Variance of a random vector multiplied by a data points matrix

Question

I have a random valued vector $R = (R_{1},R_{2},..,R_{d})^{T}$, from which I have some series of gathered observed data.

And the random variable of "return rates" defined as:

$R_{p} = p^{T}R $ , where p is a known vector $p = (p_{1},.. p_{d})^{T}$

The paper I'm reading calculates the variance of this return rate doing:

$Var[R_{p}] = Var[p_{T}R] = p_{T}Cov[R]p$

I don't understand the last step, which equality does it follow?

Answer 1

See property 3 here, which implies that for a random vector $X$ and a matrix $A$, we have $$\text{Cov}(AX) = A\text{Cov}(X) A^\top,$$ where "Cov" is notation for "covariance matrix of." You can prove this general fact directly from the definition of a covariance matrix.

Your situation is the case where $X=R$ and $A = p^\top$. Moreover, since $p^\top R$ is a random variable (rather than a random vector), its covariance matrix is a $1 \times 1$ "matrix" containing the variance of $p^\top R$.