US Equity vs US Multi-Asset Investing: Risk-Adjusted Evidence

This project compares a US equity-only portfolio against US multi-asset constructions that allocate across US equities, investment-grade bonds, and listed real estate under long-only constraints and quarterly rebalancing.

The objective is not only to compare cumulative outcomes, but to test whether diversification within the US market improves utility-relevant performance once volatility and drawdown are explicitly penalized.

Key metrics include expected return ${\mu }_p=\mathbb{E}[r_{p,t}]$, portfolio variance ${\sigma }_p^2=w^{\top }\Sigma w$, and Sharpe ratio $S_p=({\mu }_p-r_f)/{\sigma }_p$. Empirical insights appear below.

Modern portfolio theory (Markowitz 1952) frames asset allocation as a mean–variance trade-off. An investor chooses a weight vector $w$ to maximize expected utility of return while recognizing that risk is quadratic in those weights.

For a universe of US risky assets with expected-return vector $\mu$ and covariance matrix $\Sigma$, the one-period portfolio return and risk are

$$r_p=w^{\top }r,{\mu }_p=w^{\top }\mu ,{\sigma }_p^2=w^{\top }\Sigma w.$$

Long-only strategic allocation imposes $w_i\ge 0$ and ${\sum }_i{w}_i=1$. The classical Markowitz program is then

$$\underset{w}{\min }{w}^{\top }\Sigma w\text{s.t.}{w}^{\top }\mu ={\mu }_{\star },1^{\top }w=1,w\ge 0.$$

Varying the target return ${\mu }_{\star }$ traces the efficient frontier. Equity-only concentration places the investor near the highest-volatility corner; multi-asset mixes occupy interior points where diversification reduces ${\sigma }_p$ for a given expected return when correlations are imperfect.

With a risk-free asset, Tobin’s separation theorem introduces the capital market line. The Sharpe ratio of a risky portfolio becomes the slope-based ranking device

$$S_p=\frac{{\mu }_p-r_f}{{\sigma }_p}.$$

Because investors care asymmetrically about losses, we also report the Sortino ratio, which replaces total volatility with downside semi-deviation

$${\sigma }_p^{-}=\sqrt{\mathbb{E}[\min (r_p-T,0{)}^2]},S{o}_p=\frac{{\mu }_p-r_f}{{\sigma }_p^{-}},$$

where $T$ is a minimum acceptable return (often set to $r_f$ or zero). Breadth of diversification can be summarized by the diversification ratio

$$DR(w)=\frac{w^{\top }\sigma }{\sqrt{w^{\top }\Sigma w}},$$

with $\sigma$ the vector of standalone asset volatilities. Larger $DR$ indicates that portfolio risk is smaller than a volatility-weighted sum of constituents.

Relative to a US market proxy $r_m$, Jensen’s alpha from the CAPM identity is

$${\alpha }_p=r_p-[r_f+{\beta }_p(r_m-r_f)],{\beta }_p=\frac{\mathrm{Cov}(r_p,r_m)}{\mathrm{Var}(r_m)}.$$

Ending wealth for a unit initial investment compounds at the geometric mean

$$W_T=\prod _{t=1}^T(1+r_{p,t})=(1+G{)}^T,G=(\prod _{t=1}^T(1+r_{p,t}){)}^{1/T}-1.$$

Volatility drag implies that for similar arithmetic means, lower ${\sigma }_p$ typically raises geometric growth $G\approx \mu -\frac{1}{2}{\sigma }^2$, which is the economic rationale for multi-asset allocations even when raw average returns look similar.

The empirical design uses weekly total-return series for US asset-class proxies covering equity (broad market), investment-grade aggregate bonds, and listed real estate (REITs), sampled from mid-2002 through end-2017 so that the Global Financial Crisis is nested inside the history.

Index construction. Each portfolio is rebased to 100 at inception. Between rebalance dates, wealth grows at the asset-weighted weekly return;

$$I_t=I_{t-1}(1+{\sum }_i{w}_{i,t-1}{r}_{i,t}).$$

Weights are restored to strategic targets every calendar quarter, which prevents equity drift from converting a balanced mix into a disguised equity sleeve.

Portfolio menu. We evaluate five nested constructions: (i) 100% US equity; (ii) US 60/40 equity–bonds; (iii) low equity (30% equity, remainder split equally between bonds and REITs); (iv) balanced (50% equity, 25% bonds, 25% REITs); (v) high equity (70% equity, 15% bonds, 15% REITs). Cash is omitted so that comparisons isolate the risky opportunity set.

Path metrics. For each construction we compute annualized geometric return $G$, annualized volatility $\sigma \sqrt{52}$ from weekly returns, maximum drawdown

$$\mathrm{MDD}=\underset{t}{\min }(\frac{I_t}{{\max }_{\tau \le t}{I}_{\tau }}-1),$$

Sharpe and Sortino ratios, CAPM beta and alpha versus the equity-only sleeve, and rolling one-year Sharpe paths.

Inference. Sharpe differentials use the Jobson–Korkie (1981) statistic (Memmel 2003 adjustment),

$$z=\frac{{\hat{S}}_A-{\hat{S}}_B}{\widehat{\mathrm{SE}}({\hat{S}}_A-{\hat{S}}_B)},$$

with additional Mann–Whitney tests on rolling Sharpe samples and bootstrap mean differences for downside semi-deviation. CAPM alpha significance is assessed via the usual OLS $t$-ratio.

The sections that follow show the cumulative wealth paths, risk–return map, hypothesis tests, rolling Sharpe diagnostics, and drawdown paths produced by this design.

Before comparing outcomes, it helps to visualise the strategic weights applied under each construction. Equity-only is fully concentrated; low / balanced / high equity split the residual sleeve between US bonds and REITs; 60/40 is the classic equity–bond blend without listed real estate.

The stacked bar chart below labels each sleeve (US Equity, US Bonds, US REITs) so the composition differences are explicit.

Diversification requires imperfect co-movement. The correlation heatmap of US Equity, US Bonds, and US REITs over the sample shows bonds as the primary diversifier versus equities, while REITs retain a sizeable equity-like correlation.

This structure explains why balanced mixes reduce portfolio variance without requiring exotic alternative assets.

The figure tracks wealth of a $100 investment under each strategic mix. Even when equity-only posts competitive long-run growth, diversified sleeves often finish with similar or higher ending wealth while traversing a smoother path—consistent with reduced volatility drag.

Axes are labelled by date (horizontal) and wealth index (vertical). Stress episodes such as 2008 compress the equity-only index more sharply; multi-asset mixes cushion but do not eliminate drawdowns.

Placing each portfolio in $(\sigma ,\mu )$ space makes the diversification trade-off concrete. Equity-only sits at the rightmost (highest volatility) point; low-equity sits leftmost. Balanced and 60/40 mixes occupy intermediate points that typically dominate equity-only on a Sharpe-ratio basis when correlations remain below one.

Bubble size scales with Sharpe. Axis labels show annualized volatility (%) and annualized return (%). The pattern is a flattening frontier: incremental equity beyond roughly 50–60% adds volatility faster than it adds compounded return.

A labelled bar chart ranks constructions on Sharpe, CAGR, Sortino, and Calmar. Toggle the metric control to re-rank the same sleeves and see which dimensions drive the diversification case.

Stepping equity from 0% to 100% in 10-point increments (residual split between bonds and REITs) traces how return, volatility, and the reward-to-risk ratio evolve.

The dual-axis chart labels equity weight on the horizontal axis, return/vol on the left axis, and reward-to-risk on the right. The ratio peaks in the moderate-equity region and declines toward 100% equity.

We test multiple statements rather than a single headline metric. Primary alternatives ask whether US Balanced improves Sharpe and compresses drawdowns relative to equity-only, and whether high-equity multi-asset still beats pure equity on downside risk.

Null forms include $H_0:S_{\mathrm{bal}}-S_{\mathrm{eq}}\le 0$, $H_0:{\mathrm{MDD}}_{\mathrm{bal}}-{\mathrm{MDD}}_{\mathrm{eq}}\ge 0$, and $H_0:{\alpha }_{\mathrm{bal}}\le 0$. Supported alternatives require risk-adjusted improvement for the diversified sleeve.

The table below reports effect sizes, test statistics, $p$-values, and verdicts for seven hypotheses. Not every claim clears 5% significance; joint directional consistency across Sharpe, drawdown, and rolling metrics is the stronger inferential object.

Always-on sample Sharpe ratios can hide regime dependence. Rolling one-year Sharpe paths show whether the diversified edge persists outside a few lucky windows.

The labelled line chart includes all constructions. Multi-asset Sharpe remains higher and less negative through the crisis window, supporting time-consistent risk-adjusted benefits.

Companion to rolling Sharpe, the annualized volatility paths show how multi-asset sleeves compress realised risk through time—especially around the GFC—while equity-only spikes toward 25–30% annualised weeks.

Maximum drawdown is a path-dependent measure that investors experience consultatively more acutely than variance. For an index level $I_t$,

$${\mathrm{DD}}_t=\frac{I_t}{{\max }_{\tau \le t}{I}_{\tau }}-1,\mathrm{MDD}=\underset{t}{\min }{\mathrm{DD}}_t.$$

The drawdown chart (date on the x-axis, drawdown % on the y-axis) shows that equity-only reaches deeper troughs while balanced and low-equity sleeves cap losses materially.

Within this US sample, a well-designed multi-asset mix—especially near a balanced equity weight—delivers stronger Sharpe efficiency and shallower stress drawdowns than equity-only concentration, without systematically forfeiting ending wealth.

The theory implies that imperfect asset correlations create a free lunch in variance space; the empirical tests show that lunch is partially harvested under realistic quarterly rebalancing and long-only constraints.

This project is a research artifact for quantitative exploration and education; it is not investment advice.

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