Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 23 / 2 / d

For weights summing to one, collect the exact portfolio return asThe desired approximation means that $e_n\to 0$, for example in mean-square convergence. Centering the residuals gives $\mathbb{E}{e}_n=0$, but its variance isThis identifies an omission in the printed hypotheses: bounded individual variances do not control the cross terms. Indeed, take ${ϵ}_i=Z$ for every $i$, where $Z$ is a nondegenerate centered random variable with $\mathrm{Var}Z<s^2$. Equal weights $1/n<W/n$ for any $W>1$ retain $e_n=Z$ at every $n$. The claimed diversification conclusion does not follow from the printed assumptions alone.

Under the usual extra assumption that the residuals are pairwise uncorrelated and the weights are nonnegative, the advertised calculation isThe Chebyshev inequality then gives $\mathrm{Pr}(|e_n|>\delta )\le s^{2}W/(n{\delta }^2)\to 0$. Alternatively, signed weights satisfying $|w_i|\le W/n$ give the bound $s^2{W}^2/n$. These are instances of the covariance criterion for diversification of factor residuals. More generally, with that absolute weight bound, ${\sum }_{i,k}|\mathrm{Cov}({ϵ}_i,{ϵ}_k)|=o(n^2)$ suffices.

If short selling are allowed, the printed one-sided weight bound is another insufficiency. Choose $0<c<W-1$, put $w_1=-c$ and $w_i=(1+c)/(n-1)$ for $i\ge 2$, and let only ${ϵ}_1$ be a nondegenerate centered residual. For all sufficiently large $n$ the one-sided bounds hold, but $e_n=-c{ϵ}_1$ never vanishes. Thus nonnegative weights or an absolute bound is needed.

With the repaired assumptions, well-diversified portfolios become approximately exposed only to the common factors, and their residual standard deviation is $O(n^{-1/2})$ in the uncorrelated case. The averaged coefficients may depend on $n$; convergence to fixed coefficients needs additional assumptions. In the associated asymptotic arbitrage pricing theory, diversifiable risk cannot command a persistent premium in well-diversified portfolios under absence of asymptotic arbitrage, while common-factor exposure can. This does not prove an exact pricing equation for every individual noisy asset from the finite-market hypotheses alone.