Diversification With Many Assets

A Mean–Variance Analysis of U.S. Stock and Bond Returns, 1927–2022

Executive Summary

This report estimates the maximum Sharpe ratio portfolio of the SP500, long-term corporate bonds (LTC), and long-term government bonds (LTG) across two halves of the historical sample — 1927–1971 and 1972–2022. The two portfolios are dramatically different. The first half produces a heavily levered position in corporate bonds funded by a large short position in government bonds, while the second half produces a more conventional, all-positive allocation across stocks, corporates, and governments. The differences are driven by changes between the two periods — different inflation environments, a strengthening stock-bond correlation, and a narrowing of the spread between corporate and government bond risk-adjusted returns.

The most important takeaway is that mean-variance optimization is highly sensitive to its inputs: a moderate shift in the estimated LTG return between the two halves was enough to swing the optimal LTG weight by more than 60 percentage points. This sensitivity means that any single tangency portfolio reflects the past it was estimated on, not necessarily the future. For forward-looking allocation, I would hone in a mix appropriate for my time horizon, apply shrinkage to the inputs, add constraints, stress-test the portfolio against multiple historical periods, and rebalance on a schedule rather than on a whim.

1. Introduction

This report estimates the maximum Sharpe ratio portfolio across two halves of the historical sample — 1927–1971 (Period 1) and 1972–2022 (Period 2) — using the SP500, long-term corporate bonds (LTC), and long-term government bonds (LTG), with Treasury bills as the risk-free asset. After comparing the two sub-period portfolios, I discuss what their differences reveal about the sensitivity of mean-variance optimization to its inputs and how I would approach allocation over the next 20 years.

2. Data and Methodology

After estimating the expected returns and covariance matrix for the SP500, LTC, and LTG using the first half of the data (1927–1971), I computed the maximum Sharpe ratio portfolio using the closed-form tangency formula rather than Excel's Solver tool. From there, I derived the inverse covariance matrix, the optimal weights, the portfolio variance via the box method, and the resulting Sharpe ratio for the overall portfolio. I then repeated these steps using the second half of the sample (1972–2022).

3. Results

The two sets of results varied significantly. Sub-period summary statistics are reported in Appendix 1, the sample covariance matrices in Appendix 2, and the maximum-Sharpe portfolio weights and performance statistics in Appendix 3.

4. Analysis & Discussion

4.1 Comparison of the Two Sub-Period Portfolios

The two portfolios differ dramatically (see Appendix 3). Three factors explain the differences. First, the inflation environments were very different across the two periods, which directly affected bond returns. Period 1 was a low-inflation environment, producing low bond volatility and modest bond returns. Period 2 included the ‘great inflation’ and the subsequent disinflation, which drove both bond returns and bond volatility higher. Second, the stock-bond correlation changed. In Period 1, bonds were sometimes negatively correlated with stocks, which made them effective diversifiers. In Period 2, the stock-bond correlation strengthened, which compressed the diversification benefit and reduced the optimal bond weight in the overall portfolio. Third, the corporate-versus-government spread shifted. In Period 1, LTC dominated LTG on a Sharpe-per-unit-risk basis. In Period 2, LTC and LTG offered similar risk-to-reward, so both received positive weights.

4.2 Sensitivity of Mean–Variance Analysis to Inputs

The differences between the two portfolios highlight how sensitive mean-variance optimization is to its inputs, especially expected returns. The shift in average LTG return between the two halves of the sample (Appendix 1) drove a swing of more than 60 percentage points in the optimal LTG weight (Appendix 3). The same asset moved from a large short position to a moderate long position based on a moderate change in one input. This means that any single tangency portfolio is the optimal portfolio for the historical window it was estimated on, not necessarily the right portfolio going forward.

4.3 Forward-Looking Asset Allocation Framework

When allocating across these asset classes, I would start with 70/20/10 stocks/corporates/governments. This split assumes a long-horizon, growth-orientation — a sixty-year-old would allocate differently than a twenty-two-year-old college student. From there, I would apply shrinkage to the inputs: shrinking μ toward implied returns and shrinking Σ toward a structured target, both of which stabilize the resulting portfolio weights. I would then add constraints such as position caps. Next, I would stress-test the portfolio against historical periods; first running it against Period 1 to see how it performs if that environment were to repeat, then against Period 2, and against any other time-periods that could plausibly return. Finally, rather than continuously second-guessing the portfolio after committing to an investment approach, I would maintain a rebalancing schedule under predefined rules.

5. Conclusion

The two sub-period portfolios differ dramatically, driven primarily by changes in the bond market. Particularly, different inflationary environments, a stronger stock-bond correlation in the second half, and a narrowing of the corporate-versus-government risk-adjusted spread are considerable changes. These differences show that mean-variance optimization is highly sensitive to its inputs and produces unstable weights when applied mechanically to historical data. For that reason, practical asset allocation should rely on input shrinkage, disciplined constraints, and consistent rebalancing.

Appendix

Appendix 1. Sub-Period Summary Statistics

Source: SP500_and_Bond_returns spreadsheet (Canvas).

Appendix 2. Sample Covariance Matrices

Period 1: 1927–1971

Notes: Negative values are shown in parentheses.

Period 2: 1972–2022

Notes: Negative values are shown in parentheses.

Appendix 3. Maximum-Sharpe (Tangency) Portfolio: Weights and Performance

Notes: Negative weights shown in parentheses.

@misc{wisniewski2026diversification,
  author = {Wisniewski, Kyle},
  title = {Diversification With Many Assets: A Mean–Variance Analysis of U.S. Stock and Bond Returns, 1927–2022},
  year = {2026},
  month = {may},
  howpublished = {\url{https://www.kylewisniewski.com/library/diversification-with-many-assets}},
  note = {Course paper}
}