ETF Portfolio Optimization

Given the uncertainty about the future performance of financial markets, investors typically diversify their portfolios to improve the quality of their returns. In this project, I constructed a portfolio composed of two ETFs (exchange traded funds) that tracks US equity and fixed income markets. The equity ETF tracks the widely followed S&P 500 while the fixed income ETF tracks long-term US Treasury bonds.

I used optimization techniques to determine what fraction of money should be allotted to each asset (i.e., portfolio weights) by forming an optimal portfolio using the Sharpe ratio with historical data (from 2003 to 2013) and examine its performance over a test period(2014).

The data for the two assets were downloaded from Google Finance (ticker symbols: SPY, TLT) and the federal funds interest rate, representing the risk free rate, was taken from the U.S. Federal Reserve Bank website.

Data preprocessing

The data are splited into training and testing subsets. The training data is used to compute the portfolio weights and testing set is used to evaluate our portfolio. I computed the rate of return for the S&P 500 and long term treasury bonds ETF asset, and constructed a time series of data with the returns for both assets plotted.Red line for S&P500 and blue line for treasury bonds.

Examine the correlation

I computed the general correlation between the S&P500 and long term treasury bond returns in the training set and constructed a scatterplot to present the relationship. The correlation between two asset returns over the whole train set is -0.3439, and the scatter plot also indicate a negative relationship between these two assets

I computed the rolling window correlation with a window of 24 weeks of data (i.e., weeks 1 to 24, weeks 2 to 25...) and constructed a time series plot of the rolling window correlation with each point plotted on the last day of the window.

The rolling-window correlation is a better way to describe the relationship between two assets. In different economic conditions, the relationship between stock and bond can also vary: for example, in the financial crisis in 2008, the stock price decreased and people tended to buy more tresury bonds, which resulted in a larger negative rolling correlation between these two assets. On the other hand, the overall correlation describes the average relationship between two assets across the sample period, and hence is less informative.

Assumption check

The Sharpe ratio calculation assumes that returns are normally distributed. Before computing it, I check the assumption of normal distribution by construct two normal quantile plots, one for training set returns of each assets.

S&P500	US Treasury Bond

From these two figures we can see that the normality assumption seems to be satisified for both series.

Sharpe Ratio Optimization

I computed sharpe ratio for each assets on training set and create a function which takes a vector of portfolio weight and a vector of the return of both assets as input, using equation below to obtain the returns for the portfolio.

For each weight value in my vector x, the function compute the Sharpe ratio for the corresponding portfolio. I used STAT_FUNCTION() to plot the result to see which portfolio weight produces the maximum Sharpe ratio.

It can be seen that the portfolio returns reach its maximum when weights allocation for S&P 500 around 0.6 (x=0.6) is the optimal weight. The result computed by optimize() is similar: x = 0.5958418

Portfolio Evaluation

Lastly, I evaluated my portfolio using the test set data. I compared the result three strategies: investing only in the S&P500, investing only in long term treasury bonds, and investing in the combined portfolio with the optimal weight.

Still, we use red for SP500, blue for bonds, while the yellow line is for the combined portfolio with optimal weight. We find that: i) for most periods, the three time series are above 100, i.e., they mostly yield positive excess returns; ii) the bonds perform the best, following by the combined portfolio while the SP500 performs the worst.

Our Portfoilo Performance on the Test set

The excess returns index can be interpreted as follows: if you invested in $100 in at time t = 1, the index value at time T represents how much you have earned in addition to (i.e., in excess of) the risk free interest rate.

If we invested $100 in each asset (portfolio, all in long term treasury bonds, or all in S&P500) in the first week of January, 2014 , how much would we have at the end of the test set period for each asset in addition to the risk-free interest rate? Let's find out whether our portfolio perform well in the test set.

cat('sp500 only', test_set$g_t[nrow(test_set)], '\n')
## sp500 only 107.8763
cat('bonds only', test_set$g_t1[nrow(test_set)], '\n')
## bonds only 116.376
cat(' portfolio', test_set$g_t2[nrow(test_set)], '\n')
##  portfolio 111.6367

That is, we get 108 with SP500, 116 with bonds, and 112 with combined portfolio. The portfolio seems to underperform in the test set. This might because the optimal weight we get from the test set is for the entire period over different economic conditions, while the test set is just for one year, where economic condition matters more. To get a better performance, we should take the economic situation into account and design different portfolios for different situations.