I have displayed here the output from a simulation forecasting the performance of a 2 asset portfolio. The portfolio is 50% SPY (SPDR S&P 500 ETF) and 50% LQD (iShares iBoxx Investment Grade Corporate Bond ETF). A stock-and-bond portfolio is a common diversification strategy, meant to manage the risks specific to investing in stocks or bonds on their own. Historically stocks and bonds have been thought to move inversely to each other. Therefore each position hedges risk in the other position in our 50/50 portfolio.

I used data only for the past 10 years for both securities to do the simulations. I did this to implement a technique I devised to capture the covariance of the securities in the portfolio, which must be done to properly model the relationship between them. I call this technique “Day Matching”. Day Matching is the practice of choosing pairs of historical price changes that occurred on the same day of market activity in the past when walking forward through a simulation. Both arrays of daily returns (one for SPY, the other for LQD) are the same size and represent the same days in the past. So when I simulate the performance of the portfolio, I select the price changes for each security that took place on the same day (and, of course, the day itself is selected at random). Thereby, I attempt to capture any effect of correlation and covariance in the performance of the constituent securities. Not doing this would be to assume the price changes among the constituent securities day to day are totally independent of each other. That may be true, but it is better to avoid making that assumption in case it is false. I go on to compare these results to those you would calculate using Modern Portfolio Theory, and I calculate expected return and variance using both information computed using historical data, as well as information captured from the future returns forecast and compare them also.

Here’s the simulation output forecasting returns for SPY. Each simulation begins with a made-up price of $100 per share, and the simulation from there models price changes out to various time horizons. The terminal per share prices are displayed below:

Here’s the simulation output forecasting returns for LQD:

Finally, here’s the simulation output for a 50/50 portfolio of SPY and LQD:

It is important to remember these simulations are only modeling price changes, not including dividends received. This means that these forecasts may systematically underestimate the total returns because the dividends aren’t included.

Here are the formulae to calculate the 1 year expected return and 1 year expected variance of our 2 asset portfolio using Modern Portfolio Theory:

Expected Return _Portfolio = Weight _SPY * Expected Return _SPY + Weight _LQD * Expected Return _LQD

Variance _Portfolio = Weight _SPY * Standard Deviation _SPY ² + Weight _LQD * Standard Deviation _LQD ² +

2 * Rho _SPY,LQD * Weight _SPY * Weight _LQD * Standard Deviation _SPY * Standard Deviation _LQD

Standard Deviation _Portfolio = Square Root(Variance _Portfolio)

Rho = Pearson’s correlation coefficient

Below are the figures calculated from historical data to be used in the Modern Portfolio Theory calculations to estimate the expected return and the risk. All figures are for a 1 year time horizon.

We can also use our simulation program to get the input data to estimate the expected return and variance. This is done simply by running the performance forecast for a 1 year time horizon, and analyzing the output. After normalizing the output data, the results are displayed below:

As we can see, the expected portfolio performance from the calculation gives a higher expected return and also a higher expected volatility (standard deviation) than does the expected performance derived from the performance forecast. That said, to me these figures seem to roughly agree with each other, which is good confirmation that Day Matching is a worthwhile concept to implement when forecasting multi-asset portfolios via simulation.