Creating an n-factor Certainty Equivalent Discounting Formula

Question

Brealey & Myers provide a certainty-equivalent version of the present value rule, using CAPM, as follows:

$$PV_0=\frac{C_1 - \lambda_m *cov(C_1, r_m)}{1 + r_f}$$

$PV_0$ - Present Value of cash flow 1 at time 0.

$\lambda_m$ - Market price of risk = $\frac{r_m-r_f}{\sigma_m^2}$

$cov(C_1, r_m)$ - Covariance of the cash flow at time 1 with the return on the market.

I want to create an n-factor version of this same model. However, using Fama French 3-factor model as an example, the following doesn't seem to work on a toy example I've set up:

$$PV_0=\frac{C_1 - \lambda_m *cov(C_1, r_m)- \lambda_{smb} *cov(C_1, r_{smb})- \lambda_{hml} *cov(C_1, r_{hml})}{1 + r_f}$$

$\lambda_m$ = $\frac{r_m-r_f}{\sigma_m^2}$

$\lambda_{smb}$ = $\frac{r_s-r_b}{\sigma_{smb}^2}$

$\lambda_{hml}$ = $\frac{r_h-r_l}{\sigma_{hml}^2}$

Question: what am I doing wrong? Is there some way I need to adjust for the covariance amongst the factors?

===Update===

In checking my toy example again, I realized that I might in fact have the right formula above. So points/checkmarks to anyone who can either prove the above right or wrong or provide a citation to the more general form:

$$PV_0=\frac{C_1 - \displaystyle\sum_{i=1}^n\lambda_i *cov(C_1, r_i)}{1 + r_f}$$

for orthogonal risk factors $i_1,i_2,\dotsc,i_n$.

Answer 1

Alright, here's the proof (I think):

Statement of APT:

$$E(r_a)=r_f + \displaystyle\sum_{i=1}^n\lambda_i * cov(r_a, r_i)$$

Expand $E(r_a)$:

$$\frac{E(C_1)}{PV_0} - 1 =r_f + \displaystyle\sum_{i=1}^n\lambda_i * cov(\frac{C_1}{PV_0} - 1, r_i)$$

Since $PV_0$ doesn't have any covariance with $r_i$, we can reduce the above to the following:

$$\frac{E(C_1)}{PV_0} - 1 =r_f + \displaystyle\sum_{i=1}^n\frac{\lambda_i * cov(C_1, r_i)}{PV_0}$$

Rearrange:

$$\frac{E(C_1) -\displaystyle\sum_{i=1}^n\lambda_i * cov(C_1, r_i)}{PV_0} = 1 + r_f $$

And finally:

$$\frac{E(C_1) -\displaystyle\sum_{i=1}^n\lambda_i * cov(C_1, r_i)}{1 + r_f} = PV_0 $$

QED (until somebody points out a dumb error I've made).