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Four asset classes: minimum-variance and maximum-Sharpe portfolios by Merton's closed form

ToolEfficient Frontier
SourceMerton, R. C. (1972). "An Analytic Derivation of the Efficient Portfolio Frontier." Journal of Financial and Quantitative Analysis, Vol. 7, No. 4, pp. 1851–1872; Markowitz, H. (1952). "Portfolio Selection." Journal of Finance, Vol. 7, No. 1, pp. 77–91.
LevelUndergraduate

Problem

Four holdings have expected returns of 7.0, 7.5, 3.5 and 6.0 percent a year and volatilities of 16, 18, 5 and 14 percent, with correlations 0.75 (1–2), 0.10 (1–3), 0.60 (1–4), 0.10 (2–3), 0.55 (2–4) and 0.20 (3–4); the risk-free rate is 2.5 percent. Find the least volatile portfolio, the portfolio with the highest Sharpe ratio, and how far its weights move when one expected return changes by a point. The numbers are illustrative, not forecasts.

Inputs to Enter

  • Typed estimates: US stocks 7.0 / 16, International stocks 7.5 / 18, Bonds 3.5 / 5, Real estate 6.0 / 14 (expected return / volatility, percent per year)
  • Correlation matrix rows: 1, 0.75, 0.10, 0.60; 0.75, 1, 0.10, 0.55; 0.10, 0.10, 1, 0.20; 0.60, 0.55, 0.20, 1
  • Risk-free rate 2.5 %; long-only; no maximum weight
Open the Efficient Frontier tool (loaded with this example)

Answers

  • Sums over the inverse covariance matrix (percent units): A = 0.15695, B = 0.64377, C = 0.041911, D = BC − A² = 0.0023495
  • Minimum variance: return A / C = 3.74 %, volatility √(1 / C) = 4.88 % (below the 5 % of the bonds alone); weights 4.8 / 0.9 / 92.7 / 1.6 %
  • Maximum Sharpe: return (B − A r) / (A − C r) = 4.82 %, volatility 6.67 %, Sharpe ratio √(B − 2 A r + C r²) = 0.348; weights 15.2 / 13.5 / 61.4 / 9.9 %, all positive, so the long-only search returns the closed form
  • Equal weight: 6.00 % at 10.73 %, Sharpe 0.326; risk parity: 4.83 % at 6.73 %, Sharpe 0.346 (weights 13.3 / 12.0 / 59.2 / 15.5 %)
  • Sensitivity: US stocks +1 point takes its weight from 15.2 to 33.2 %; bonds −1 point takes theirs from 61.4 % to zero

Notes

The tool's default inputs are this case; press Submit. Choose Shorting allowed to see A, B, C and D printed with the closed-form equations. The case is the tool's own default, checked against numpy and scipy (test_efficient_frontier.php), not an example printed in either paper.