Portfolio Risk & Return: Combining Two Assets

You’ve met correlation and the portfolio-variance formula in the abstract. Now we put two real assets in a blender and watch what comes out. Pick a spicy growth stock and a calm bond fund, hold some of each, and two very different things happen to the two numbers you care about. The return of the mix behaves exactly as your gut expects — it lands neatly between the two. The risk of the mix does something almost magical: it can drop below either a simple average would predict, and sometimes below the calmer asset on its own. That gap between “what a naive average says” and “what actually happens” is the entire free lunch of diversification — and it has an exact equation. This lesson is where return-math and risk-math part ways.

Expected return is a weighted average

Start with the easy half. A portfolio is just a set of weights — what fraction of your money sits in each asset — and those weights must sum to 1 (you can’t invest 130% of your cash unless you borrow, which we’ll save for later). Call the weight in asset $A$ simply $w_A$, with $w_B = 1 - w_A$.

The portfolio’s expected return is the weighted average of the pieces:

$\mathbb{E}[R_p] = \sum_i w_i\,\mathbb{E}[R_i] = w_A\,\mathbb{E}[R_A] + w_B\,\mathbb{E}[R_B].$

This is linear in the weights — double a weight’s effect by doubling the weight, no surprises. Think of it like mixing paint by the bucket: 60% of a 12%-return bucket and 40% of a 6%-return bucket gives you a return that sits proportionally between them. No cancellation, no bonus, no penalty.

Worked example. Asset A (a stock) expects 12%; Asset B (a bond) expects 6%. Hold 60% A, 40% B:

$\mathbb{E}[R_p] = 0.60 \times 12\% + 0.40 \times 6\% = 7.2\% + 2.4\% = 9.6\%.$

Sweep the weight and the return marches in a straight line between 6% and 12%:

Each 1% you shift from bond to stock adds exactly the same $0.06$ percentage points of return. Perfectly linear.

Risk is NOT a weighted average

Here’s the trap that sinks beginners. If 60/40 of two returns is a weighted average, surely 60/40 of two volatilities is too? It is not. Variance has a cross term that the naive average completely ignores. For two assets:

$\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2\,w_A w_B\,\rho\,\sigma_A\sigma_B,$

where $\rho$ is the correlation between A and B. The first two terms are each asset’s own variance, scaled by the square of its weight. The last term — the cross term — is where correlation enters, and it’s the hero of the story. When $\rho < 1$, that cross term is smaller than it would be if the assets moved in lockstep, and the whole sum lands below the naive blend.

The wrong thing to compute is the weighted average of the volatilities:

$\sigma_{\text{naive}} = w_A\sigma_A + w_B\sigma_B \quad (\textbf{wrong unless } \rho = 1).$

That formula is only correct in the one case where diversification gives you nothing — perfect positive correlation. For anything less, the real number is smaller.

Worked example. Same assets, now with risk: $\sigma_A = 28\%$, $\sigma_B = 14\%$, correlation $\rho = 0.2$. Weights 60% A, 40% B. First the cross-term ingredient:

$\text{covariance term inputs: } \rho\,\sigma_A\sigma_B = 0.2 \times 28 \times 14 = 78.4.$

Now plug into the variance formula (working in percent-squared):

Take the square root: $\sigma_p = \sqrt{351.23} \approx 18.74\%$.

Now compare to the naive blend: $\sigma_{\text{naive}} = 0.6 \times 28 + 0.4 \times 14 = 16.8 + 5.6 = 22.4\%$.

You held exactly the same 60/40 mix and got 3.66 percentage points less risk than the naive average claimed — for free, just because the two assets don’t move in perfect step. The return was still a clean 9.6%. That’s the free lunch in one calculation.

The bullet curve

So return moves in a straight line as you shift weight, but risk bows below the naive average. Put those two facts on the same chart — risk on the x-axis, return on the y-axis — and you don’t get a line connecting the two assets. You get a curve that bulges to the left, called the bullet (or the portfolio frontier for two assets). Drag the weight below and watch the marker trace it.

Why a curve and not a line? Trace the two coordinates separately as $w_A$ sweeps $0 \to 1$:

• The y-coordinate (return) moves linearly — equal steps for equal weight changes, exactly the straight ladder from the first table.
• The x-coordinate (risk) does not — it dips below the straight line because of that $\rho < 1$ cross term, deepest somewhere in the middle where both weights are sizeable.

A point whose height climbs steadily but whose horizontal position pulls inward sweeps out a curve bowing toward the low-risk (left) side. The more sub-1 the correlation, the deeper the bow. If $\rho$ were exactly 1, the cross term wouldn’t shrink, the risk would equal the naive blend at every weight, and the bullet would collapse into a straight line between the two assets — no diversification, no curve.

The minimum-variance portfolio

That leftmost tip of the bullet has a name: the minimum-variance portfolio (MVP) — the single mix with the lowest possible volatility you can build from this pair. It’s the “calmest” portfolio available, and it often sits at a more extreme weight than you’d guess. There’s a closed-form weight for it:

$w_A^{\min} = \frac{\sigma_B^2 - \mathrm{Cov}_{AB}}{\sigma_A^2 + \sigma_B^2 - 2\,\mathrm{Cov}_{AB}}, \qquad \mathrm{Cov}_{AB} = \rho\,\sigma_A\sigma_B.$

Worked example. Our numbers: $\sigma_A = 28$, $\sigma_B = 14$, $\rho = 0.2$. First the covariance:

$\mathrm{Cov}_{AB} = 0.2 \times 28 \times 14 = 78.4.$

Now the weight (variances are $28^2 = 784$ and $14^2 = 196$):

$w_A^{\min} = \frac{196 - 78.4}{784 + 196 - 2(78.4)} = \frac{117.6}{823.2} \approx 0.143.$

So the minimum-variance portfolio holds about 14.3% in the stock and 85.7% in the bond. Notice it’s not 100% bond — even though the bond is the calmer asset, sprinkling in a little stock actually lowers total risk, because the stock’s wobble partly cancels the bond’s. That’s the counterintuitive punchline: the safest mix usually contains a dash of the risky asset.

Let’s check its return and risk. Return: $0.143 \times 12\% + 0.857 \times 6\% \approx 6.86\%$. Its volatility (plug $w_A = 0.143$ into the variance formula) works out to about 13.6% — below the bond’s own 14%. The MVP is less risky than either asset alone.

The MVP splits the bullet into two arcs. Everything above the MVP is the efficient part of the frontier: for each of those portfolios, no other mix gives more return at the same risk. Everything below it is dominated — for any lower-arc portfolio there’s an upper-arc one with the same risk but higher return, so no rational investor would ever pick it. When people say “the efficient frontier,” they mean that bright upper arc, starting at the MVP and running up to the high-return asset.

Scaling to many assets

Two assets gave us one cross term. The pattern generalizes, but the bookkeeping explodes. With $N$ assets the variance is a double sum over every pair:

$\sigma_p^2 = \sum_i \sum_j w_i w_j\,\mathrm{Cov}(R_i, R_j).$

When $i = j$ that’s an asset’s own variance; when $i \neq j$ it’s a pairwise covariance, counted twice. This grid of all the covariances is the covariance matrix — the complete risk DNA of the portfolio.

Here’s the part worth internalizing: the number of covariance terms grows much faster than the number of assets. With $N$ assets you have $N$ variances but $\tfrac{N(N-1)}{2}$ distinct pairwise covariances. For 10 assets that’s 10 variances vs. 45 covariances; for 100 assets it’s 100 vs. 4,950. The covariances swamp the variances. So in a large portfolio, your total risk is governed far more by how your holdings move together than by how jumpy any single one is. Diversification doesn’t kill an asset’s own variance so much as it averages away the parts that aren’t shared — which is exactly why a basket of correlated-but-not-identical assets can be so much calmer than its parts. We’ll trace the full many-asset frontier in the next lesson; for now, just hold onto the idea that covariances, not variances, run a big portfolio’s risk.

Pitfalls

A few ways this goes wrong in practice:

• Averaging the volatilities. The single most common error: treating $\sigma_p$ as $w_A\sigma_A + w_B\sigma_B$. That overstates your risk (it’s the $\rho = 1$ case) and makes diversification invisible. Always go through variance and the cross term.
• Forgetting the cross term’s sign. The cross term carries the sign of $\rho$. Positive correlation adds risk; negative correlation subtracts it — a negatively correlated pair can drive portfolio risk dramatically low, even to zero in the idealized $\rho = -1$ case. Drop or mis-sign that term and your whole risk estimate is off.
• Trusting your correlation estimate too much. $\rho$ is estimated from noisy past data, and it drifts — correlations notoriously spike toward 1 in a crash, exactly when you wanted diversification most. The math is exact; the inputs are not.
• Letting weights drift. Your 60/40 doesn’t stay 60/40. When the stock rallies it becomes, say, 70/30 on its own, quietly raising your risk above what you signed up for. Holding a target mix requires periodic rebalancing — selling the winner back down to weight.

Key Takeaways

Next up: the efficient frontier — the same logic with many assets at once, where the bullet becomes a whole region and only its upper edge is worth holding.