Setup
Suppose there are n independent securities, each priced at 1.
The payoff Xi of each is drawn from a normal
distribution N(μ = 3, σ² = 1): an expected payoff of
3 (a 200% expected return over the price of 1) and a
variance of 1, so negative draws are vanishingly rare. There are also
n individuals, each with an endowment of 1.
- Case 1. Each individual buys one security on her own. Her payoff is just that one security's draw, Xi.
- Case 2. All n individuals pool their endowments
to buy every security, then share the total return equally. Each gets
(1/n) Σ Xi.
The expected payoff is identical in both cases (E = 3). But the
variance is dramatically different — and that gap is the entire reason
mutual funds, insurance, and other financial intermediaries exist.
Case 1 · realized X1 per trial
Empirical (30 trials): mean — ·
variance —
Theoretical: mean 3.00 · variance 1.00
Case 2 · realized (1/n) Σ Xi per trial
Empirical (30 trials): mean — ·
variance —
Theoretical: mean 3.00 · variance 0.10
(= 1 / n)
Push n higher. Case 2's path flattens toward the dashed
E[X] = 3 line — the law-of-large-numbers intuition behind
diversification. Both options have the same expected return, but Case 2
carries a fraction of the risk. We'll formalise this in Chapter 8
(financial structure) and Chapter 12 (insurance).