Two independent assets

Two independent bets, each with a volatility of 10%. Put half your money in each, and the volatility is not 10% but 7.07%. Nothing about either bet changed. Where did almost a third of the risk go?

L3.1.01: Portfolio return and risk showed that the variance of a two-asset portfolio has three parts: each asset on its own, plus a cross term that depends on the correlation. The cleanest case to study first is the one where the cross term vanishes: the two assets are independentTwo returns are independent if knowing one tells you nothing about the other. Independence implies zero correlation, but zero correlation does not imply independence., so their correlation is zero.

Truly independent returns are rare in markets: most assets feel the same economy. They are more common elsewhere in a financial group. RiskAtoms Insurance's motor policies in one country and its house policies in another, or two trading strategies driven by unrelated signals, come close. For our equities and government bonds the universe's correlation is −0.10, close to zero; in this lesson we set it to exactly 0 to see the mechanism clearly. L3.1.03: Two correlated assets puts it back.

Why risk falls

With zero correlation, the portfolio variance is just the two own terms, $w_1^2\sigma_1^2 + w_2^2\sigma_2^2$. The weights are squared. Hold half of each and each asset contributes only a quarter of its variance, not half. That is the whole trick: a weight below 1 shrinks the variance faster than it shrinks the money. Two equally volatile, independent assets held half and half have a variance of half of either one, so a volatility of $\sigma/\sqrt 2$: 10% becomes 7.07%. That fall, compared with the weighted average, is the diversification gainThe fall in volatility from combining assets, compared with the weighted average of their volatilities..

Equal weights or the best weights?

Half and half is a natural start, but it is not the lowest-risk mix unless the two are equally volatile. Take global equities (16%) and government bonds (6%), now independent. Half and half gives a volatility of 8.54%: well below the 11% average, but above the bonds' 6% on their own. The minimum-varianceThe mix of assets with the lowest possible volatility. mix puts more in the calmer asset: about 12.3% in equities and the rest in bonds, for a volatility of 5.62%.

Notice two things. First, the lowest-risk mix is less volatile than the bonds alone, even though it adds an asset almost three times as volatile. Second, it also has a higher expected return: 3.93% against 3.50%. An all-bond portfolio is beaten on both counts. That is a first glimpse of an inefficient portfolio, the subject of L3.1.05: The efficient frontier.

Many independent assets

Now spread the money equally over $n$ independent assets, each with volatility $\sigma$. Each weight is $1/n$, each contributes $\sigma^2/n^2$ of variance, and there are $n$ of them: the variance is $\sigma^2/n$ and the volatility $\sigma/\sqrt n$. This is the square-root-of-n ruleWith n independent assets of equal volatility σ held in equal amounts, the portfolio volatility is σ/√n., the same arithmetic that makes an insurer's results steadier the more independent policies it writes.

The rule has a catch: halving the risk takes four times as many assets, so the gain slows quickly. And it needs true independence. With any common correlation the volatility stops falling at a floor, as L2.6.05: Diversification and its limits showed. Independence is the best case diversification can offer.

Start from the two-asset variance of L3.1.01: Portfolio return and risk with $w_1 = w$, $w_2 = 1 - w$ and $\rho_{12} = 0$. The variance of a sum of independent random variables is the sum of their variances (M-C11: Combining random variables):

To find the lowest-risk weight, treat the variance as a function of $w$ and find where its slope is zero, using the derivative of M-B01: Derivatives.

The last step, line by line. Write $S = \sigma_1^2 + \sigma_2^2$, so $w^* = \sigma_2^2/S$ and $1 - w^* = \sigma_1^2/S$:

It is a minimum, not a maximum: the second derivative is $2(\sigma_1^2 + \sigma_2^2) > 0$. And because $1/\sigma_*^2$ is bigger than either $1/\sigma_i^2$, $\sigma_*$ is below both volatilities.

Many assets. With $n$ independent assets every cross term is zero, so

The lowest-risk weights generalise too: $w_i \propto 1/\sigma_i^2$ and $1/\sigma_*^2 = \sum_i 1/\sigma_i^2$. With more than two weights a single derivative is not enough; L3.1.04: The minimum-variance portfolio brings in the tool for it, Lagrange multipliers.

Start at half and half in equities and bonds and press “Jump to the lowest risk”. Then make asset 1 riskier, up to 40%: the best weight in it shrinks, but never to zero. Finally set both volatilities equal and check that the lowest point sits at exactly 50%.

Global equities ($\sigma_1 = 16\%, \ \mu_1 = 7\%$) and government bonds ($\sigma_2 = 6\%, \ \mu_2 = 3.5\%$), correlation set to 0. Try each step first.

Capital rules decide how much credit a firm gets for risks being unrelated, and they are cautious about it. Solvency II's standard formula combines an insurer's risk modules with a fixed correlation matrix (Annex IV of Directive 2009/138/EC). In it, life and non-life underwriting risk have a correlation of 0, so a composite insurer such as RiskAtoms Insurance gets the full independent-assets benefit between those two; most other pairs carry 0.25 or 0.5. Banks get less: Basel market risk rules aggregate with prescribed correlations and then stress them (L2.5.03: Correlation and diversification). Both come back in M4.3: Market risk capital (FRTB) and CVA and M4.6: Solvency II and insurance regulation.

• With zero correlation, σp² = w₁²σ₁² + w₂²σ₂²: squared weights shrink each variance faster than the money.
• Two equally volatile, independent assets held half and half: σ/√2, about 29% less risk.
• The lowest-risk mix is w* = σ₂² / (σ₁² + σ₂²): inverse-variance weights. Its volatility is below both assets'.
• For the group's pair at ρ = 0: 12.3% in equities gives 5.62% volatility and 3.93% expected return, beating bonds alone on both.
• n independent, equally volatile assets in equal amounts: σ/√n. Halving the risk needs four times the assets, and any common correlation sets a floor.

Independent returns

Two returns are independent if knowing one tells you nothing about the other. Independence implies zero correlation, but zero correlation does not imply independence.

Diversification gain

The fall in volatility from combining assets, compared with the weighted average of their volatilities.

Minimum-variance mix

The mix of assets with the lowest possible volatility.

Inverse-variance weights

Weights proportional to one over each asset's variance: the lowest-risk mix when the assets are independent.

Square-root-of-n rule

With n independent assets of equal volatility σ held in equal amounts, the portfolio volatility is σ/√n.