1. From a letter sent in 1703 by Gottfried Leibniz (1646–1716), a German mathematician and philosopher, to Jacob Bernoulli (1654–1705), a Swiss mathematician who made significant contributions in the field of probability.
2. Many reasons exist for an investment not to be held to maturity, some due to the investor and some to the issuer. A few of these reasons include the instrument’s maturity is longer or shorter than the investment horizon, the instrument is paid back early, or the issuer recalls the instrument.
3. Aked and Ko (2017) also point out that investors should be aware that although the average volatility over a decade is low, which is meaningful for generating expectations of return, over a 10-year period investors will experience, on average, 14.9% volatility each year.
4. We use nominal returns because the bond yield is stated in nominal terms and includes an expected market inflation rate. The relationship of yield to the real return of bonds is much weaker because the market-implied inflation rate at the purchase date could be vastly different from realized inflation over the 10-year horizon. Also, as noted earlier, if the bond was a zero-coupon bond, held for 10 years, the return would be the same as the starting yield, a perfect correlation.
5. For implementation, we replace the minimum expected return with the 5th percentile rank and the maximum expected return with the 95th percentile rank, consistent with Asness, Ilmanen, and Maloney (2015).
6. Although we readily concede that no backtest can ever truly be out of sample when using data from the past known to the modeler at the time the model is created.
Aked, Michael, and Amie Ko. 2017. “Time Diversification Redux.” Research Affiliates (August).
Arnott, Robert, and Peter Bernstein. 2002. “What Risk Premium Is ‘Normal’?" Financial Analysts Journal, vol. 58, no. 2 (March/April):64–85.
Asness, Cliff, Antti Ilmanen, and Thomas Maloney. 2015. “Market Timing Is Back in the Hunt for Investors.” Institutional Investor (November 11).
Baetje, Fabian, and Lukas Menkhoff. 2016. “Equity Premium Prediction: Are Economic and Technical Indicators Unstable?” DIW Berlin Discussion Paper No. 1552 (February 24). Available at SSRN.
Bernstein, Peter. 1996. Against the Gods: The Remarkable Story of Risk. New York, NY: John Wiley & Sons, Inc.
Brightman, Chris, Jim Masturzo, and Jonathan Treussard. 2014. “Our Investment Beliefs.” Research Affiliates (October).