Adding a riskier asset can lower your portfolio’s risk

1. Can the correlations all hold at once?

Yes. Three correlations can hold together when the determinant of their correlation matrix is positive:

det = 1 + 2(ρEB)(ρER)(ρBR) − ρEB² − ρER² − ρBR²
    = 1 + 2(0.2)(0.5)(0.1) − 0.04 − 0.25 − 0.01
    = 1 + 0.02 − 0.30 = 0.72  > 0

The leading minors are 1, then 1 − 0.2² = 0.96, then 0.72. All are positive, so the matrix is valid. As a numerical check, its eigenvalues are 0.489, 0.932 and 1.578, which sum to 3 as they must.

You can also see how much room there is. Given the other two, each correlation must lie within ρ₁ρ₂ ± √((1−ρ₁²)(1−ρ₂²)):

All three are well inside their ranges.

2. Does some of the riskier asset lower each pair’s volatility?

Why the test works: start fully in the lower-volatility asset L and move a small weight w into the higher-volatility asset H. The pair’s variance is (1−w)²σL² + w²σH² + 2w(1−w)ρσLσH. At w = 0 its slope is 2σL(ρσH − σL). That slope is negative exactly when ρ < σL/σH.

To show how big the effect is, I also found the lowest-volatility mix of each pair. Its weight in the riskier asset is w* = (σL² − ρσLσH)/(σL² + σH² − 2ρσLσH), and its variance is σL²σH²(1−ρ²)/(same denominator). In %² units:

• Equities–bonds: w* = (36 − 21.6)/(360 − 43.2) = 14.4/316.8 = 4.5% equities. σ² = 36×324×0.96/316.8 = 35.35, so σ = 5.95%, versus 6.00% for bonds alone.
• Equities–real assets: w* = (144 − 108)/(468 − 216) = 36/252 = 14.3%. σ² = 144×324×0.75/252 = 138.86, so σ = 11.78%, versus 12.00%.
• Real assets–bonds: w* = (36 − 7.2)/(180 − 14.4) = 28.8/165.6 = 17.4%. σ² = 36×144×0.99/165.6 = 30.99, so σ = 5.57%, versus 6.00%.

3. Comparison with published estimates

Two caveats apply to all of this:

• Horizon. Only State Street publishes a one-year forecast. The other sources forecast 10 years or longer.
• Compound returns. Where a source gives a compound (geometric) return, I converted it to arithmetic with arithmetic ≈ geometric + σ²/2.

The sources:

• State Street, Q3 2026 forecasts. Figures are as of June 30, 2026, and cover 1-year, 3–5-year and 10+-year horizons. Returns are arithmetic averages, gross of fees, in local currency. For volatility I used their ordinary annual standard deviation, not their long-horizon figure.
• Verus 2026 Capital Market Assumptions. These are 10-year forecasts as of September 30, 2025. Volatility is long-run history adjusted for autocorrelation, and correlations are what was realized over the past decade.
• Vanguard Capital Markets Model. The June 30, 2026 run: 10-year, nominal, compound returns in US dollars.
• J.P. Morgan 2026 Long-Term Capital Market Assumptions, covering 10–15 years (press release, summary). That edition came out on October 20, 2025. I didn’t find a 2027 edition.

Global equities: 7% return is inside the range

• State Street: MSCI ACWI at 7.0% for one year and 6.8% for 10+ years.
• Verus: ACWI at 7.3% arithmetic, 6.0% geometric.
• J.P. Morgan: ACWI at 6.7%, compound. Converting: 6.7 + 16.6²/200 = 8.1%. I used Verus’s 16.6% volatility because I didn’t find J.P. Morgan’s.
• Vanguard: no global figure. It gives 4.2–6.2% for US equities, 4.5–6.5% for developed markets outside the US, and 2–4% for emerging markets.
  • I weighted the midpoints by Verus’s ACWI weights of 67/22/11: 0.67×5.2 + 0.22×5.5 + 0.11×3.0 = 5.0% compound.
  • Converting: 5.0 + 1.38 = 6.4% arithmetic. This combination is mine, not Vanguard’s.

The range is 6.4–8.1%, so your 7% is not flagged.

Global equities: 18% volatility is above the range (flagged)

State Street gives 14.8% and Verus gives 16.6%. Yours is 1.4 to 3.2 points higher. State Street’s figure is in local currency, so it leaves out the currency risk an unhedged dollar investor takes on the non-US part.

Bonds: 3% return is below every source (flagged)

• State Street: US Aggregate at 4.3% for one year, and US cash at 3.8%.
• Verus: 4.8% arithmetic.
• J.P. Morgan: 4.8%. Converting: 4.8 + 4.9²/200 = 4.9%.
• Vanguard: 3.8–4.8% as of late 2025. The June 2026 run raised this modestly, but the page doesn’t state the new figure.
• Starting yield:
  • An Aggregate index fund showed a 5.00% yield to worst at August 31, 2026.
  • Treasury yields rose across the curve in September, and by late September the index’s average yield to maturity was about 5.5%.

The range is 3.8–5.5%, and your 3% is 0.8 to 2.5 points below it.

What 3% implies: a one-year return is roughly the yield minus duration times the change in yield. The index’s duration is 5.88 years. So:

yield 5.0%:  Δy = (5.0 − 3.0)/5.88 = 0.34 points
yield 5.5%:  Δy = (5.5 − 3.0)/5.88 = 0.43 points

Your 3% is therefore roughly a forecast that yields rise 0.3 to 0.4 points over the year. This ignores roll-down, convexity and changes in credit spreads.

Your 3% is also below the cash forecasts: State Street’s 3.8%, and cash yields of about 4%. If your explorer includes a risk-free rate, bonds will have a negative excess return.

Bonds: 6% volatility is above the range (flagged)

State Street gives 5.1% and Verus gives 4.9%.

In yield terms, dividing by the 5.88-year duration:

• Yours: 6/5.88 = 1.02 points of annual yield movement.
• The sources: 4.9/5.88 = 0.83 and 5.1/5.88 = 0.87.

Duration is now above its long-term average of 4.99 years, so volatility figures based on long-run history may understate today’s sensitivity to yields.

Real assets

No source forecasts your mix, so I built it from Verus’s components. Verus is the only source here that gives both volatilities and correlations. I used equal thirds and built two versions: one with listed real estate (REITs) and one with private core real estate.

Verus’s inputs:

• Returns and volatility: REITs 8.8% / 18.0%; core real estate 7.9% / 11.7%; global infrastructure 9.5% / 16.5%; commodities 7.7% / 15.8%.
• Correlations among the components: REITs–infrastructure 0.8, REITs–commodities 0.4, infrastructure–commodities 0.5; core real estate–infrastructure 0.2, core real estate–commodities 0.2.
• Correlations with global equity: REITs 0.8, infrastructure 0.8, commodities 0.4, core real estate 0.3.
• Correlations with US core bonds: REITs 0.5, infrastructure 0.4, commodities 0.0, core real estate 0.1.

The listed version:

σ² = (1/9)[18.0² + 16.5² + 15.8² + 2(0.8)(18.0)(16.5) + 2(0.4)(18.0)(15.8) + 2(0.5)(16.5)(15.8)]
   = (1/9)[324 + 272.25 + 249.64 + 475.2 + 227.52 + 260.7] = 1809.31/9 = 201.0  →  σ = 14.2%
ρ with equities = [16.6 × (0.8×18.0 + 0.8×16.5 + 0.4×15.8)/3] / (16.6 × 14.18) = 187.7/235.4 = 0.80
ρ with bonds    = [4.9 × (0.5×18.0 + 0.4×16.5 + 0×15.8)/3] / (4.9 × 14.18) = 25.48/69.48 = 0.37

The private version:

σ² = (1/9)[136.89 + 272.25 + 249.64 + 77.22 + 73.94 + 260.7] = 1070.6/9 = 119.0  →  σ = 10.9%
ρ with equities = [16.6 × (3.51 + 13.2 + 6.32)/3] / (16.6 × 10.91) = 127.4/181.1 = 0.70
ρ with bonds    = [4.9 × (1.17 + 6.6)/3] / (4.9 × 10.91) = 12.69/53.46 = 0.24

Private real estate is valued by appraisal, so it looks smoother than it is. Verus de-lags its returns for correlations and sets its volatility at 65% of REIT volatility. Treat the private version as the low end. Verus rounds its correlations to one decimal, so the derived values are uncertain by about ±0.05.

Real assets: 5% return is narrowly below the range (flagged)

• State Street, one year: REITs 4.3%, commodities 6.4%, with no infrastructure figure. Average: (4.3 + 6.4)/2 = 5.35%.
• J.P. Morgan: US core real estate 8.2%, global core infrastructure 6.5%, commodities 4.6%.
  • Average: 19.3/3 = 6.43% compound. Adding 14.18²/200 = 1.0 gives about 7.4%.
  • J.P. Morgan’s private-bank summary lists slightly different figures, 8.0%, 7.1% and 4.1%. I used the press release.
• Verus: listed version (8.8 + 9.5 + 7.7)/3 = 8.67%; private version (7.9 + 9.5 + 7.7)/3 = 8.37%.

The range is 5.35–8.67%. Your 5% is 0.35 below the low end, and the low end is the only one-year source.

Real assets: 12% volatility is inside the range

Your 12% falls within the 10.9–14.2% derived above. For reference, State Street’s components are REITs 17.9%, commodities 17.0%.

Correlations: all three are below the published values (flagged)

• Equities–bonds, yours 0.2: Verus gives 0.4 between global equity and US core bonds, realized over the past decade. It’s the only published number I found, so the “range” is a single point.
• Equities–real assets, yours 0.5: derived range 0.70–0.80.
• Bonds–real assets, yours 0.1: derived range 0.24–0.37.

Summary

How much your results depend on the flagged inputs

Step 2 changes for two of the three pairs if you use the published correlations.

• Equities–bonds: 0.4 is above the 0.333 threshold, so adding equities to bonds would no longer lower volatility.
• Equities–real assets: 0.70–0.80 is above 0.667, so the same is true there.
• Bonds–real assets: 0.24–0.37 is still below 0.5, so that result holds.

Portfolio volatility is understated. As a measuring point only, not a suggestion, take equal thirds:

σ² = (1/9)[324 + 36 + 144 + 2ρEB(18)(6) + 2ρER(18)(12) + 2ρBR(6)(12)]
Yours:              (1/9)[504 + 43.2 + 216 + 14.4]  = 777.6/9 = 86.4   → 9.30%
Published, listed:  (1/9)[504 + 86.4 + 344.3 + 52.8] = 987.5/9 = 109.7 → 10.48%
Published, private: (1/9)[504 + 86.4 + 304.1 + 34.1] = 928.6/9 = 103.2 → 10.16%

Your correlations understate this mix’s volatility by 0.9 to 1.2 points.

The low-risk end of the frontier looks worse under your numbers. The equal-thirds mix returns (7+3+5)/3 = 5.00% with your bond forecast. With the published bond range it would return (7+3.8+5)/3 = 5.27% to (7+5.5+5)/3 = 5.83%. Bonds’ return per unit of volatility is 3/6 = 0.50 under your numbers and 4.8/4.9 = 0.98 under Verus’s.

4. A scenario for an equity decline

The evidence

• Correlations among risky assets rise in declines. Longin and Solnik studied monthly data on the five largest stock markets from 1958 to 1996. They found correlation rises in bear markets but not in bull markets, and that it follows the market’s direction rather than its volatility. In 2008, commodities fell 37% alongside global equities, which fell roughly 40%.
• The stock–bond correlation depends on what causes the decline.
  • AQR finds that stocks and bonds react in opposite directions to growth news and in the same direction to inflation news. Their simple model explains about 70% of the long-run variation in the US stock–bond correlation.
  • The correlation was positive in the 1970s through the 1990s and negative for the first two decades of this century.
  • In 2022, the Aggregate index fell 13%, its first double-digit loss, while stocks also fell.
  • In 2008, bonds were the best-performing asset class, up 5.2%.
• Commodities behave differently in inflation shocks. AQR found commodities have had low correlation with both stocks and bonds, and diversified more when inflation uncertainty was high. So in an inflation shock, the commodity third of your real assets partly offsets rate-driven losses.
• Caution about measuring stress correlations. Even when the true correlation is constant, correlation measured during volatile periods comes out higher than in calm ones. If you also raise volatilities in the scenario, don’t also take correlations estimated from high-volatility periods, or you count the effect twice.

Proposed correlations

There are two versions, because the sign of the stock–bond correlation depends on the cause of the decline.

How I chose the numbers:

• 0.85: above the published normal-period range of 0.70–0.80, following Longin and Solnik. The exact amount is my judgment.
• 0.4 in A: listed real estate’s realized 0.5 with bonds, pulled down by commodities’ 0.0.
• The B values: bonds rose while real assets fell in 2008. The sizes are my judgment.

Checking the scenarios

Both can hold at once:

A: det = 1 + 2(0.5)(0.85)(0.4) − 0.25 − 0.7225 − 0.16 = 1 + 0.34 − 1.1325 = 0.2075 > 0
B: det = 1 + 2(−0.2)(0.85)(0)  − 0.04 − 0.7225 − 0    = 0.2375 > 0

The step-2 test under each:

Equal-thirds volatility under each:

A: (1/9)[504 + 108 + 367.2 + 57.6] = 1036.8/9 = 115.2 → 10.73%
B: (1/9)[504 − 43.2 + 367.2 + 0]   = 828/9    = 92.0  → 9.59%
Yours, for comparison: 9.30%

Equity volatility also tends to rise in a decline, which lowers the step-2 thresholds. For example, at 25% equity volatility the equities–bonds threshold would be 6/25 = 0.24.

5. What I assumed

• Returns: arithmetic, nominal, total return, before fees, in US dollars, unhedged, for October 14, 2026 to October 2027.
• Indexes: global equities = MSCI ACWI; bonds = Bloomberg US Aggregate.
• Real assets: equal thirds of real estate, infrastructure and commodities. You didn’t say whether the real estate is listed or private, so I ran both versions.
• Compound to arithmetic: arithmetic ≈ geometric + σ²/2, using the source’s own volatility where it gives one, and Verus’s where it doesn’t.
• Bond break-even: return ≈ yield − duration × change in yield, ignoring roll-down, convexity and credit spreads.
• Vanguard global equity: the 5.0% figure is my weighting of Vanguard’s regional ranges using Verus’s ACWI weights.
• Published correlations: Verus’s realized correlations over the past decade, rounded to 0.1. They are not forward-looking forecasts.
• Option-implied volatility: I didn’t use it, because I couldn’t find a current one-year global figure.