Extending the History of Instruments

At Bertram Solutions, we believe in a data-driven approach to investing. Unfortunately, some instruments have not been around long enough to allow simulations covering a wide range of economic situations. In this post, we show how we extend the history of instruments to include those past periods.

Many of our model portfolios use Exchange Traded Funds (“ETFs”) to achieve their target allocation. The main advantage of using ETFs is that they allow us broad diversification, while at the same time keeping minimal investment amounts low. As an example SPY, the SPDR S&P 500 ETF, enables us to diversify across 500 stocks in increments of about $280 – unthinkable with buying individual stocks.

ETFs are a relatively new development and part of the so-called index revolution, which John C. Bogle started in 1975. Here are a few inception dates:

When backtesting our model portfolios, we like to go back at least one recession period, or to the beginning of 2008. For portfolios with trading activity mostly limited to recession periods, we need to go further. Covering two recession periods would require us to go back to the beginning of 1999. If we could go back to 1970, we would be able to include the last seven recessions. As the inception dates above show, this often won’t be possible using real market data. Instead, we need to find suitable substitutes.

Stock Market Index ETFs

Typical stock market ETFs track an index, most often the S&P 500 index. As the ETF invests in the constituent stocks, it will not only participate in their price appreciation but also the dividends paid. Therefore, to replicate the performance of an index-based stock ETF, we need to use the applicable total return index. Most often these indices are readily available, see here for the S&P 500.

The following chart compares the total return index, to an SPDR’s S&P 500 ETF:

spy spxtr

We see that the that ETF is tracking the total return index exceptionally well, with the ETF lagging behind the index by about 0.16% per year. We can explain most of this deviation with the ETF’s expense ratio of 0.0945% per year. By accounting for the fees, we were able to reduce the error to only 0.06% per year. Based on these results, total return indices are more than appropriate substitutes for index-based stock market ETFs, and our data feed provides history back to 1936.

Bond ETFs

Bonds are the second most crucial component of many portfolios. Just like for the stock market, there are also total return indices available. However, their history often does not date back far enough. Our data feed provides the following total return indices:

  • S&P US Treasury Bond Total Return Index, available since 12/29/1989
  • Dow Jones Equal Weight US Corporate Bond Total Return Index, available since 12/31/1996
  • US Aggregate Bond Total Return Index, available since 04/30/2002

These periods are much shorter than what we would like to see, so we need to take a different approach. Bonds can be valued using a present value approach, in which we discount future cash flows to determine today’s value. Here is the concept:

  • yesterday, we invested in a bond at par value and with a coupon matching yesterday’s yield
  • we receive interest at the coupon rate, pro-rated for a single day
  • the bond now trades at a market price, based on today’s yield rate
  • we adjust our total return index by the accrued interest, as well as the change in valuation, and start over

Typical bond ETFs are diversified funds, holding many different bonds with varying yields, and remaining maturities. We modeled the aggregate bond market ETF as follows:

  • 30% Corporate Bonds
    • 7.5% with AAA rating, and 11 years remaining maturity
    • 22.5% with BAA rating, and 11 years remaining maturity
  • 70% Treasury Bonds
    • 44.1% with 2 years remaining maturity
    • 15.4% with 10 years remaining maturity
    • 10.5% with 30 years remaining maturity

Here is a chart comparing our simulated data against iShares’ Aggregate Bond ETF:

agg sim

There are a few periods, during which our simulated total return deviates from the ETF. These deviations are caused both by an accumulation of simulation errors and by not including enough variety of maturities in our simulation. However, over the long term, our synthetic data track the ETF reasonably well. Using this approach, we can simulate the total bond returns back to 1977.

Inverse 3x Leveraged ETFs

Just like we did for stock market ETFs, we can also determine the total return of inverse leveraged ETFs from the underlying total return index. Unlike the stock market ETFs, we need to apply some additional steps, though:

  • yesterday, our 3x Bear index closed at Q
  • on yesterday’s close, we initiated a short position in the total stock market with an exposure of 3x Q
  • book P&L based on yesterday’s exposure, but at today’s closing price of the underlying index
  • receive money-market interest for a single day over an amount of 4x Q
  • pay fees for Q, pro-rated for a single day
  • reset, and start over

Here is a chart, comparing our simulated 3x Bear Index to ProShares UltraPro Short S&P 500 ETF:

spxu sim

We are surprised to see how well this simple model tracks the ETF, given that these ETFs are complex products. Using this model, we can simulate 3x Bear ETFs back to 1936.


Concluding, we find that we have successfully extended the history of various ETF instruments. Our models track available ETFs very well, and we are confident that these models are accurate enough, to serve as valid substitutes for the respective ETFs before their inception dates. Using these models, we can extend our simulation ranges, and test our model portfolios over a broader range of economic situations. Doing so is a crucial step in improving the quality of our model portfolios.

We very much hope that this post provided some valuable insight behind the scenes of our simulations. At Bertram Solutions, we pride ourselves in doing independent research, and this post is a good example of our approach to analyzing investments.