Optimal Portfolio Choice

Two risky assets, one decision

Last lesson we computed portfolio mean and variance for any weighting of two assets. Now we ask: which weighting is best?

In mean-variance framing, “best” means: for a given expected return, lowest variance; or equivalently, for a given variance, highest expected return. The set of weights that achieve this trade- off optimally traces out a curve in (risk, return) space.

For two assets, as we vary $w_1$ from 0 to 1:

• At $w_1 = 0$: portfolio is 100% asset 2; we sit at $(\sigma_2, E[R_2])$.
• At $w_1 = 1$: portfolio is 100% asset 1; we sit at $(\sigma_1, E[R_1])$.
• In between: a curve that depends on $\rho_{12}$.

The expected return is linear in the weight, but the variance is not:

$$E[R_p] = w E[R_1] + (1-w) E[R_2], \quad \sigma_p^2 = w^2\sigma_1^2 + (1-w)^2\sigma_2^2 + 2w(1-w)\rho_{12}\sigma_1\sigma_2.$$

The cross term carries $\rho_{12}$. Lower $\rho_{12}$ shrinks it, so the curve bows further to the left (lower risk for a given return). At $\rho = -1$ the curve kinks and can reach $\sigma = 0$: two perfectly negatively correlated assets can be combined into a riskless position.

Drag the correlation slider and watch the frontier bow. Then note how the risk-free rate picks out a single tangent portfolio:

E[R₁]: 12.0%σ₁: 20.0%E[R₂]: 7.0%σ₂: 10.0%Correlation ρ: 0.20Risk-free Rf: 3.0%

Tangent Sharpe

0.55

Tangent: weight in asset 1

37%

Min-variance σ

9.6%

Drag correlation ρ down toward -1 and watch the blue frontier bow left: the same expected return becomes reachable at lower risk. That leftward bow is diversification. The green capital market line, drawn from the risk-free rate through the tangent portfolio, is the best risk-return trade-off any investor can reach by mixing the risk-free asset with that one tangent portfolio.

Baseline: asset 1 is higher risk and return (12%, σ 20%), asset 2 lower (7%, σ 10%), correlation 0.2, risk-free rate 3%. Lower ρ toward -1 to see the bow deepen; the red tangent point (set by the risk-free rate) is the highest-Sharpe risky mix.

The efficient frontier with many assets

Now consider many assets. For each level of expected return, find the portfolio weights that minimize variance. The locus of those minimum-variance portfolios is the efficient frontier.

Above the minimum-variance point on the frontier: every additional unit of risk gets you a positive bump in expected return — these are “efficient” portfolios.

Below the minimum-variance point: you could get the same risk at higher return; these are dominated.

No rational investor would hold a portfolio below the minimum- variance point. So the relevant set is the upper half of the frontier.

Adding a risk-free asset

Now imagine you can also hold an asset with zero risk (Treasury bills, $\sigma = 0$, return $R_f$). Combine the risk-free asset with any risky portfolio $P$ at weights $w$ and $1-w$:

• $E[R_{\text{combined}}] = w R_f + (1-w) E[R_P]$
• $\sigma_{\text{combined}} = (1-w) \sigma_P$ (since $\sigma_f = 0$)

Eliminate $w$: $\frac{\sigma_{\text{combined}}}{\sigma_P} = 1 - w$, so combined return is a linear function of combined risk, starting at $R_f$ when $w = 1$.

The line has slope $(E[R_P] - R_f) / \sigma_P$ — the Sharpe ratio of portfolio $P$. Higher Sharpe = steeper line = better trade-off.

The tangent portfolio

Among all risky portfolios $P$ on the efficient frontier, one has the highest Sharpe ratio. The straight line from $R_f$ through that portfolio just touches (is tangent to) the efficient frontier.

That tangent portfolio $T$ is the unique best mix of risky assets — every investor should hold some combination of the risk-free asset and the tangent portfolio, varying only the proportion based on their personal risk tolerance:

• Conservative investors: more $R_f$, less $T$.
• Aggressive investors: less $R_f$, more $T$. Possibly even borrow at $R_f$ to hold more than 100% in $T$ (leverage).

But the risky portion is the same for everyone — it’s $T$.

The two-fund separation theorem

Two implications, often summarized as the two-fund separation theorem: with a risk-free asset, every investor’s optimal portfolio is a combination of:

1. The risk-free asset.
2. The market portfolio $T$.

That’s it. Two funds suffice for every investor. (This is the theoretical foundation for index investing: just buy the market and adjust your risk-free allocation to your taste.)

Who is $T$ in practice?

In theory, $T$ is the portfolio every investor wants to hold. In equilibrium, the total dollars of every risky asset must be held by someone. So $T$ must equal the value-weighted market portfolio — every traded asset, weighted by its market capitalization.

In practice, a broad index like the S&P 500 (large US stocks) or the MSCI ACWI (all-country world index) is a reasonable proxy for $T$. The S&P 500 isn’t exactly the theoretical market portfolio (it excludes bonds, real estate, private equity, foreign stocks), but it’s the most commonly used approximation.

Why this matters for valuation

The cost of equity for a project or firm — the discount rate we’ll use in DCFs going forward — comes from the asset’s expected return in equilibrium. Because in equilibrium every investor holds the market portfolio, the relevant risk for any single asset is its contribution to the market portfolio’s variance — its covariance with the market. That’s what CAPM (next lesson) formalizes.

A common caveat

The mean-variance framework assumes investors care only about mean and variance of returns. This is exactly right under two conditions: returns are normally distributed (no skew, no fat tails), or investors have quadratic utility. Both conditions are violated in practice — financial returns have fat tails and investors are risk-averse beyond just variance.

For undergraduate work, mean-variance gives the right qualitative intuition. For more advanced treatment, downside risk, value-at- risk, and expected-shortfall models supplement it. Berk and DeMarzo stick with mean-variance throughout; that’s enough for FIN 3610.