{"id":19374,"date":"2022-11-02T19:27:25","date_gmt":"2022-11-02T18:27:25","guid":{"rendered":"https:\/\/quantpedia.com\/?p=19374"},"modified":"2025-06-04T14:20:24","modified_gmt":"2025-06-04T12:20:24","slug":"how-to-replicate-any-portfolio","status":"publish","type":"post","link":"https:\/\/vvv.quantpedia.com\/es\/how-to-replicate-any-portfolio\/","title":{"rendered":"How to Replicate Any Portfolio"},"content":{"rendered":"\n

Would you like to see the performance of your portfolio 100 years back in history? Do you want to analyze the risk of your strategy under 100 years of real historical scenarios? All of these, and much more, will be soon (in a few days) available for <\/strong>Quantpedia Pro<\/strong><\/a> subscribers. How? We will explain today how we can model a 100-year history of your portfolio<\/strong>.<\/p>\n\n\n\n

Replicating Portfolios with Factors<\/strong><\/span><\/h2>\n\n\n\n

When reading the title of this article, the first question that might come to your mind is why would anyone want to replicate a portfolio with risk factors. Nevertheless, there are many reasons why replicating a portfolio with factors is beneficial.<\/p>\n\n\n\n

For example, as ETFs were only developed in the 1990s, only very limited historical time windows are available to analyze their performance<\/strong> in extreme market conditions. Also, many strategies rely on ETFs, so their histories are usually only 20 years long, often much shorter. Having the ability to replicate any portfolio with factors with longer history might provide us with countless valuable information.<\/p>\n\n\n\n

Another reason for replicating might be an interest in someone else\u2019s portfolio. What factors drive my competitor\u2019s returns?<\/strong> On the other hand, you might also want to find out to which factors is your own portfolio most sensitive. Whatever the case might be for you, having a more extended data history is always beneficial.<\/p>\n\n\n\n

Therefore, we examined 20 factors and used them to replicate various portfolios in the following steps:<\/p>\n\n\n\n

  1. In the first step, we synchronize the factor and portfolio dates to allow for further calculations.<\/li>
  2. Secondly, we use multi-factor regression analysis in combination with Akaike\u2019s Information Criterion (AIC) to find the explanatory factors of a portfolio and their weights. We apply the procedure to the available history of one\u2019s input portfolio.<\/li>
  3. Thirdly, we check our fit quality by visualizing equity curves of both the original and factor portfolios for the available history of an input portfolio.<\/li>
  4. Lastly, we extend the history of a portfolio to 100 years by modelling an input portfolio via factors with rich data history, created based on Quantpedia\u2019s unique methodology.<\/li><\/ol>\n\n\n\n

    We will now dive deeper into the methodology in the following sections.<\/p>\n\n\n\n

    100 Years of Daily Factor Data<\/strong><\/span><\/h2>\n\n\n\n

    First of all, we had to choose carefully our factor universe \u2013 i.e. what will be our building block for modelling portfolios and strategies. In this choice we had to take into account both:<\/p>\n\n\n\n

    1. Having enough uncorrelated and representative market factors for various asset classes<\/li>
    2. Long-term Data availability for such underlying factors<\/li><\/ol>\n\n\n\n

      However, finding factor data with a 100-year history is almost impossible. Thus, we had to get creative and produce our own data series. With the exception of a few factors, we combined multiple data sources to obtain historical data from 1926. We list the factors and a short description of the methodology for obtaining the data below.<\/p>\n\n\n\n

      U.S. 10-Year Bonds (US10Y)<\/strong><\/span><\/h3>\n\n\n\n

      We described the process of creating a 100-year history of U.S. 10-Year Bonds in detail in our data primer: Extending Historical Daily Bond Data to 100 Years<\/a>. To obtain the 100-year history, we combined three data sources:<\/p>\n\n\n\n

      1926 \u2013 1962: Monthly US 10-Year Bond Yields<\/p>\n\n\n\n

      1962 \u2013 2002: Daily US 10-Year Bond Yields<\/p>\n\n\n\n

      2002 \u2013 2022: IEF ETF (iShares 7-10 Year Treasury Bond ETF)<\/p>\n\n\n\n

      From 1926 to 1962, we worked with monthly yields, and from 1962 to 2002, with daily yields. Firstly, we transformed the bond yields into total returns. Once we calculated the returns, the second challenge was to transform the monthly returns from 1926 to 1962 into daily ones.<\/p>\n\n\n\n

      We achieved that by extrapolating the daily volatility from the US 3-month T-bills, our unique method we call a \u201cVolatility proxy extrapolation\u201d. In simple terms, we copy the daily volatility from the 3-month T-bills, and plug it in between two monthly data points of US 10-year Treasuries.<\/p>\n\n\n\n

      As mentioned above, this is only a short summary of the methodology. If you are interested in the specific details of any of the steps, please see Extending Historical Daily Bond Data to 100 Years<\/a>.<\/p>\n\n\n\n

      \"\"<\/p>\n\n\n\n

      U.S. 2-Year Bonds (US2Y)<\/span><\/h3>\n

      Accordingly to the methodology for the US 10-year bonds, we created 100-year history also for 2-year bonds. We combined the following sources:<\/p>\n

      1926 – 1934: we used monthly 3M rates and monthly 5y rates, interpolated 2y yield and interpolated daily price series from 2y rate<\/p>\n

      1934 – 1940: we used monthly 3M dealers and 3-5y Notes monthly data, interpolated 2y yield and interpolated daily price series from 2y rate (https:\/\/fraser.stlouisfed.org\/title\/banking-monetary-statistics-1914-1941-38\/part-i-6408<\/a>) pages 460-462<\/p>\n

      1941 – 1962: we used 9-12M issues market yield a 3-5y issues, interpolated 2y yield and interpolated daily price series from 2y rate (https:\/\/fraser.stlouisfed.org\/title\/banking-monetary-statistics-1941-1970-41<\/a>) pages 697-703,<\/p>\n

      1962 – 1976: we interpolated daily 2y yield from 1y\/5y daily rates from FRED and calculated price series<\/p>\n

      1976 – 2002: we used daily data from FRED (https:\/\/fred.stlouisfed.org\/series\/DGS2<\/a>)<\/p>\n

      2002 – 2022: SHY ETF (iShares 1-3 Year Treasury Bond ETF)<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      U.S. 20+ Year Bonds (US20Y)<\/span><\/h3>\n

      Again, accordingly to the methodology for the US 10-year bonds, we created 100-year history also for 20+ year bonds. We combined the following sources:<\/p>\n

      1925 – 1941: we used 10-year yield as proxy with a duration fitting 25y bonds (as TLT ETF has) + we then interpolated monthly data to daily<\/p>\n

      1942 – 1961: we used 20-year yield from FRED (https:\/\/fred.stlouisfed.org\/series\/M13058USM156NNBR<\/a>), then extrapolated 25-yield yield from 10y a 20y, created price series with duration fitting 25y bonds + we then interpolated monthly data to daily<\/p>\n

      1962 – 1976: we used 20-year yield from FRED (https:\/\/fred.stlouisfed.org\/series\/DGS20<\/a>), then extrapolated 25-yield yield from 10y a 20y, created price series with duration fitting 25y bonds + we then interpolated monthly data to daily<\/p>\n

      1976 – 1986: we used 30-year yield from FRED (https:\/\/fred.stlouisfed.org\/series\/DGS30<\/a>), then extrapolated 25-yield yield from 20y a 30y, created price series with duration fitting 25y bonds + we then interpolated monthly data to daily<\/p>\n

      1987 – 1993: daily price data calculated from 25-y yield interpolated from 10-y and 30-y daily yields<\/p>\n

      1993 – 2002: daily price data calculated from 25-y yield interpolated from 20-y and 30-y daily yields<\/p>\n

      2002 – 2022: TLT US (iShares 20+ Year Treasury Bond ETF)<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      World Ex US Bonds
      <\/span><\/h3>\n

      To obtain a 100-year history of World Ex US Bonds, we combined the following data sources:<\/p>\n

      1926 – 1980: we used monthly data from FRED on the 10-year yield of the following countries – UK, GER, JAP, FRA, ITA, NET, CAN, AUS, CHINA, KOREA, and GDP-weighted them into the synthetic 10-year ex-US yield. Countries were added to the portfolio as the data became available. We then interpolated monthly data into daily and created a total return price series. That price series was then converted into the USD currency by using our data series on the <\/span>dollar factor<\/span><\/a>.<\/span><\/p>\n

      1980 – 2007: we used the same methodology as for the 1926-1980 period, but this time, we do not need to perform interpolation as sufficient daily yield data are available.<\/span><\/p>\n

      2007 – 2022: BWX US (SPDR\u00ae Bloomberg International Treasury Bond ETF<\/span>)<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      Corporate Bonds Spread (BAA CORP)<\/strong><\/span><\/h3>\n\n\n\n

      To obtain a 100-year history of Corporate Bonds, we combined the following data sources:<\/p>\n\n\n\n

      1926 \u2013 1985: Monthly Baa Corporate Bond Yield<\/p>\n\n\n\n

      1986 \u2013 2002: Daily Baa Corporate Bond Yield<\/p>\n\n\n\n

      2002 \u2013 2022: spread between LQD ETF (iShares iBoxx $ Investment Grade Corporate Bond ETF) and IEF ETF (iShares 7-10 Year Treasury Bond ETF)<\/p>\n\n\n\n

      Similarly to U.S. 10-Year Bonds, we applied daily volatility proxy extrapolation to the monthly returns for the first data source. Only this time, we used the beta-adjusted equity market returns as the source of volatility. The beta was calculated so that the volatility of the equity market matched the volatility of bonds.<\/p>\n\n\n\n

      During first two periods, we had to again transform bond yields into total returns, in the same fashion as was the case with US Treasury yields described above. To better understand our entire data methodology, we advise reading Extending Historical Daily Bond Data to 100 Years<\/a>.<\/p>\n\n\n\n

      Finally, we utilize the corporate bonds data in the form of a spread against US Treasuries. This way we are able to isolate the credit spread effect and include it separately, in addition to a \u201ccurve\u201d effect represented by US Treasuries.<\/p>\n\n\n\n

      \"\"<\/p>\n\n\n\n

      U.S. Equities (US EQUITIES)<\/strong><\/span><\/h3>\n\n\n\n

      The construction of the U.S. Equities factor was fairly straightforward. We simply combined Fama & French market factor (1926 \u2013 1993) from Fama & French data library<\/a> and SPY (SPDR S&P 500 ETF Trust) ETF\u2019s daily returns (1993 \u2013 2022).<\/p>\n\n\n\n

      \"\"<\/p>\n\n\n\n

      US Equity Sectors\u2019 Spreads (Energy, Technology, Utilities, Financials)<\/strong><\/span><\/h3>\n\n\n\n

      The data for market factors were obtained from Fama & French data library<\/a> specifically from the 12 Industry Portfolios [Daily]. We used:<\/p>\n\n\n\n