# How is CAPM used to price an asset once it has been used to derive the assets expected return?

As I understand it (correct me if I'm wrong) the theoretical price of an asset should be the present value of all future cash-flows that it is expected to yield, discounted at the risk-free rate.

I am currently reading some notes on the CAPM which quote the key result of this model to be the equation of the security market line: $$E_i = r + \beta_i (E_M - r)$$

where:

• $$E_i$$ is the expected return on the security in question, security $$i$$ (over a single fixed time period)
• $$r$$ is the risk-free rate of return
• $$\beta_i$$ is the beta of security i
• $$E_M$$ is the expected return on the market portfolio



My confusion stems from the following statement:

The security market line can be used to estimate the prospective return that an asset should offer given its systematic risk, providing the economy is stable. This return can then be used to discount projected future cash flows and so price the asset.

Why would you discount the expected future cash flows from a security at the estimated rate of return offered by it?

Edit: Having written out this question, it has occurred to me that the theoretical price of an asset would not be the present value of all future cash-flows that it is expected to yield, discounted at the risk-free rate, but instead discounted at some risky discount-rate.

Is the expected return on the asset, as estimated by CAPM, equal to the risky discount rate that should be used to discount these expected cash flows?

• Yes, you're right. The cash flows of a firm are risky (uncertain). You cannot discount them at the risk-free rate. Your discount rate needs to reflect the (systematic) risk. The CAPM is one way of estimating the risk-adjusted return (=required return of equity holders = discount rate) which you need for discounting future cash flows (dividends). Jul 30, 2020 at 11:28
• In theory, the CAPM is a static, one-period equilibrium model. Let $D_1$ be the (random/risky) cash flow in the next period. Then, the return is $R=\frac{D_1-P_0}{P_0}$. Thus, the price is given by $P_0=\frac{\mathbb{E}[D_1]}{1+\mathbb{E}[R]}=\frac{\mathbb{E}[D_1]}{1+R_f+\beta(\mathbb{E}[R_m]-R_f)}$. Jul 30, 2020 at 11:48

# Estimating the Discount Rate

As indicated in the comments, you would use the CAPM (or another equity factor model) to model stock returns $$r_{it}$$ beyond the risk-free rate $$r_f$$ as a function of returns on a market index $$r_{Mt}$$: $$r_{it} - r_f = \alpha_i + \beta_i (r_{Mt} - r_f) + \epsilon_{it}.$$

Once we have estimates $$\hat\alpha_i$$ and $$\hat\beta_i$$, we can use the average market index return $$\bar{r}_M$$ to estimate the expected return of asset $$i$$: $$\hat{r}_i - r_f = \hat\alpha_i + \hat\beta_i (\bar{r}_M - r_f),~\text{or} \\ \hat{r}_i = r_f + \hat\alpha_i + \hat\beta_i (\bar{r}_M - r_f).$$

### Whither Alpha?

The decision about whether or not to use $$\hat\alpha_i$$ in the above can lead to debate. Some people will treat $$\hat\alpha_i$$ as correcting for trends and so will set $$\hat\alpha_i=0$$.

Others want to allow for the possibility that asset $$i$$ has some non-risk-factor related return. (Perhaps the firm has a unique patent or product.) In that case, including $$\hat\alpha_i$$ discounts cashflows further -- which makes us less happy when such a firm pays us dividends instead of reinvesting the money.)

### Why Use a Model?

Why not just measure the average stock return? That is typically too noisy. Using a model gives less noisy estimates, lets us correct for trends we do not know will continue (setting $$\hat\alpha_i=0$$), and ensures our pricing is tied to risk factors (or proxies for risk factors). The last part is important; we don't want a model that says we can get returns beyond $$r_f$$ for taking no risk.

# Why Discount at This Rate?

We discount dividends at this rate for many reasons. First, riskier cashflows need to be discounted at a higher rate than $$r_f$$ to account for their risk. Second, the rate we use is because of opportunity cost: If the firm pays dividends, the value of those cashflows should be discounted to account for the opportunity cost of keeping the cash in the firm. Had we kept the cash in the firm, it would have grown at a rate $$\hat{r}_i$$. Therefore, that is the appropriate discount rate.

# Discounting Dividends

Once we have an estimate of $$\hat{r}_i$$, we then use that to discount dividends paid to shareholders, possibly accounting for dividends growing at some rate $$g$$.

Typically, we assume equilibrium and so we pull things out of the single-period model and into a perpetuity form. This is not perfect, but multi-stage models tend to be hard to estimate since you also need to decide when transitions between stages occur.

If we characterize the futures dividends by the expected dividend in a year, we will value the equity as: $$P_0 = \frac{E(D_1)}{\hat{r}_i},~\text{or} \\ P_0 = \frac{E(D_1)}{\hat{r}_i-g}$$ if dividends grow at rate $$g$$.

### Timing of Dividends

Many firms do not pay dividends annually. However, doing all of the accounting for that here would be messy. I'll just say that we need to account for the time to the next dividend and payouts that may not be annual.

# Uncertainty of Estimation

There is one problem (and now we are going beyond what you asked): our estimates of $$\hat{r}_i$$ are uncertain. There is uncertainty in the estimate $$\bar{r}_M$$ of average market index returns; and, there is uncertainty in the estimates $$\hat\alpha_i$$ and $$\hat\beta_i$$. There might even be uncertainty if we have to estimate the dividend growth rate $$\hat{g}$$.

In these cases, we need to consider the variance of our estimates $$\sigma^2_{\hat{r}_i}={\rm var}(\hat{r}_i)$$ and $$\sigma^2_{\hat{g}}={\rm var}(\hat{g})$$.

Since $$\hat{r}_i$$ and $$\hat{g}$$ appear in the denominator, the uncertainty does not cancel out as to whether we under- or over-estimated. Consider if dividends were \$10 annually and our discount rate were estimated at 10%. Then we would price the stock at $$\frac{\\\10}{0.1}=$$\$100. However, if we were off by 1% in one way or the other, the stock might be valued at $$\frac{\\\10}{0.09}=$$\$111.11 or $$\frac{\\\10}{0.11}=$$\$90.91 -- so \$11.11 higher or \$9.19 lower.

The discount rate being lower has a larger effect than the discount rate being higher. Therefore, uncertainty about our discount rate means we need to adjust our valuation higher. You can do this by simulation or there is a closed-form solution. That is a bit complicated, but Chapter 13 in A Quantitative Primer on Investments with $$R$$ covers that and all of the above issues as well.