CAPM Discount Rates, Cashflow Risk, and Jensen's Inequality

# CAPM, DCF, and Jensen's inequality


# CAPM, DCF, and Jensen's inequality












One way to value a cashflow is to first calculate the expected return from CAPM, and then use the expected return to discount the future cashflows.

The problem here is that the expected return from CAPM is an average $\mathrm{E}[R]$, and therefore, by Jensen's inequality:

$$\mathrm{E}\left[\frac{1}{R}\right] \ge \frac{1}{\mathrm{E}[R]}$$

Then I wonder why people use $\frac{1}{\mathrm{E}[R]}$ to discount cashflows.

To clarify, I will use an example.Let's assume there is only one cashflow of \$100 a year from now I want to value. According to CAPM, the expected return is $$ \bar{R}_a = R_f + \beta_a (\bar{R}_m - R_f) $$

Note that the return $R$ is defined as $R=\frac{S_{t+1}}{S_t}$.

Now the problem here is that $\bar{R}_a$ represents an average return of the following year. The actual return $R_a$ is a random variable that is not known right now. In other words, $\bar{R}_a=\mathrm{E}[R_a]$, but $R_a$ is random, and will realize different values in different alternative universes.

However, what I usually see is that we value the \$100 cashflow a year from now as $$\frac{100}{\bar{R}_a}=\frac{100}{\mathrm{E}[R_a]} $$

But shouldn't it be valued as $$\mathrm{E} \left [ \frac{100}{R_a} \right] $$?

But according to Jensen's inequality: $$\mathrm{E} \left [ \frac{100}{R_a} \right] \ge \frac{100}{\mathrm{E}[R_a]} $$

## Answer by Quantifeye (score 3, accepted)

https://quant.stackexchange.com/a/25896

What you are missing is that the cashflow itself is also a random variable. We assess the risk related to that cashflow by relating it to a linear measure of risk that is expressed in terms of variance and covariance... by happy accident this turns out to be beta, and if the CAPM actually works, turns out making our lives easier.

If you rewrite the CAPM in terms of prices instead of returns you get something that looks like

https://en.wikipedia.org/wiki/Capital_asset_pricing_model#Asset_pricing

$$P_0=\frac{1}{1+r_f}\left[\mathrm{E}(P_T) - \frac{ \mathrm{Cov}(P_T;R_M)(\mathrm{E}(R_M)-r_f)}{\mathrm{Var}(R_m)}\right]$$

So you are not discounting with an expected value...

What this formula tells us is that the expected present value of the cashflow is not the right price for a risk averse investor. The Risk averse investor requires an additional risk premium...

## Answer by Quantifeye (score 0)

https://quant.stackexchange.com/a/25884

I think that Jensen inequality in this context is relevant to averaging return over time. And that when discounting over multiple periods, convexity needs to be accounted for.

That is to say that we cannot simply discount over multiple periods by using $$\frac{1}{(1+\sum{R_t})}$$ as a discount factor. We have to really use $$\frac{1}{\prod{(1+R_t)}}.$$

The CAPM is a single period model. So we use it to value a single cash flow, we form an ex-ante estimate of beta that relates to that specific cash flow, and that is commensurate with the risk of that particular cash flow. And this beta applies to a particular horison. The resulting discount rate calculated then relates to that particular time horizon.

When Discounting a stream of cash flows we have to do this for each cash flow in order to be consistent with the single period nature of a model such as the CAPM.

If we make the simplifying assumption that all cash flow betas are the same then we could get around the problem you mention by consistently working with continuously compounded rates so that $$\frac{1}{\mathrm{exp}(\sum{R_t})}=\frac{1}{\prod{\mathrm{exp}{(R_t)}}}.$$

Using such a simplifying assumption is problematic however.

When valuing a single cash flow you need an expected return for the market proxy over that particular period, as well as a risk free rate proxy over a particular period and an estimated forward looking beta over a particular period.

I suppose one could try to use the Macaulay duration of the cash flows as the "average" time period for determining the desired inputs. But then we really are starting to stray significantly from the intended use of the CAPM.

Another question however is the validity of the CAPM in determining discount rates, but that is a different discussion all together I guess.

I am not sure if I answered the question since I am not completely sure what the question is.

## Answer by Charles Fox (score 0)

https://quant.stackexchange.com/a/41949

You are correct that something is missing: ambiguity risk. Two projects may have the same $\hat{\beta}$, but you could be more confident in your estimation of $\beta$ for one of the projects. In a single period model, all else equal you would likely prefer the one you were more knowledgeable about (ambiguity aversion). In multi-period, you might value learning and choose the uncertain project.

In the standard model, the market risk premium and $\beta$ are treated as known constants rather than random variables.