# Binomial correlation measure in the trivariate case

I have a question about the binomial correlation measure at page 530 in Hull(2009), Options futures and other derivatives (7th Edition) which is defined for the bivariate case as:

$$\beta_{AB}(T)=\frac{P_{AB}(T)-Q_A(T)Q_B(T)}{\sqrt{[Q_A(T)-Q_A(T)^2][Q_B(T)-Q_B(T)^2]}}$$

where $$P_{AB}$$ is the joint probability of $$A$$ and $$B$$ defaulting between $$0$$ and $$T$$, and $$Q_i(T)$$ is the cumulative probability that company $$i$$ will default by time $$T$$.

If I want to extend to the trivariate case, should I have something which is related to partial correlation? Something like this:

$$\beta_{AB,C}(T)=\frac{P_{AB}(T)-Q_{AC}(T)Q_{BC}(T)}{\sqrt{[Q_{AC}(T)-Q_{AC}(T)^2][Q_{BC}(T)-Q_{BC}(T)^2]}}$$

where $$Q_{ij}(T)$$ is the joint cumulative probability that company $$i$$ and $$j$$ will default by time $$T$$.