VaR estimate.
Here is some code for that.
As you can see, the 95% expected shortfall estimate one point seven one is a bit smaller than the true value two point zero six for a normal in this case. More data would really be needed to get a better estimate.
7. Let's practice!
Now it is time to work through the exercises for the international equity portfolio example.