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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. 🙂

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. 🙂

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  1. Singapore Recession says


    That’s a cool explanation video.


  2. 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. 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. 🙂

    (Thanks for the link today, too!)

  4. 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. 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. 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.

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