How to find the correlation coefficient between two technologies when those are sub correlated?

Question

Suppose I have two power generating technologies, Coal and Oil, of which the generating cost components(total generating cost = Capital cost+Fuel cost+Variable O&M cost+Fixed O&M cost) are correlated in the following manner:

                                         Oil
                      Fuel           Variable O&M             Fixed O&M
           Fuel       0.48              0                       0
Coal    Var O&M       0                 0.7                     0.1
      Fixed O&M       0                 0.1                     0.7

Suppose Technology A is COAL and Technology B is OIL. How do I find the correlation between the total costs of the two technologies, that is COAL and OIL (one correlation coefficient) when three categories of costs are already correlated as given above? I have the weights of cost components of each total technology cost and the standard deviation of each cost component of the two technologies as well.

Answer 1

Let me update your table to reduce confusion by making different variable names unique:

                                         Oil
                     Fuel.O          VarO&M.O               FixedO&M.O
           Fuel.C     0.48              0                       0
Coal     VarO&M.C     0                 0.7                     0.1
       FixedO&M.C     0                 0.1                     0.7

You have given 9 entries of a correlation matrix, but there should be a further 6 subdiagonal elements, being the correlations within Coal and within Oil.

For further simplicity:

$C_f$ Fuel.C
$C_v$ VarO&M.C
$C_x$ FixedO&M.C
$O_f$ Fuel.O
$O_v$ VarO&M.O
$O_x$ FixedO&M.O

$C = C_f + C_v + C_x$

$O = O_f + O_v + O_x$

$$\operatorname{Cov}(C,O) = \operatorname{Cov}(C_f + C_v + C_x,O_f + O_v + O_x)$$

$$= σ_{C_f}⋅σ_{O_f}⋅0.48+ σ_{C_v} ⋅σ_{O_v}⋅0.7+ σ_{C_x}⋅σ_{O_x}⋅0.7\\ + σ_{C_v}⋅σ_{O_x}⋅0.1+ σ_{C_x}⋅σ_{O_v}⋅0.1$$

Using the "no correlation within" specified in comments:

$σ_C= \sqrt{σ_{C_f}^2+ σ_{C_v}^2+ σ_{C_x}^2}$
$σ_O = \sqrt{σ_{O_f}^2+ σ_{O_v}^2+ σ_{O_x}^2}$

$\text{Corr}(C,O)= \text{Cov}(C,O)/ (σ_C⋅σ_O)$

From there, it's impossible to simplify further, but the question states that the remaining quantities $σ_{C_f}, ..., σ_{O_v}$ are all known so it's a simply matter of substitution from here.