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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-32/1/c/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 32 / 1 / c / ii / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Let

M

denote the displayed matrix of excess-return coefficients. If it were invertible, choose

v

with

M v = (1, 0, 0)^{t}

. Hold

v_{i}

units of the three risky investments and short

\sum_{i} v_{i}

units of the risk-free asset. Initial net cost is zero. The terminal payoff is

\sum_{i} v_{i} (r_{i} - R) = \sum_{i} v_{i} (a_{i} - R) + (\sum_{i} v_{i} b_{i}) f_{1} + (\sum_{i} v_{i} c_{i}) f_{2} = 1.

This is a guaranteed arbitrage, contradicting the assumption. Hence

det M = 0.

The contradiction does not require guessing probabilities for the factors: invertibility would allow simultaneous cancellation of both random exposures while keeping a nonzero constant gain.

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