### Get new articles by email:

Oblivious Investor offers a free newsletter providing tips on low-maintenance investing, tax planning, and retirement planning.

Join over 20,000 email subscribers:

# Volatility Increases Return: Mathematical Proof

A couple weeks ago I mentioned that–when dollar-cost-averaging into an investment–volatility actually helps your returns. And at the time, I had created a few hypothetical scenarios to illustrate the idea.

Today I want to show mathematically exactly why volatility can be so beneficial. Don’t worry. At just 3 minutes in length, it doesn’t get too grisly with math. ðŸ™‚

Having re-watched the video after making it, perhaps I jumped to the conclusion too quickly at the very end. So here’s a more thorough explanation:

In the final form of the equation, we have (1-x^2) in the denominator. When X (our volatility measure) increases from zero to one, this (1-X^2) amount decreases toward zero. And as our denominator gets smaller, the total value of the fraction increases, thereby meaning that the other side of the equation–shares purchased–increases as well.

And (all other things being equal) if you get more shares for your money, your return goes up.

Let me know if you have questions, if anything wasn’t clear (visually in the video), or if anything was confusing in my explanation. Alternatively, if there’s something I completely missed in my analysis, feel free to point that out as well. ðŸ™‚

### New to Investing? See My Related Book:

 Investing Made Simple: Investing in Index Funds Explained in 100 Pages or Less
Topics Covered in the Book:
• Asset Allocation: Why it's so important, and how to determine your own,
• How to to pick winning mutual funds,
• Roth IRA vs. traditional IRA vs. 401(k),
• Click here to see the full list.

A Testimonial:

"A wonderful book that tells its readers, with simple logical explanations, our Boglehead Philosophy for successful investing." - Taylor Larimore, author of The Bogleheads' Guide to Investing

1. Singapore Recession says

Hey,

That’s a cool explanation video.

Ren

2. Miranda says

Great, informational video. My friends and relatives have been surprised that I’m not more worried about my retirement account. I just shrug and say, “My money is buying more right now. Later, I’ll be very happy.”

Also, I’ve been doing a little more buying on the dips lately for my other investments. It’s been good, and I’ve even made a little extra scratch when I sell on the rallies.

3. Monevator says

Nice video Mike, I’ve just posted it up on Monevator.

What about a lump sum? My understanding is because the market tends to rise, it’s better most of the time to invest a lump sum right away (a few times you’ll lose a lot, but generally over the very long-term you’ll win more times – although you may well argue you’d rather potentially lower returns in exchange for removing the risk of investing at a market peak).

Anyway, might be another video idea for you. ðŸ™‚

(Thanks for the link today, too!)

4. Mike says

Monevator, that’s a great question. (In fact, just yesterday, a friend in that very situation asked me what I would suggest.)

I think I agree with you. At least, I know that when I have a lump sum, I’ve always just invested it right away rather than spreading it out. My line of thinking is exactly what you stated: most of the time, the market rises. So to keep money out is to essentially bet against it.

On the other hand, I definitely don’t think it would be unreasonable for somebody to try and spread it out over a few weeks or months given precisely how volatile things are lately.

5. Kevin says

A caveat could be that the lump sum you’re investing “right away” should be the kind of lump you can afford to take some risk with, and definitely as diversified as possible to lessen the risk.

6. Mike says

Yep, I agree completely, Kevin. To do it as a lump sum definitely exposes you more heavily to the impact of market volatility.

And I’d always agree that it’s wise to be widely diversified, whether DCAing or investing a lump sum.

Disclaimer: By using this site, you explicitly agree to its Terms of Use and agree not to hold Simple Subjects, LLC or any of its members liable in any way for damages arising from decisions you make based on the information made available on this site. The information on this site is for informational and entertainment purposes only and does not constitute financial advice.

Copyright 2023 Simple Subjects, LLC - All rights reserved. To be clear: This means that, aside from small quotations, the material on this site may not be republished elsewhere without my express permission. Terms of Use and Privacy Policy

My Social Security calculator: Open Social Security