Diversification, correlation and the two-asset portfolio

◈ 9 cards

Two stocks beat one because they do not move together: the σ formula with ρ inside, equal to the weighted average only at ρ = +1; and the firm-specific risk that diversification removes versus the market risk that stays.

The formula with ρ inside

Cobalt Ridge Mining has a standard deviation of 18 %; Halton Dairy has 12 %. A portfolio puts 60 % in Cobalt Ridge and 40 % in Halton. Its return is the weighted average (Lesson 13.2). Its risk is not:

$σ_{p} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} σ_{1} σ_{2} ρ$

The last term carries the correlation coefficient $ρ$ between the two return series, from −1 (they move exactly opposite) through 0 (unrelated) to +1 (they move in lockstep). A mining company and a dairy co-op are driven by different things; suppose $ρ = 0.3$ :

=SQRT(0.6^2*0.18^2+0.4^2*0.12^2+2*0.6*0.4*0.18*0.12*0.3) = 0.130684 — 13.07 %.

The weighted average of the two σs is =0.6*0.18+0.4*0.12 = 15.60 %. The portfolio is less risky than the average of its parts — by 2.5 points — and nothing was given up on return. That is diversification: the gain from holding things that do not move together.

Run ρ from +1 to −1

ρ	σₚ	Gap below 15.60 %
+1.0	15.60 %	0
+0.5	13.84 %	1.76
+0.3	13.07 %	2.53
0	11.82 %	3.78
−0.5	9.37 %	6.23
−1.0	6.00 %	9.60

At $ρ = + 1$ the square root collapses to $(w_{1} σ_{1} + w_{2} σ_{2})$ exactly — the weighted average, no benefit: two shares that always move together are one share. Every ρ below +1 gives a σₚ below the average, and the lower the correlation the larger the benefit. At $ρ = - 1$ the formula becomes $∣ w_{1} σ_{1} - w_{2} σ_{2} ∣$ = |10.8 − 4.8| = 6.00 %, and with the right weights it could reach zero — a perfect hedge, which is why Module 12's futures hedge works: a short future is −1 correlated with the physical.

In Excel, ρ between two columns of returns is =CORREL(a,b); the covariance is =COVARIANCE.S(a,b), and $ρ = cov / (σ_{a} σ_{b})$ .

Two kinds of risk

Add a third share, a tenth, a thirtieth. Each new holding brings its own idiosyncrasies — a strike, a recall, a discovery — and those firm-specific shocks, being uncorrelated across companies, cancel more and more as the portfolio grows. This is unsystematic (diversifiable) risk, and a portfolio of roughly 20–30 shares across industries has removed most of it. What is left is the risk every share bears together — recessions, rate rises, the market's mood — systematic (market) risk. It is the floor the curve falls to and cannot go below, because you cannot diversify away the economy. Only systematic risk earns a premium (Lesson 13.4): the market will not pay you for a risk you could have removed for free.

Recompute

At 50 / 50 with ρ = 0.3: =SQRT(0.5^2*0.18^2+0.5^2*0.12^2+2*0.5*0.5*0.18*0.12*0.3) = 12.22 %. Cold: diversification removes unsystematic risk; systematic risk remains; the weighted average of σ is reached only at ρ = +1.

The diversification curve as a table of points: each share has σ 18 % of which 9 % is market risk, and the firm-specific parts are uncorrelated. The firm-specific component falls with √n; the market component is the floor no number of shares removes.