Two assets, then many

Chapter 3 fixed the weights and varied the correlation. This chapter fixes the correlation and varies the weights, which is the decision an investor actually faces.

The curve

Equity at 11% expected return and 14% standard deviation; debt at 7% and 4%; correlation 0.2. Move the equity weight from nothing to everything:

Equity weight	Expected return	Standard deviation	Weighted-average risk
0.0	7.00%	4.000%	4.00%
0.1	7.40%	4.115%	5.00%
0.2	7.80%	4.654%	6.00%
0.3	8.20%	5.494%	7.00%
0.4	8.60%	6.519%	8.00%
0.5	9.00%	7.655%	9.00%
0.6	9.40%	8.860%	10.00%
0.8	10.20%	11.387%	12.00%
1.0	11.00%	14.000%	14.00%

The second column is a straight line. Each step of 0.1 in weight adds exactly 0.40 points of expected return. Chapter 2's linearity, visible.

The third column is not. The steps are 0.115, 0.539, 0.840, 1.025, 1.136, 1.205 — they start tiny and grow. Risk accelerates as you add equity, which is the shape of the whole subject.

And the fourth column is what a naive investor would predict, and it is wrong everywhere except the two endpoints. The gap peaks around 30% equity at 1.51 points.

The curve bends left

Plot risk on the horizontal axis and return on the vertical, and the points above do not lie on a line from the debt point to the equity point. They bow out to the left — less risk for the same return than a straight line would give.

The amount of bowing is set entirely by the correlation. At $\rho = +1$ the curve straightens into exactly that line. As $\rho$ falls, the bow deepens. Chapter 3's table is this fact seen at a single weight.

This is the only picture in portfolio theory worth carrying in your head, because everything that follows is about finding the best point on a curve of this shape.

The result that persuades people

Look at the first two rows again.

Equity weight	Expected return	Standard deviation
0.0	7.000%	4.000%
0.0253	7.101%	3.985%

Adding 2.5% equity to an all-debt portfolio raises the expected return and lowers the risk. Not a trade-off — an improvement on both axes at once.

This is not a trick and it is not marginal to the theory. It follows directly from the cross term: at low equity weights the $w^2\sigma_e^2$ penalty is negligible because $w$ is squared, while the diversification benefit from a correlation below 1 arrives immediately. For a short stretch the second effect dominates.

The point where it stops dominating is the minimum-variance portfolio. For two assets:

$w_1^{\text{mv}} = \frac{\sigma_2^2 - \rho_{12}\sigma_1\sigma_2}{\sigma_1^2 + \sigma_2^2 - 2\rho_{12}\sigma_1\sigma_2}$

Here that gives $w_{\text{equity}} = 0.0253$, a portfolio with 3.985% risk — fractionally below the 4.000% of holding debt alone.

The honest size of the effect. It is 0.015 percentage points of risk and 0.10 points of return. It is real, it is free, and it is small — and it gets presented as though it were the main benefit of diversification, which it is not. The main benefit is the whole curve bowing left, not the sliver at the end of it.

But the direction is what matters. It establishes that "this asset is riskier than what I hold, so adding it makes me riskier" is false as a general statement. Risk is a property of the combination. A holding can only be called risky relative to the portfolio it is going into — which is chapter 1's thesis, now demonstrable rather than asserted.

Where the minimum-variance point sits

It depends on the correlation, and not in the direction people guess:

$\rho$	Equity weight at minimum variance	Portfolio standard deviation
+0.8	−0.333	8.94%
+0.3	0.105	9.79%
0.0	0.200	8.94%
−0.3	0.258	7.66%

(computed with $\sigma_1 = 20\%$, $\sigma_2 = 10\%$, so the weights differ from the equity/debt example above)

At a high correlation the minimum-variance weight goes negative — the best way to reduce risk is to short the riskier asset against the safer one, because at $\rho = 0.8$ they are close enough to substitutes that one hedges the other. That is a real result and an unavailable one for most individuals, which chapter 5 takes up.

Many assets: a region, not a curve

With two assets, every portfolio lies on one curve. With three, each pair traces a curve, and portfolios holding all three fill in the space between them. With many assets the set of achievable risk-return combinations is a solid region.

Markowitz's description of what to do with it is the definition of the next chapter:

the natural approach for an economics student was to imagine the investor selecting a point from the set of Pareto optimal expected return, variance of return combinations, now known as the efficient frontier.

"Pareto optimal" is the operative phrase. Among all the points in the region, most are beaten by some other point on both axes at once. The ones that are not are the frontier.

What many assets cost you

Before the theory gets elegant, the practical obstacle should be stated, because chapter 12 is largely about it.

To solve an $n$-asset problem you need every mean, every variance and every pairwise covariance:

Assets	Means	Variances	Covariances	Total estimates
2	2	2	1	5
10	10	10	45	65
50	50	50	1,225	1,325
500	500	500	124,750	125,750

The covariances grow with the square of the number of assets, so the inputs explode while the data available to estimate them does not.

And every one of those numbers is an estimate. The Quantitative methods subject's chapter on sampling is the relevant background: each estimate carries a standard error, and an optimiser fed 1,325 noisy numbers will happily find the portfolio that exploits the noise. The theory is exactly correct and the inputs are not, which is the tension the rest of the subject is written under.

Working the problem

All debt at 7% and 4%; equity at 11% and 14%; correlation 0.2.

Step 1 — write the variance as a function of the equity weight $w$.

$\sigma_p^2 = w^2(196) + (1-w)^2(16) + 2w(1-w)(0.2)(14)(4)$

$= 196w^2 + 16(1 - 2w + w^2) + 22.4(w - w^2) = 189.6\,w^2 - 9.6\,w + 16$

Step 2 — find the minimum. A parabola in $w$ with a positive leading coefficient bottoms out at $w = -b/2a$:

$w = \frac{9.6}{2 \times 189.6} = 0.0253$

Step 3 — evaluate both ends.

	Expected return	Variance	Standard deviation
$w = 0$	7.000%	16.000	4.000%
$w = 0.0253$	7.101%	15.879	3.985%

What happens as you move from 0% to 2.53% equity: return rises by 0.101 points and risk falls by 0.015 points. Both improve. Beyond 2.53% risk starts rising again, and from there on every further point of equity is a genuine trade.

Three things I would say to someone holding only debt.

The free part is tiny. A fortieth of the portfolio in equity, for fifteen thousandths of a point of risk reduction. Nobody should rearrange their affairs for that.

The useful part is just past it. At 10% equity the portfolio returns 7.40% for 4.115% risk — 0.40 points more return for 0.115 points more risk. That is a trade at better than three to one, and it is available because the curve is still nearly flat there. The good trades are near the left end, not in the middle.

And the input that drives all of it is the one you know least. The 0.2 correlation decides the shape. The Risk subject's chapter on correlation is about how that number behaves when it is tested — and chapter 3 noted that measured correlations move with the window. A result that depends on a correlation estimate deserves to be checked at a worse correlation before it is acted on. At $\rho = 0.5$ the same calculation still gives an improvement, but a smaller one; at $\rho = 1$ there is none at all.

The point

Vary one weight and expected return moves in a straight line while risk traces a curve that bows left, with the bowing set entirely by the correlation. Because the own-variance penalty is squared in the weight while the diversification benefit is not, a small holding of a much riskier asset can raise a safe portfolio's return and lower its risk at the same time — 2.5% equity in an all-debt portfolio, here, worth 0.10 points of return and 0.015 of risk. The effect is real, free and small; the significant benefit is the shape of the whole curve, and the best trades sit near its left end. With many assets the achievable set becomes a region whose upper-left boundary is the efficient frontier — and whose inputs grow with the square of the number of holdings, every one of them estimated.