Understanding Geometric Mean Return for the Yield Curve [CFA Level 1 Notes #33]

Alright, FINALLY I get to talk about the Yield Curve!!!

The Yield Curve is a huge deal.

One reason — this stuff can absolutely trip you up even in Level 2.

But we’re not just grinding for a certification, are we!??!? This matters in finance, like, for real. So let’s actually nail it down~

There are 4 things introduced here, but 2 of them are basically the same thing, so in the end we only really need 3 things locked in :-)

OK OK OK OK OK OK OK OK OK OK OK OK OK.

In my earlier posts, even just for finance stuff — and bonds especially — I kept saying: you have to understand the principle of what a geometric mean return is.

https://blog.naver.com/gdpresent/222305107421

Everything about returns (geometric mean, compound returns, continuous compounding, lognormal distribution) — I finally wrapped all my probability theory studies hahahahahaha finished in 117 pages total!!!!!!!!!! That… blog.naver.com

I kept linking that post over and over,

but reading all of it drags you into continuous compounding and lognormal distributions and — yeah, a bit much for right here.

So I’ll just lob out the one piece I actually need — the geometric mean part — and then we dive in.

The arithmetic mean….. the one where you just mindlessly add everything up and divide by how many things you added —

that everyday arithmetic mean we all use —

what does it actually mean????

Say I earned money day by day like this:

+10,000 won + 20,000 won + 10,000 won + 30,000 won + 50,000 won + 10,000 won + 20,000 won

Say that’s one week’s worth of earnings.

Then the natural thought pops up:

“On average, how much money per day should I think of myself as having made this week?”

With that thought, you say: if I got x ten-thousand-won each day for 7 days,

+10,000 won + 20,000 won + 10,000 won + 30,000 won + 50,000 won + 10,000 won + 20,000 won

what’s the value of x that makes this work out the same?

And you go, “Ah~ so I earned about 21,428 won per day over the week~” and call it done.

But flip this to return rates and — BZZT. Wrong.

Say over one week of stock investing

you got returns of 1%, 2%, 1%, 3%, 5%, 1%, 2%,

you must NOT think “oh cool, I made about 2.14% per day on average”!!!!!!!!!!!!

Because the moment we’re talking in % units, your account doesn’t grow by addition — it grows by multiplication.

So “how much did I earn per day on average” has to be the average of multiplication.

(If we were talking not in % but in the raw won-amount the account went up or down, arithmetic mean would’ve been totally fine.)

So what principle do we use here?

Let me talk about rectangles for a sec.

I had a farm. It was 9 wide, 3 tall.

This time I bought the surrounding land to expand, and now it’s 18 wide, 9 tall —

meaning I stretched it 2x horizontally and 3x vertically.

“Aha!!!! So on average I expanded by (2+3)/2 = 2.5x —”

BZZT. Wrong. You must NOT say the width and height got scaled up by 2.5x on average!!!!!

So what do you say instead!!!!

You say: both sides got multiplied by the same factor x. Solve for x.

The width and height expanded by an average of √6 times!!!!!

(I’m gently, gently, gently ramping up your feel for the geometric mean here. About to make it nauseating in a sec, so grab the principle of “average of multiplication” NOW!!!)

When we talk about returns — 10%, -10%, whatever —

we’re talking about something that grows by multiplication.

So: an account with 100,000 won earns +10% on day 1 and -10% on day 2.

You must NOT say the average return was 0%!

The account went $100{,}000 \to 110{,}000 \to 99{,}000$. It ended at 99,000 won.

So when you ask “what % did I earn (or lose) on average per day?” —

this person saw +10% / -10%, and instead of breaking even, on average they lost about 0.501% per day.

(There’s actually a hard inequality for this — the AM-GM inequality — saying the geometric mean is always ≤ the arithmetic mean.)

Let’s check —

imagine the person right next to them who actually did lose 0.501% every single day.

$$100{,}000 \times (1 - 0.00501)^2 \approx 99{,}000$$

Right!?!??!?!?!?!?

So CAGR — the reason we calculate the annualized average return this way:

is what I want to walk through.

Let’s go.

Starting point: $V(t_0) = 100{,}000$ won.

Exactly 3 years later, at $t_n$, the account is worth $V(t_n) = 2{,}500{,}000$ won.

There are roughly 252 trading days in a year, so let’s just say 1 year = 252 days.

Over 3 years, there’d be winning streaks and losing streaks,

zig zag zig zag zig zag zig zag zig zag zig zag zig zag — and 100,000 won turned into 2,500,000 won.

To work it out, I first have to explain what’s actually going on under the hood —

that’s just compound returns, right?????????????

Let’s look.

Say at the exact 1-year mark, the account that started at 100,000 won is now 500,000 won.

That’s the result of stacking up 252 days.

Say the return on day 1 was $r_1$. How’s $r_1$ actually computed?

It’s: account on day 1 divided by account on day 0 — meaning, the account on day 1 grew by a factor of $r_1$ compared to day 0. Right?

$$r_1 = \frac{V(t_1)}{V(t_0)}$$

So the return on day 2 would be computed the same way.

$$r_2 = \frac{V(t_2)}{V(t_1)}$$

Same story all the way through day 252.

$$r_{252} = \frac{V(t_{252})}{V(t_{251})}$$

The 252-day process where 100,000 won becomes 500,000 won over a year goes like this:

$$V(t_{252}) = V(t_0) \cdot r_1 \cdot r_2 \cdots r_{252}$$

Right?!?!?!?

If you actually plug in each $r_i$ using its $V$ expression —

$$V(t_{252}) = V(t_0) \cdot \frac{V(t_1)}{V(t_0)} \cdot \frac{V(t_2)}{V(t_1)} \cdots \frac{V(t_{252})}{V(t_{251})}$$

You see it? Cancel, cancel, cancel???????????????

$$V(t_{252}) = V(t_0) \cdot \frac{V(t_{252})}{V(t_0)}$$

OK but that’s not actually the point I wanted to make.

The point is:

$$V(t_{252}) = V(t_0) \cdot r_1 \cdot r_2 \cdots r_{252}$$

this is what matters.

$t_0$ to $t_n$ is exactly 3 years.

So over $252 \times 3 = 756$ days,

100,000 won became 2,500,000 won:

$$V(t_n) = V(t_0) \cdot r_1 \cdot r_2 \cdots r_{756}$$

OK OK OK OK OK OK OK OK OK OK OK OK OK.

Here’s where we ask the question:

“I multiplied my account by 25 over 3 years — on average, by what factor per year should I think of myself as having multiplied it?!?!?!?!??!?!?”

That number is CAGR.

And boom, the formula falls right out.

$$V(t_n) = V(t_0) \cdot (1 + \text{CAGR})^{t_n - t_0}$$$$\text{CAGR} = \left(\frac{V(t_n)}{V(t_0)}\right)^{\frac{1}{t_n - t_0}} - 1$$

That’s what we’re writing!!!!!

$$\text{CAGR} = 25^{1/3} - 1 \approx 1.9240177$$

So on average, you earned about 192.40177% per year.

(Wow. What a dream number….. heh heh heh.)

And now it should also click why that last piece — “$t_n - t_0$ is the number of years” — is there?!?!?!?!?!?!

Phew.

So this is exactly why —

in the Performance Analysis Report I made:

https://blog.naver.com/gdpresent/222479855152

Asset allocation performance visualization (60:40, all-weather portfolio) [My CFA Studies #3.] In the previous post https://blog.naver.com/gdpresent/222451139125 The reason why you should do asset allocation in finance… blog.naver.com

the number is called CAGR (Compound Annual Growth Rate / annualized geometric mean return). That’s the principle behind it.

Ahh….. I was supposed to mention this briefly as a lead-in to the Yield Curve,

and somehow I ended up writing a whole post about returns…

This is what “going off on a tangent” means I guess (cries) (cries) (cries) (cries) (cries) (cries)

Sorry…..

I got a little too excited………

hahahahahahahahahahahahahahahaha

haha.

Ended up being a pretty fun little arithmetic session………

Next post, for real, we get into the Yield Curve.

I am con!!!!!!vinced!!!!!!! that this understanding of the geometric mean return

will clearly, definitely pay off for you —

it’ll carry you all the way through Level 2!!!!!!!!

Honestly, this geometric mean stuff — I should’ve written it at the very start of Fixed Income.

It would’ve been the right call,

but I was lazy and just plowed ahead…………..

Couldn’t dodge it forever though, so here we are, writing it now….. heh heh heh.

Hehehehehehehe — I should pin this at the top of the first Fixed Income post and tell everyone to read it first~

Originally written in Korean on my Naver blog (2021-12). Translated to English for gdpark.blog.