Capital Asset Pricing Model (CAPM) [Corporate Finance I Studied #12]

Now the content here is CAPM (Capital Asset Pricing Model).

But before getting into CAPM proper: when we were discussing the Efficient Frontier in the immediately previous post, there were some things I didn’t mention, a few things I deliberately saved to talk about here. I’ll mention those before going in.

Everything is already determined

First, all the stocks that exist in the world are already determined. This means the (Previous, Markowitz) Efficient Frontier is already, already determined. Let’s say it’s determined like this.

Hand-drawn axes with μ going up and σ going right, and the curve of the previous (Markowitz) efficient frontier

But, after we derive the Efficient Frontier like this, we also thought about a risk-free asset, right???? And since this risk-free asset is also already determined as one that exists in the world,

The same axes with the risk-free rate μ_f marked as a red dot on the μ axis

I’ll say the risk-free rate of the risk-free asset is determined like this. Then the New Efficient Frontier created by this predetermined efficient frontier and the risk-free interest rate will also be determined.

Um.. so here’s how the New one was derived: we said there’s a Frontier that maximizes each individual’s utility more than the Efficient Frontier does, and that Frontier, we could say, is the one where the straight line drawn from that point is ’tangent’ to that curve — that’s what maximizes human utility. And the line drawn from this point that’s tangent to that curve is uniquely determined, right???? It’s already determined as 1.

The green New Efficient Frontier line from μ_f touching the previous efficient frontier at one point, with μ_m on the μ axis and σ_m on the σ axis

The Market Portfolio

That is, some portfolio that belongs to the Efficient Frontier and builds the New Efficient Frontier, with an expected return and standard deviation of return of (σ_m, μ_m) , is uniquely determined — that’s what I’m saying.

We’ll call this portfolio the Market Portfolio!!!!

Then this New Efficient Frontier can be expressed in equation form like this :)

The New Efficient Frontier as an equation: μ = ((μ_m − μ_f)/σ_m) σ + μ_f

Ah but, the $\mu$ and $\sigma$ without subscripts that are being decided now are the expected return and standard deviation of return of the “risky + risk-free” portfolio, right???? So I’ll use the subscript $p$ to make the meaning more firm.

The same equation with the subscript p: μ_p = ((μ_m − μ_f)/σ_m) σ_p + μ_f

But here’s the thing: would it even be possible to construct such a portfolio?????

Finding one would be hard. Agh!!! But!!! There’s something similar to that!!! No no, something that can be seen as similar in form: the composite stock price index — things like the S&P 500, or, here in Korea, the KOSPI 200!!

Why!! The composite stock price index can’t be said to move by reflecting the movements of all individual stocks in the world, but still, it’s an index that bundles together the really big ones and represents the overall movement of that bundle. So thinking of the expected return and standard deviation of return of the composite stock price index as (σ_m, μ_m) … that’s the majority opinion, and there are actually companies using this exact principle, so it seems hard to call it an outlandish assumption.

CAPM and its assumptions

Then now we can head to CAPM (Capital Asset Pricing Model). First, as it’s written in English, CAPM is the Capital Asset Pricing Model. So it can be described as “a model that predicts the expected return commensurate with the risk of an asset (when the market is in equilibrium, where supply and demand match)”.

That is, for an individual asset we want the expected return $\mu$ that goes with its standard deviation of return $\sigma$, and to say ‘for an individual asset’ more directly, I’ll write $\mu$ and $\sigma$ with the subscript $i$ as I explain. And what are we going to predict it with? As mentioned above, we’ll predict it based on the composite stock price index.

Uengh…T_T what’s this all of a sudden…T_T

Before anything else, there are assumptions embedded in the model above, so I’ll mention those first.

All investors participating in the market decide whether to invest or not using $\mu$ and $\sigma$ as measures, and all investors are risk-averse investors.
The market is a perfect capital market. (No transaction costs like taxes, fees, etc.), and also individual investors are so numerous that each individual cannot affect the market price.
All investors have the same probability distribution for future asset returns (homogeneous expectation)
All investors can freely borrow and lend at the risk-free rate of return.
The investment period is a single period.
All publicly tradable financial assets are investment targets. (Excluding non-financial assets such as human resources)
The market is in equilibrium

From the New Efficient Frontier to CAPM

And now I’ll rewrite the New Efficient Frontier equation from before.

The New Efficient Frontier equation again: μ_p = ((μ_m − μ_f)/σ_m) σ_p + μ_f

The meaning of this equation can actually be viewed like this, right???? You can read it in the order of 1, 2, 3 haha

The equation μ_p = ((μ_m − μ_f)/σ_m) σ_p + μ_f with three handwritten notes: 1, the expected return of my portfolio, made of the market portfolio and the risk-free asset; 2, start from the risk-free rate as the base and ask how much more we need on top of it; 3, that much is proportional to my portfolio’s risk, and the proportionality constant is (μ_m − μ_f)/σ_m

So my question is as follows.

The same equation once more: μ_p = ((μ_m − μ_f)/σ_m) σ_p + μ_f

Since the expected return and standard deviation of return of a portfolio composed of risky and risk-free assets are determined like this, for the return and risk (standard deviation of return) of an individual asset

The same line for an individual asset: μ_i = ((μ_m − μ_f)/σ_m) σ_i + μ_f, with μ_i and σ_i in red

If we shing! plug in the risk of the individual asset as input, then $\mu_i$ pops shing~ out — couldn’t we just use it like this~~~?

This is my question.

But here’s the thing. That equation with the subscript $p$ above was the New Efficient Frontier, and its target was the “efficient portfolio”. And the risk of an efficient portfolio is actually such that non-systematic risk totally flies away thanks to the diversification effect, and the only risk left is systematic risk (things like war, financial crisis — the overall… absolutely unremovable risk). That is, since the risks of individual assets “can be eliminated”, attention shouldn’t be focused on them; attention should be focused on how an individual asset’s risk is affected as the systematic risk changes.

So σ_p → σ_i it shouldn’t just go in pow! with only the name changed. σ_i This guy has to be adjusted by “how sensitively does it respond to the risk of the market portfolio that has only systematic risk (that is, to the systematic risk)” and then plugged in.

Therefore, not this one:

Not this: μ_i = ((μ_m − μ_f)/σ_m) σ_i + μ_f, with μ_i and σ_i in red

but this one:

$But this: μ_i = ((μ_m − μ_f)/σ_m) (ρ_{i,m} σ_i) + μ_f, with the correlation coefficient in blue$

That is, we adjust σ_i by its correlation coefficient with the market portfolio and then plug it in — that’s my conclusion.

This equation is exactly CAPM.

If I tidy the equation up a bit,

Step by step: μ_i = ((μ_m − μ_f)/σ_m) ρ σ_i + μ_f, with ρ written as Cov(R_i, R_m)/(σ_i σ_m), simplifies to ((μ_m − μ_f)/σ_m²) Cov(R_i, R_m) + μ_f

CAPM: μ_i = ((μ_m − μ_f)/σ_m²) Cov(R_i, R_m) + μ_f

written like this, I feel like we can sense more directly what CAPM is trying to say!!!!

The same equation with handwritten notes: 1, an individual asset’s expected return; 2, start with the risk-free rate as the base; 3, the part we get on top depends on the covariance with systematic risk, and the proportionality constant is (μ_m − μ_f)/σ_m²

(The notation is wrong. Please read what’s inside Cov as R, not $\mu$!)

Beta

Here I’ll introduce the “stock’s beta”. The covariance of an individual stock with systematic risk, normalized by σ_m² , is called the stock’s beta β_i , they say.

The definition of beta: β_i = Cov(R_i, R_m)/σ_m²

Um, then β_m would be 1, right???? That is,

β_i can be said to mean “how sensitive is it to systematic risk????”, using β_m = 1 as the benchmark.

So here’s that equation from before:

CAPM again: μ_i = ((μ_m − μ_f)/σ_m²) Cov(R_i, R_m) + μ_f

It’s CAPM, all right, but when we meet CAPM elsewhere, rather than using this equation,

The usual form: μ_i = ((μ_m − μ_f)/σ_m²) Cov(R_i, R_m) + μ_f, which equals (μ_m − μ_f) β_i + μ_f

this one is directly used a lot!!!!

The Security Market Line

One more thing!!!! Taking the x-axis as the individual asset’s beta and the y-axis as the individual asset’s expected return, and drawing the above equation

Hand-drawn graph of μ_i against β_i: a rising straight line that starts at μ_f on the μ axis, with the equation μ_i = (μ_m − μ_f) β_i + μ_f and the label Security Market Line: SML

is called the Security Market Line: SML!!!!!!

What does the SML mean?????

With the characteristics of an individual stock $\text{stock}_i$, once its sensitivity to the systematic risk that (at the time) exists in the world is determined, the predicted $\mu_i$ goes up in a straight line with beta.

It can also be said like this.

The excess-return form μ_i − μ_f = (μ_m − μ_f) β_i with two handwritten notes: the left side is the excess return over the risk-free rate, called the risk premium; the right side is the market risk premium times the sensitivity β, which decides whether we get more or less

Side note: β_p = Σ w_i β_i; this is not a single stock’s β, a portfolio’s β is computed like this

Two ways to use CAPM

Let’s briefly touch on using CAPM and move on. Apparently there are 2 ways to use CAPM, so this will be a post that introduces, just roughly, what those two ways are. The first can be named ex-ante CAPM.

Whatever I hold — an individual asset, a portfolio, whatever — to use CAPM I need μ_f , the risk-free interest rate, and for that we can get a proxy by using the CD rate, or the government bond rate, or the deposit rate, and so on.

Then the market portfolio…. Above we said we’d substitute with something similar, like the S&P 500, S&P 100, KOSPI 200, KOSDAQ… something like that, right??? That is, find out the relationship (covariance) between the individual asset and that index, and use my asset’s return and risk (standard deviation of return)~~~

But you know? The principle seems… right, and it’s easy enough, but actually… there can be many cases where it doesn’t make sense… If someone says those proxies ‘can’t be trusted’, well…. there’s nothing to say. So the second method can be named ex-post CAPM.

Anyway, let’s first organize the use of ex-ante CAPM, where we mobilize any values we can bring in. To start with,

what changed in this equation is that it became $p$ instead of $i$. Because we got the conclusion above that we can also predict the return of a portfolio composed of several individual assets, not just an individual asset, I changed it like this.

Then, the values we have to estimate from the values scattered widely in the world add up to 3 fellas: μ_f the risk-free interest rate, μ_m the expected return of the market portfolio, and β_p the beta of the portfolio I hold. As mentioned earlier, for the risk-free interest rate we said to use government bonds or the deposit rate or such things, and for the expected return of the market portfolio we said to substitute with the expected return of the composite stock price index.

How do you extract the expected return of the composite stock price index?????

We can calculate it by drawing up scenarios that predict the composite stock price index through a probability distribution, and then computing the expected value of the return from them. Like this.

Suppose the current stock price index is 1000, and the risk-free interest rate is 2%.

Scenario	Probability	Index
Boom	50%	1100
Status quo	30%	1000
Recession	20%	980

These are the probability of a boom, the probability of staying the same, and the probability of a recession — and if it’s a boom we expect the index to be 1100, if it’s a recession we expect it to be 980. Let’s say we predict the future like this.

Then, next to it, I’ll write out a table that calculates the ‘return ( R_m )’ at each of those times once more, in red.

Scenario	Probability	Index	Return
Boom	50%	1100	100/1000 = 10%
Status quo	30%	1000	0/1000 = 0%
Recession	20%	980	-20/1000 = -2%

So we can compute the expected return.

Well, something like this?

Then now we also have to estimate the portfolio’s beta. If the stocks in the portfolio are $\text{stock}_1$ through $\text{stock}_n$, we estimate each stock’s $\beta_i$ and then

The beta of a portfolio: β_p = Σ_i w_i β_i

we have to calculate it like this as the last step.

For now, let’s say there’s only stock A in the portfolio, so stock A’s beta is the portfolio’s beta.

Then the table that contains our own predictions for the future of stock A should be written like this.

Scenario	Probability
Boom	50%	10%	15%	2%
Status quo	30%	0	5%	2%
Recession	20%	-2%	-5%	2%

Looking at this table, we can calculate the ’expected’ return ($\mu$) for each. μ_f = 2% This is obvious. μ_m = 4.6% This one we calculated before. μ_A = 8% This one, if you do it, comes out like this. σ_m² = 0.2964% But since we need to find beta, let’s say we’ve found this too.

Now the portfolio’s beta — ah, since right now there’s only one stock —

The beta of stock A: β_i = Cov(R_i, R_m)/σ_m²

can be found like this, rightttt.

Argh, then we also have to find the covariance. But since we have a definition for covariance, it’s not that hard a calculation.

The covariance worked out: Cov(R_m, R_A) = Σ p_i (R_m − μ_m)(R_A − μ_A) = 0.5(0.1 − 0.046)(0.15 − 0.08) + 0.3(0 − 0.046)(0.05 − 0.08) + 0.2(−0.02 − 0.046)(−0.05 − 0.08) = 0.00402

Therefore beta is $\frac{0.00402}{0.002964} = 1.356$. Something like this, we can find it this way.

Having estimated it like this, the expected return of the portfolio containing stock A that I hold is

and when we plug in and substitute, it comes out as $0.02 + 0.026 \times 1.356 = 0.055256$ — like this, we can calculate it.

Then we can drawww the Security Market Line

The Security Market Line for stock A: it starts at 0.02 on the μ axis and passes through the point with β 1.356 and μ 0.055, labelled SML

like this — and that’s it for ex-ante CAPM.

Ex-post CAPM

But there are people who say the discussions we’ve had so far can’t be trusted. Because the tables about the future…. can’t be trusted…..? So now the ex-post CAPM story comes up, and I’ll take a really light look at how it’s handled.

Ex-post CAPM just mobilizes past data that has come and gone (using a regression model) to estimate the beta value backwards. Clearly the value $\beta$ is supposed to represent the ‘sensitivity’ between the market portfolio and my portfolio. So, using historical data over some period, when R_m was some value, at those times R_A was what — pair them up as ordered pairs and then stamp points like crazy on a coordinate plane like this,

Hand-drawn axes with R_m going right and R_A going up, and a scatter of small red dots rising to the right

gathering the past data and having a look at it, suppose the points got plotted like this.

The same scatter with a green best-fit straight line running through the dots

Then (through regression analysis) we pull out one ‘straight line’ where the error is smallest, and the slope of that line is used directly as the portfolio’s beta.

In our data, the x-axis is $R_m$ and the y-axis is $R_A$. So if you think of $x$ in Linear Regression in Finance (Derivatives I Studied #3) as $R_m$ and $y$ there as $R_A$, you can confirm that the slope of the green line in the figure above becomes the definition of beta:

The definition of beta: β_i = Cov(R_i, R_m)/σ_m²

Suppose we’ve found the portfolio’s beta like that.

We also have to ram in these two things for CAPM to be complete, and this equation

rewritten like this, pulling out a ton of historical risk-free interest rate and excess return data, estimating the $\mu_A$ value, which is the average value of $R_A$, (after judging whether using the estimated beta value is reasonable or not~) — that’s how ex-post CAPM gets completed~

Heads-up. The figures here are the original notes, slips and all. A few things to watch for:

In the handwritten formula for an individual asset, the covariance is between the returns, $Cov(R_i, R_m)$, as the text says; $\mu_i$ and $\mu_m$ are expected values, numbers without a covariance.
In the two portfolio versions of the equation, the beta should be the portfolio’s, $\mu_p = (\mu_m - \mu_f)\beta_p + \mu_f$; the pictures keep $\beta_i$ from the single-asset line.