Sources of Various Risks

So we can earn excess returns from certain stocks, bonds, or some mix of the two in a portfolio.

Excess return — remember? — is how much your return departs from the risk-free rate.

And the risk-free rate is what you get from parking your money in ultra-safe stuff. T-bills, T-notes, T-bonds, that family.

Why are those things “ultra-safe”???

Because they’re issued by the U.S. Treasury. And the U.S. Treasury cannot go bankrupt.

The dollar is the world’s reserve currency, and if things ever got really desperate, the Treasury could just… print money. So yeah — the Treasury cannot go bankrupt.

Which means these are ultra-safe assets.

Which means the returns on them count as risk-free. (And, as safe as they are — the returns are gonna be low. No free lunch.)

Anyway. When you put money in other securities,

you can pick up higher returns than what those guys give you,

and the gap between your return and the risk-free rate — that’s the excess return.

Say I dump some money somewhere and get back 13%, while at that moment the T-bill is paying 3%.

Then 10% is my excess return.

So — why does excess return happen in the first place?!?!!!!

I want to break it into 3 buckets.

First: when the market is hot, excess returns can happen.

Let’s call market conditions

$$R_M$$

Now, even if the market as a whole is doing well, some companies are gonna be super sensitive to that,

and some companies are barely gonna feel it.

(Capital M? Yeah, you can just think of it as the M in “market.”)

So each company — each security — gets pushed around by the market by a different amount.

We’ll write that amount of pushing-around as

$$\beta$$

(If beta is bigger than 1, the company moves more more more more than the market does,

and if $0 < \beta < 1$, the company is kinda chill about market swings. Stocks with $\beta < 1$ are called defensive stocks, apparently.)

OK but is excess return only about the market?

Nah. There can be excess returns — or anti-excess-returns, the bad kind — from stuff that’s totally specific to one company.

Each individual company has its own little dramas that can move its return.

Like, the CEO suddenly turns into Steve Jobs. Or the factory burns down overnight.

Wild stuff. It happens.

We write this kind of risk as

$$e$$

This $e$ is called firm-specific risk,

and we say its expected value is 0.

Which makes sense — you can’t really sit there and assign an expected value to something genuinely unpredictable. By definition, you can’t predict it. So we shrug, write $E[e] = 0$, and move on.

And there’s one last source.

We write it as

$$\alpha$$

and call it security alpha.

Definition: “the expected return on a stock that’s above and beyond whatever return the market index movement is dragging along with it.”

OK so isn’t this kind of like… this scenario?

Imagine some microorganism-related stock in the market suddenly pops —

and because of that, companies making fish tanks for growing microorganisms, companies making lab equipment, etc.,

couldn’t a chain reaction kick off?

The difference from $R_M$ is: $R_M$ is something the whole market is feeling,

but security alpha isn’t market-wide — it’s coming from influence on a specific slice of the market.

Make sense, right?

Oh, and one more thing about $\alpha$ — security alpha is a specific constant.

Why?! Because it only kicks in under specific conditions, it’s hard to call it some kind of statistical quantity.

$\alpha$ is a constant!!!!

So if I throw all of that into one expression,

it’s a linear sum:

$$R = \alpha + \beta R_M + e$$

Now, it’s worth thinking about how likely a return like this is to actually happen,

and that “likelihood” — the value called standard deviation — is the point in a Gaussian where the probability density drops to $1/e$ of its peak,

so let’s think about the standard deviation of this $R$.

If you turn this thing

$$\sigma^2(R) = \beta^2 \sigma^2(R_M) + \sigma^2(e)$$

into variance language…

it would be cleaner to do it with $\langle \cdot \rangle$ — bra-ket notation, the kind you see in stats —

but eh, I’ll just go with intuition.

$$\sigma^2(R) = \beta^2 \sigma^2(R_M) + \sigma^2(e)$$

I wrote it out in this kinda elaborate way,

but the whole point I’m trying to make is just this one line.

$$\sigma^2(R) = \beta^2 \sigma^2(R_M) + \sigma^2(e)$$

Honestly the conclusion of this whole post is: there exists a coefficient called $\beta$!!!!

There’s this thing called $\beta$, and it tells you how sensitive you are to swings in market conditions!!!!

“Let’s get to know this thing!!” — that was the whole goal of the post.

But once I started talking about beta, I felt like I had to introduce all the other stuff hanging out next to it,

and… this is what came out.

Alright, moving on.

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Originally written in Korean on my Naver blog (2016-04). Translated to English for gdpark.blog.