Diversification

Speculation takes risk. Hedging eliminates it. Insurance buys away the bad half of it. Diversification minimises it — and the workbook is careful about what that means: diversification reduces both return and risk, but reduces risk more than return, so that risk per unit of return falls.

That is a lower bar than it sounds, and it is the honest one. You do not get something for nothing. You give up the chance of the single best outcome in exchange for never being exposed to the single worst.

The mechanism is correlation, not count. Ten holdings that all rise and fall together are one holding wearing ten hats. Two that move independently do more for you than twenty that do not.

For two positions of equal size, with standard deviations σ₁ and σ₂ and correlation ρ between them, the combined risk is not the average of the two — it is less, and how much less depends entirely on ρ.

For n equally weighted positions that each have standard deviation σ and share a common correlation ρ with one another, the arithmetic collapses to something very revealing:

Portfolio variance = σ² × [ 1/n + (1 − 1/n) × ρ ]

As n grows, the 1/n term vanishes and the whole expression tends to σ²ρ. The risk you can diversify away disappears; the risk you share with everything else does not, however many holdings you add. That floor is the subject of systematic risk, which argues the limit in full — this page is about the technique that gets you down to it.

Three decisions make a diversification programme, and only one of them is "how many":

1. Across what? Currencies, counterparties, maturities, geographies, industries. Diversifying across ten stocks in one sector diversifies almost nothing.
2. How correlated? The lower the pairwise correlation, the steeper the benefit per holding added.
3. In what weights? Equal weights extract the most benefit from equally risky, equally correlated holdings; concentrating 80% in one of them throws most of it away.

Two equally weighted positions:
σp = √( 0.25σ₁² + 0.25σ₂² + 0.5 × ρ × σ₁ × σ₂ )

n equally weighted, equal σ, common ρ:
σp = σ × √( 1/n + (1 − 1/n)ρ )

Diversification floor as n → ∞:
σp → σ × √ρ

An Indian software exporter bills Rs 100 crore equivalent of receivables and currently invoices everything in US dollars. The three-month standard deviation of its rupee realisation is 4% — about Rs 4 crore of uncertainty.

It restructures new contracts to invoice half in USD and half in EUR, each with the same 4% volatility.

If USD and EUR move closely together against the rupee (ρ = 0.60):

σp = √(0.25×16 + 0.25×16 + 0.5×0.60×4×4)
   = √(4 + 4 + 4.8) = √12.8 = 3.58%   → Rs 3.58 crore

Risk falls 10.6%, from Rs 4 crore to Rs 3.58 crore.

If the two are far less related (ρ = 0.20):

σp = √(4 + 4 + 1.6) = √9.6 = 3.10%   → Rs 3.10 crore

Risk falls 22.5% — more than twice the benefit, from exactly the same two-way split. The correlation did the work, not the splitting.

Now hold ρ at 0.60 and keep adding currencies:

The first extra currency buys 0.42 percentage points. Going from four to eight buys 0.13. Everything beyond the floor of 3.10% is unreachable — no number of currencies removes the part of the risk they all share, which here is the rupee itself. That is why the exporter still needs to hedge.

Chapter 2 (Foreign Exchange Derivatives), section 2.1, lists diversification as the fourth approach to price risk, with the precise wording "reduces both return and risk but in such a way that risk is reduced more than return". Expect a one-line recall question distinguishing it from hedging and insurance — diversification is the only one of the four that reduces return as well as risk, and the only one that does not need a derivative at all.

Chapter 2 also notes empirically that changes in exchange rates have very low correlations with foreign equity and bond returns, which is the diversification argument sometimes advanced against hedging currency exposure — and which the workbook then rebuts.

• Diversification reduces return too. It is the only one of the four approaches that does. A question offering "reduces risk without reducing return" is describing something else.
• It is not hedging. Hedging offsets a specific exposure with a specific contract; diversification never eliminates a single exposure, it dilutes all of them.
• Count is the weak lever, correlation is the strong one. Doubling the number of holdings from four to eight in the table above bought less than adding the second one did.
• There is a floor, and it is σ√ρ. For the limit and why market risk survives any amount of diversification, see systematic risk — do not try to argue it from the number of holdings.
• Correlations rise in a crisis. The ρ you diversified against is measured in calm markets; the one that applies when you need it is higher, which is precisely when the diversification disappoints.
• A single derivative position is never diversification, however clever the strategy.