Deriving a Multi-Factor Certainty-Equivalent Present Value
Summary
The document asks how to extend a CAPM certainty-equivalent present-value formula to several priced risk factors. The proposed expression subtracts each factor’s price of risk multiplied by the cash flow’s covariance with that factor, then discounts the adjusted cash flow at the risk-free rate. An answer motivates this form by applying the Arbitrage Pricing Theory expected-return relation to an asset whose payoff is a future cash flow divided by its present value, then rearranging the equation.
The derivation conveys the covariance-based intuition for pricing systematic risk across multiple factors. It is presented as a tentative proof, however, and leaves assumptions implicit, including a correctly specified factor pricing relation and treatment of expected rather than realized cash flows. The question raises factor covariance, but the response does not address it directly. In general, factor covariances matter when estimating factor risk prices; the displayed sum is not justified simply by assuming the factors are orthogonal.
Key ideas
- A multi-factor valuation adjusts expected cash flow for its covariance with priced risk factors.
- The adjustment uses each factor’s risk price multiplied by the cash flow’s factor covariance.
- The proposed derivation follows by applying an APT expected-return relation and rearranging.
- The answer does not explain how correlated factors affect estimation of their risk prices.
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Full text
# Creating an n-factor Certainty Equivalent Discounting Formula
# Creating an n-factor Certainty Equivalent Discounting Formula
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 by MikeRand (score 2, accepted)
https://quant.stackexchange.com/a/3055
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.