3.12 The Capital Asset Pricing Model (CAPM)
The CAPM will provide us with a discount rate (“R” or “k”), which is relative to the amount of “market” (or “systematic”) risk a portfolio has. The greater the market/systematic risk, the greater will the return (“R”) be according to this model. Further, the model depicts a linear and positive relationship between risk and return.
The model also assumes that investors are “rational” and that the market is “efficient.” If investors were not rational, they would not demand more return for taking on more risk. The risk/return relationship is thus positive. If markets were not, by and large, “efficient,” prices would not reflect this rationality; market prices would be random and not reflect accurate “intrinsic values.” Intrinsic Value has to do with what the true value of a stock should be.
The return (“R”) to the investor will be the discount rate (k), which was used when doing Capital Budgeting. Remember that the return to an investor is the economic cost, which the corporation must provide. R and K are two sides of the same coin. Once again, Return to the investor is the economic cost to the corporation. The investor expects a certain return which must then be provided by the corporation. If the corporation does not meet the investor’s expected or “required” return, s/he will sell the stock and invest in a better alternative, or vote the company’s management out. Let’s look at the math of this linear model.
The CAPM indicates that a portfolio’s “required” or ‘expected” return (“R_{P}”) is equal to – in the manner of a linear equation – the risk-free rate of interest (“R_{F}”) plus a “risk premium” (“R_{M} – R_{F}”) as follows:
R_{P} = R_{F} + (R_{M} – R_{F}) β_{P }
The risk-free rate of return represents an investment that presents no risk. Its Beta would be Zero. That means that the actual return will equal the expected return. There is no deviation or variability of actual return from expectations. As Beta increases, however, so too does risk, i.e., variability of return, so that the actual return may either exceed or fall short of its expected return.
Again, this is a linear model and linear equation. At this point, some readers may prefer to go directly to the “Diagram of the CAPM” (below), and then to return here for explanation and insight.
The variables in the CAPM equation (above) correspond to the standard linear formula: y = mx + b although the order of the constants and variables is altered for the CAPM. Here is the correspondence of the contents of the two contrasting formulae:
Linear Equation | Definition | CAPM Equivalent | |
y | Dependent Variable | R_{P} | Variable |
m | Slope | R_{M}-R_{f} | Constant |
x | Independent Variable | β_{P} | Variable |
b | Vertical Intercept | R_{F} | Constant |
The CAPM provides a dependent variable, “R_{P}” (or just R) that represents the investor’s “required (portfolio) return.” That is to say, given the formula, that a rational investor acting in an efficient market would not accept less than “R” as his portfolio return given the correlative risk of the investment; to do otherwise would be irrational. S/he therefore requires a return equal to R.
One may also think of R alternatively as the expected return one anticipates. Again, should the investor’s return fall short of his expectations, s/he may sell the stock and purchase another stock that is expected to meet his/her “requirements.” S/he can also vote out the firm’s management. To “expect” here is to “require.”
The required return, R, derived from the CAPM, is used as the discount rate, which we import into the dividend discount model. Given the Dividend Discount Model (DDM) and the concept of the Time Value of Money (TVM), as R, the discount rate rises, price goes down.
The risk-free rate (R_{P}) represents the CAPM’s vertical intercept. (Try drawing this now, with risk or Beta on the horizontal axis, and R_{P} along the horizontal, somewhat about the point of origin.) The risk-free asset has, at the risk of sounding redundant, no risk, but will nonetheless provide some positive return.
The risk-free asset is also referred to as the “Zero Beta Asset” simply because there is no (market) risk. As it has no risk, and as the horizontal axis denotes risk and is measured by Beta (while the vertical axis denotes return), R_{F} will reside on the vertical axis. Zero risk of the horizontal axis will be noted at the diagram’s point of origin. Even though there is no risk, the risk-free asset will provide a positive return or else no one would invest in it.
The phrase “Market Risk Premium” (“MRP”) refers to the incremental market return (R_{M}) above the risk-free rate of return (R_{M}-R_{f}). This premium corresponds to the market’s risk-level, which exceeds zero by a certain amount (MRP on the vertical; risk on the horizontal axis). You may (wish again to) skip ahead and view the CAPM diagram on the following pages.
The MRP is defined as “R_{M} – R_{F}.” Note that we use “R_{M},” the return on the “market portfolio.” If you invest in the market as a whole (e.g., by buying an index fund) or choose an investment whose risk equals that of the market’s, you will earn “R_{M},” i.e., the market return, which is equal to the risk-free rate of return plus the market-risk premium, i.e., R_{F} + (R_{M} – R_{F}).
When the investor chooses to invest in a portfolio other than the “market portfolio” (or in an asset whose riskiness differs from the market’s) his/her return will be either greater or less than the market return as defined by the slope of the line. The investor’s investment risk will be either to the right or left of the market’s risk, and his/her return will then “travel” along the CAPM line – up or down. One last time, the relationship between market risk and return is linear. The line is referred to as the “Security Market Line (SML).”
Beta
You may ask why R_{M} – R_{F} does not look mathematically like the slope. Slope should look like Δy / Δx, but here there clearly is no denominator! If you think of R_{M} – R_{F} as “Rise,” or as the increment along the vertical axis above the risk-free intercept, then what is the value of the denominator? Where is it in the CAPM?
Remember that R_{M} – R_{F }is the incremental return, the Market Risk Premium (MRP) awarded the investor for taking on any risk (the denominator noted on the horizontal axis) greater than zero – and thus earning a return greater than RF. In order to better understand the denominator, which is the risk to which we refer, let us first turn to the mathematical definition of risk – “Beta.”
β_{P }= (σ_{P} ÷ σ_{M}) × r _{P, M}
This formula says that the risk of a security (or portfolio of securities, as the case may be) is equal to the absolute volatility (in statistics, this is called the “standard deviation”) of return of the portfolio (σ_{P}), relative to (divided by) the market volatility (σ_{M}); we can define “market” as also referring to the “system,” as it is often called. If you are invested in the market portfolio itself (which you can do by buying an index fund), this ratio will be equal to one, i.e., (σ_{P} ÷ σ_{M}) = 1. The numerator and denominator will be the same.
Some securities will have more or less volatility than the overall market and thus greater or less than the market Beta (β_{M}) on our graph of the linear formula. Beta represents market/systematic risk, or the risk of a portfolio, relative to the “system,” the “market.” If your portfolio has more than the market risk, σ_{M}, the ratio will exceed one and vice versa.
Imagine that a portfolio has a standard (or average) deviation (of return) of plus and minus 10%. This means that on some occasions it may return as much as 10% more or less than its mean/expected return. That’s volatility! Risk then comes to mean the extent to which the return actually earned may differ from expectations (i.e., both better and worse than expected). Imagine also that the market’s volatility is 5%. The stock you chose is twice as volatile as the market! However, that is not all.
The ratio of standard deviations (σ_{P} ÷ σ_{M}) is then multiplied by the correlation coefficient of the portfolio’s and the market’s return (r _{P, M}) in order to place the relationship in a time-sensitive context. In short, correlation serves to pair up both the direction and extent of the portfolio’s return movement with the overall market’s. If, for instance, the portfolio in question is rising while the market is falling, the said counter-volatility will have the effect of reducing the portfolio’s relative risk; the “r” correlation coefficient could then be as low as -1, indicating (perfectly) negative correlation. Beta thus becomes a measure of relative volatility, rather than absolute. Remember, Beta is the only variable in the CAPM.
Getting back to risk as the denominator, if we establish that Beta is a relative measure of risk, and that the portfolio’s return is relative to the market return for a given level of portfolio risk, then the market’s risk is relative to itself and perfectly correlated with itself – a redundancy, more formally known as an “Identity,” or “Truism”! Of course, this must also be technically mathematically true.
β_{M} = (σ_{M} ÷ σ_{M}) × r _{M, M} = 1
If the market Beta is equal to one, then the denominator beneath “R_{M} – R_{F},” is also one and, as such, need not be explicitly articulated as the denominator to R_{M} – R_{F} in the CAPM formula. The “Rise” will be R_{M} – R_{F} and the “Run” will be one – Beta minus Zero.
Similarly, if a stock’s Beta is greater than one, the stock is relatively more volatile than the market. If Beat is less than one, then it is less volatile than the market. Even if it is less volatile than the market, the stock may still be volatile in an absolute sense, if the market too is quite volatile. Because the world is so small, it is virtually impossible to find securities that are negatively correlated with one another. Correlations thus tend to fall between slightly above zero and one, the latter of which indicates perfect positive correlation.
The definition of “market” is in the eyes of the beholder, or should I say researcher. In reality, the “market portfolio” is defined, most often, as the Standard & Poor’s 500 Index, but one may choose any market measure that is suitable.
To summarize, the required return is equal to the risk-free rate plus a risk premium, which is the market return less the risk-free rate modified by the slope of the SML and the investor’s choice of Beta. A picture is worth a thousand words, so let’s diagram an example (see below).
One more thing first. Recall that the “R” in the CAPM is imported into the Dividend Discount Model as its discount rate. We now see that if the required return goes up (and prices go down), the economic costs to the corporation also go up. Fewer capital projects will be accepted because they are now more costly and thus will not satisfy capital budgeting requisites and expected corporate growth will go down. There will be less capital investment in general.