How to find $\phi$, that denotes the correlation of signals among informed traders?

Question

Since I do not have an answer on Quantitative Finance in my question I cross-post here the problem to tag some other categories

The following assumptions are part of the paper of Back, Chao and Willard and I can not solve for the statistic that is denoted as $\phi$ in the sequel. I would be glad if anyone could help me. Below i set the assumptions and the equations of interest

Suppose that in the market, there are $N\geq 1$ informed agents, who trade a risky asset continuously in the time interval $[0,1)$. Each agent $i$ receives a mean-zero signal $\tilde{s}^i$ at time 0. We assume the signals and the liquidation value of the asset have a nondegenerate joint normal distribution that is symmetric in the signals. Symmetry means that the joint distribution of the asset value and the signals $\tilde{s}^1,...,\tilde{s}^N$ is invariant to a permutation of the indices $1,...,N$. Let $\tilde{v}$ denote the expectation of the liquidation value conditional on the combined information of the informed traders. By normality, $\tilde{v}$ is an affine function of the $\tilde{s}^i$. By rescaling the $\tilde{s}^i$ if necessary, we can assume without loss of generality that

\begin{equation}\tilde{v}=\bar{v}+\Sigma^{N}_{i=1} s^i\end{equation} for a constant $\bar{v}$. For simplicity, we assume $\bar{v}=0$. Let \begin{align}\phi=\frac{var(\tilde{v})}{var(N\tilde{s}^i)}\end{align}

The statistic $\phi$ is a measure of the quality of each agent’s information. Specifically, it is the $R^2$ in the linear regression of $\tilde{v}$ on $\tilde{s}^i$, that is, it is the percentage of the variance in $\tilde{v}$ that is explained by the trader’s information.

Letting $\rho$ denote the correlation coefficient of $\tilde{s}^i$ with $\tilde{s}^j$ for $i\neq j$, one can compute $\phi$ for $N>1$ as

\begin{equation}\phi=\frac{1}{N}+\frac{N-1}{N}\rho\end{equation}

If $\phi=1$, then either $N=1$ or the $\tilde{s}^i$ are perfectly corellated. In either case each informed trader has perfect information about $\tilde{v}$.

My questions are the following

1. what does it mean intuitively "a nondegenerate joint normal distribution" and in particular I would like to understand the term nondegenerate.
2. What does it mean "invariant to indices" ?
3. the liquidation value is equal to the sume of the signals, does this come from the assumption that it it an affine function of the $\tilde{s}^i$?
4. How do we find that measure $\phi$? is it from the linear regression of $\tilde{v}$ on $\tilde{s}^i$?
5. How $\phi$ is tranformed to \begin{equation}\phi=\frac{1}{N}+\frac{N-1}{N}\rho\end{equation}

Here it is a link from the paper

Answer 1

1. A degenerate joint normal is distribution is one in which you cannot find a PDF for the distribution. They assume you can. (The covariance matrix is invertible).

2. Let $f(s_1,s_2\dots,s_n,v)$ be the distribution. If I was to exchange $s_1$ for $s_2$, $f(s_2,s_1,\dots,s_n,v) = f(s_1,s_2\dots,s_n,v)$, the distribution does not change. And you can exchange as many indices (signals) as you'd like.

3. The assumption that the distribution is jointly normal implies $\tilde{v}$ is an affine function of $\tilde{s}_i$, which then implies the sum of signals.

4. Regress $\tilde{v}$ on $\tilde{s}_i$ and calculate $R^2$ (the Coefficient of determination).

5. This is something I hope someone else can answer.

Answer 2

Well, I will try to answer 4.

We know that the asset liquidation value $\tilde{v}$ is an affine function of the singals thus we have that $$\tilde{v}=\bar{v}+\sum_{i=1}^{N}\tilde{s}^i\Rightarrow \tilde{v}=\bar{v}+N\underbrace{\frac{\sum_{i=1}^{N}\tilde{s}^i}{N}}_{\tilde{s}^i}\Rightarrow\tilde{v}=\bar{v}+N\tilde{s}^i$$ where the $\tilde{s}^i$ is the average singal that is a sufficient statistic to infer the liquidation value of the asset conditioning on it instaed of the individual signal since this is also driven by the assumption that the signals and the liquidation value of the asset have a nondegenerate joint normal distribution that is symmetric in the signals. Hence the expectation of the liquidation value conditional on the combined information of the informed traders is given by the projection theorem to be (projecting $\tilde{s}^i$ on $\tilde{v}$):

$$\mathbb{E}[\tilde{v}|\tilde{s}^i]=\mathbb{E}[\tilde{v}]+\frac{\mathbb{C}ov(\tilde{v},\tilde{s}^i)}{\mathbb{V}ar(\tilde{s}^i)}\left(\tilde{s}^i-\mathbb{E}(\tilde{s}^i)\right)\Rightarrow\mathbb{E}[\tilde{v}|\tilde{s}^i]=\bar{u}+\frac{\mathbb{C}ov(\tilde{v},(\tilde{v}-\bar{v})/N)}{\mathbb{V}ar(\tilde{s}^i)}\tilde{s}^i\Rightarrow\\ \mathbb{E}[\tilde{v}|\tilde{s}^i]=\bar{v}+\frac{\mathbb{V}ar(\tilde{v})}{N^2\mathbb{V}ar(\tilde{s}^i)}\sum_{i=1}^{N}\tilde{s}^i\Rightarrow\mathbb{E}[\tilde{v}|\tilde{s}^i]=\bar{v}+\underbrace{\frac{\mathbb{V}ar(\tilde{v})}{\mathbb{V}ar(N\tilde{s}^i)}}_{\beta^{i}}\sum_{i=1}^{N}\tilde{s}^i$$

where $\beta^{i}$ denotes the beta coefficient of the linear regression of $\tilde{v}$ on $\tilde{s}^i$, that coincide with the $R$-square coefficient and as a consequence

$$\phi=\frac{\mathbb{V}ar(\tilde{v})}{\mathbb{V}ar(N\tilde{s}^i)}$$