Extension of Stock Valuation Formula to Account for Uncertainty
The valuation of stock was computed in the previous section as $$P_0/R_0 = f m \exp(-r T) /y + f_d m \frac{1-\exp(-r T)}{r}$$ where $y$ is the numeraire discount rate and $r$ is the discount rate adjusted by growth $g$, dilution $d$, and average fx and cpi variation: $$r:= y - (g + d\ln fx +d\ln cpi -d)$$
The formula above assumes that $r$ is known. It should be enhanced to take uncertainty of $r$ into account, as $r$ depends on growth, dilution, fx and cpi.
We now evaluate the two terms of the above formula under random $r$ by computing an expectation of the two valuation terms over normally distributed values of $r \sim N(\mu,\sigma)$.
Discounted Future Value Term Under Growth Uncertainty
The first term in the valuation formula corresponding to terminal value can be computed using the MGF of the gaussian integral. $$ E(\exp(-r T)) = \exp(-\mu T + \frac{1}{2} \sigma^2 T^2)$$ This first term shows that stock valuation increases with growth variance for a given average log revenue growth.
Dividend Term Under Growth Uncertainty: Closed Form
The second term is more involved, we want to calculate $E(f(T))$
with $f(T) := \frac{1-\exp(-X T)}{X}$ and $X \sim N(\mu,\sigma)$
We derive $f$ w.r. $T$: $$f'(t) = \exp(-X t)$$ Taking the expectation: $$E(f'(t)) = \exp\left(-\mu t + \frac{1}{2} \sigma^2 t^2\right) = \exp\left( \frac{1}{2}\left(\sigma t - \frac{\mu}{\sigma}\right)^2\right) \exp\left(-\frac{1}{2}\frac{\mu^2}{\sigma^2}\right) $$ Going back to our initial problem: $$E(f(T)) = E(f(0)) + \int_0^T E(f'(t)) dt$$ This integral can be expressed from the imaginary error function erfi or Dawson function: $$\textrm{erfi}(x) := \frac{2}{\sqrt{\pi}} \int_0^x \exp(t^2) dt = \frac{2}{\sqrt{\pi}} \exp(x^2) D(x)$$ We get, using the erfi function: $$E(f(T)) = \exp\left(-\frac{1}{2}\frac{\mu^2}{\sigma^2}\right) \frac{\sqrt{\pi}}{2}\frac{\sqrt{2}}{\sigma} \textrm{erfi}\left(\frac{1}{\sqrt{2}}\left(\sigma T - \frac{\mu}{\sigma}\right)\right)$$ Using the Dawson function $D$ for better normalisation : $$E(f(T)) = \exp\left(-\mu T + \frac{1}{2} \sigma^2 T^2\right) \frac{\sqrt{2}}{\sigma} D\left(\frac{1}{\sqrt{2}}\left(\sigma T - \frac{\mu}{\sigma}\right)\right)$$
This formula seems to perform ok with the scikit implementation of $D$ when $\sigma>2 \mu$, but does not perform well compared to a simple numerical integration when $\sigma < 2 \mu$, which is the domain that interests us.
Conclusion
Whether we use a closed form or numerical integration, both the first term and second term increase with $\sigma$. This can be proved given the closed form of the first term, and the convexity w.r. to $X$ of the function $\frac{1-\exp(-tX)}{X}$.
What we observe is that the uncertainty adjustment can be very large when $\sigma > \mu$, which often happens, it corresponds to a version of the Siegel paradox: a risk neutral investor would rather hold many stocks with 50% chance of halving or doubling each rather than stocks with much less volatility that only have a chance of going up or down 5%, because the expectation of the first bet is much higher. Increasing the number of high risk stock bets diversifies away the idiosyncratic risk of holding any single stock.
As shown, the volatility of a growth depends on the FX and CPI used as a monetary base to express revenue growth. At this point, we have two remarks:
- the conventional wisdom of value investing is to chose boring stocks, actually, using volatile growth stocks boosts performance
- it remains to be seen what the appropriate numeraire is for stocks, as a USD numeraire will lower US stock volatility, a foreign stocks bias may appear. Alternatively, a basket of currency could be used to avoid handicapping USD compared to EUR and JPY stocks. Another possibility, given that CPI data are underestimated for political reason, is to use Gold as numeraire. Finally, If the numeraire is a total return SPX index, stocks whose revenue growth follow the fortunes of the SPX get penalized.