The question:
Suppose in a single period investment problem we may divide our wealth between n assets and that the return on the ith security is given by
$r_i = \alpha + \beta_i\theta + \epsilon_i,$ $i = 1, ..., n$,
where $\alpha,$ $\beta_i$ are constants, and $\theta$ and $\epsilon_i$ are random variables. $\theta$ represents some underlying common economic variable (such as overall economic growth) that influences the return on all the securities. The $\epsilon_i$ are random variables assumed to be uncorrelated with each other and with $\theta$. Suppose that the $\epsilon_i$ have zero mean and common variance $\sigma_\epsilon^2$.
Denote the mean and variance of $\theta$ by $\mu_\theta$ and $\sigma_\theta^2$. Determine the covariance matrix $\Sigma$ of the returns, and verify that $\beta =(\beta_1, ..., \beta_n)$ is one of its eigenvectors.
What can you say about the other eigenvectors? Hence find the inverse $\Sigma^{-1}$. Find the portfolio that has minimal risk, and determine its variance and expected return. Let $\beta_i$ = $1$ for all $i = 1,2,... ,n$. Treating $\mu_\theta$, $\sigma_\theta^2$ and $\sigma_\epsilon^2$ as fixed, let $n$ tend to infinity, and comment on the minimal variance portfolio. Interpret this in terms of diversification as an investment strategy.
Basically, I have done all of this question except everything after computing $\Sigma^{-1}$. I have the following formula for $\Sigma^{-1}$:
$\Sigma^{-1}$ = $aI + b$, where $a = \displaystyle\frac{1}{\sigma_\epsilon^2}$ and $b = \left(\displaystyle\frac{1}{\sigma_\epsilon^2 + \sigma_\theta^2\beta^T\beta} - \displaystyle\frac{1}{\sigma_\epsilon^2}\right) \displaystyle\frac{1}{\beta^T \beta}$,
which I'm almost certain is correct. The problem is, that is pretty messy! So when I want to compute the minimal variance portfolio which is given by
$w = \displaystyle \frac{\Sigma^{-1}1}{1^T\Sigma^{-1}1}$, so I can compute the mean return. The variance is just $\displaystyle \frac{1}{1^T\Sigma^{-1}1}$, but this is still messy.
I get the motivation behind the question, which is that the $\epsilon_i$ represent risk due to unforeseen circumstances where as $\theta$ is risk felt by everyone in the market, and that by letting $n \rightarrow \infty$ I can in essence "diversify away" the contribution of risk due to the $\epsilon_i$, but I cannot show this explicitly, which is frustrating.
So, does anybody have smart ways of calculating the minimum variance portfolio, weights, variance etc?
I understand you have three questions:
1) check your matrix inverse - I pass on that one for now - my answer's gonna be long enough without that.
2) what are smart ways to calculate the minimum variance portfolio, weights, variance "etc." - this depends on the circumstances. There may be shortcuts, there may be not. Here we might have some with regards to your question 3:
3) how to show that you can diversify away the risk contribution of $\epsilon_i$
Answer to 3)
I would start by first calculating the portfolio variance in general, using the one-factor model you have been given:
$r_i=\alpha_i+\beta_i\theta+\epsilon_i$
the return of a portfolio is the weighted sum of the returns of the assets in the portfolio:
$r_p=\Sigma_iw_ir_i=\Sigma_iw_i(\alpha_i+\beta_i\theta+\epsilon_i)$
Now as the $\epsilon_i$ are uncorrelated with each other, and with $\theta$, the variance of $r_p$ is:
$\sigma_p^2=\Sigma_iw_i^2\beta_i^2\sigma_{\theta}^2 + \Sigma_iw_i^2\sigma_{\epsilon_i}^2$
where $\Sigma_iw_i^2\sigma_{\epsilon_i}^2$ is the contribution to risk due to the $\epsilon_i$ as you termed it ($\sigma_{\theta}^2$ is the variance of the common factor). In the following I refer to $\Sigma_iw_i^2\sigma_{\epsilon_i}^2$ as "unsystematic risk". And as you say the $\epsilon_i$ are supposed to have a common variance $\sigma^2_{\epsilon}$, this becomes:
$\Sigma_iw_i^2\sigma_{\epsilon}^2=\sigma_{\epsilon}^2\Sigma_iw_i^2$
This approaches zero when $\Sigma_iw_i^2$ approaches zero.
You can see, that just having $n$ approach infinity is actually not sufficient for that expression to approach zero. You could for example have one very large $w_i$ and grow $n$ by reducing all the other weights. But that would not be "diversification" in a common sense. So to show the intuition behind diversification, assume for example, that the weights are all equal, which is sometimes referred to as "naive diversification".
Then you can write the unsystematic risk as:
$\sigma_{\epsilon}^2\Sigma_iw_i^2=\sigma_{\epsilon}^2n\cfrac{1}{n^2}=\cfrac{\sigma_{\epsilon}^2}{n}$
which approaches zero as $n$ grows to infinity.
The portfolio with the smallest unsystematic risk is not necessarily the portfolio with the smallest total risk. However, here it says $\beta_i=1$ for all $i$. As $\beta_p$ is just the weighted sum of the $\beta_i$, it will always be $1$, so that all you can do to adjust risk is changing the unsystematic risk, and hence your minimum variance portfolio will contain minimal unsystematic risk.
With a fixed number of assets, you can't have zero unsystematic risk - (unless you define the common factor as the market portfolio return, with the market portfolio containing all assets - in which case you would have strictly speaking violated the assumption that the unsystematic return parts are uncorrelated).
So what is the minimum-variance portfolio under the assumptions given here, if there's "only"... a finite number of assets:
With $V_i$ being the value of the portfolio's position in asset $i$ and $V_p$ being the portfolio value:
$w_i=\cfrac{V_i}{V_p}$
$\Sigma_i w_i^2=\Sigma_i\left(\cfrac{V_i}{V_p}\right)^2=\cfrac{1}{V_p^2}\Sigma_i V_i^2$
The portfolio value is a given constant, so minimizing $\Sigma_i w_i^2$ and hence portfolio variance corresponds to minimizing $\Sigma_i V_i^2$.
Now you can of course do the 1st derivative and derive a condition for the variance-minimizing weights, or you just reason as follows:
Let's say one would initially just diversify naively, i.e. each position is of the same size $V$. And then try to improve the allocation "incrementally", by raising the weight of one asset and reducing the weight of another one.
The impact on the sum of squared weights due to raising the weight of one asset is:
$(V+x)^2-V^2=x^2+2Vx$
The impact on the sum of squared weights due to reducing the weight of one asset is:
$(V-x)^2-V^2=x^2-2Vx$
The net impact of reducing one and increasing the other is:
$x^2+2Vx+x^2-2Vx=2x^2$
Which is positive. So by deviating from the equal-weights allocation, you will increase the variance (under the assumptions here: same $\beta$ and same $\epsilon$ for all assets).
This proves, that under the specific assumptions here, the minimum variance portfolio given a finite number of assets, is the naively diversified portfolio.
(And from the first part above we know that under the given assumptions, the variance of this portfolio will decrease with rising $n$ (and approach zero when $n$ grows to infinity))
(btw, this may have been a question for https://quant.stackexchange.com/)