Maximum Sharpe · at most 40% in any asset · optimal 5 assets · monthly, Nov 1996 to Oct 2026 · Ledoit–Wolf covariance · risk-free T-Bills
Summary
The weights to hold, and what they are expected to deliver
Expected return
7.0% a year
Historical mean returns
Volatility
8.3%
Ledoit–Wolf covariance
Sharpe ratio
0.57
Against T-Bills at 2.3%
Diversification ratio
1.52
2.8 assets in effect
Largest weight
40.0%
Bonds
The weights
40%
38%
22%
Bonds 40.0%US Equity 38.3%Gold 21.7%
Asset
Weight
Expected return
Volatility
Share of risk
Bonds
40.0%
4.0%
6.1%
11.2%
US Equity
38.3%
10.0%
16.9%
64.4%
Gold
21.7%
7.4%
16.5%
24.4%
Real Estate
–
9.1%
20.5%
–
Intl Equity
–
7.1%
17.6%
–
At most 40% in any asset. Historical mean expected returns and the Ledoit–Wolf covariance, annualised by 12; the volatilities and shares of risk come from the covariance. Real Estate and Intl Equity are not held.
Risk
Where the risk comes from, and how the assets move together
Where the risk comes from
WeightShare of risk
US Equity is 38% of the weight but 64% of the risk; Bonds is 40% of the weight and 11% of the risk. An asset's share of risk is its weight times its covariance with the portfolio, over the portfolio's variance. Real Estate and Intl Equity are not held.
Correlation
US Equity
Intl Equity
Bonds
Gold
Real Estate
US Equity
1.00
0.82
−0.02
0.09
0.68
Intl Equity
0.82
1.00
−0.04
0.16
0.64
Bonds
−0.02
−0.04
1.00
0.21
0.20
Gold
0.09
0.16
0.21
1.00
0.10
Real Estate
0.68
0.64
0.20
0.10
1.00
From the covariance of the monthly returns, Nov 1996 to Oct 2026.
Risk and return
Where the weights sit among the portfolios the limits allow
The efficient frontier
The weightsEqual weightEach asset
The frontier holds the portfolios with the largest expected return for each volatility, under the same limits. The dashed line starts at T-Bills, 2.3%, and passes through the weights; its slope is their Sharpe ratio.
History
How the weights would have done on the returns they were fitted to
On the history it was fitted to
The weights 763Equal weight 817
Return
7.0% a year
Equal weight 7.3%
Volatility
8.2%
Equal weight 11.1%
Sharpe ratio
0.58
Equal weight 0.48
Maximum drawdown
−25.8%
Equal weight −37.7%
In sample: the weights were chosen knowing these returns, so this flatters them. Held from the start and rebalanced quarterly, without trading costs. To test the method on returns it had not seen, backtest it with a holdout or walk-forward validation.
Method
How the weights were found
How the weights were found
Method
Maximum Sharpe: the largest expected return in excess of the risk-free rate per unit of volatility, solved as a quadratic programme after the Charnes–Cooper transformation.
Data
US Equity, Intl Equity, Bonds, Gold and Real Estate; T-Bills as the risk-free asset. Monthly returns, Nov 1996 to Oct 2026.
Estimates
Historical mean expected returns and the Ledoit–Wolf covariance, annualised by 12.
Limits
At most 40% in any asset.
Solver
Interior point, optimal after 10 iterations.
Every figure is computed by Finalytics 1.0.0-rc.1 from the assets' returns; the same inputs give the same report, identified by the input digest cb784a063906.
