Returns and Volatility

In a Morningstar post Taylor Larimore noted that:

An ounce of higher return is worth more than a pound of lower volatility.
That's such a slick observation I couldn't help myself so I ...
>You wanted to illustrate it, right?
Well ... uh, yes, to illustrate that changes in Average Return are more significant than changes in Volatility.

In the chart below, we change the Average Return by 2%
(from 5% to 7%, without changing the Volatility of 24%)
and note that the Annualized Return increases by 2% as well
... very nearly

On the other hand ...

In the chart below, we change the Volatility by 5%
(from 35% to 30%, without changing the Average Return of 11%)
and note that the Annualized Return increases by just 1.6%
... very nearly

In fact, if the Volatility is v%, we can expect the change in Annualized Return to be roughly v% of the change in Volatility.

>Huh?
In the above example, the Volatility is roughly v% = 32.5% ... changing, as it does, from 35% to 30%.
Hence we can expect the change in Annualized Return to be roughly 32.5% of the 5% change in Volatilty
... and that means 32.5% x 5% or roughly 1.6% ... as indicated in the chart

>I assume you can prove all this, eh?
That the Annualized Return increase is roughly equal to the Average Return increase?
And that it's just a percentage of the Volatility increase - the percentage being roughly v%?
Yes, I can prove it. Would you like to see the proof?

>Certainly not! Besides, why all those roughly thingys ?
Nothing is precise in the shadowy world of finance.
Indeed, I assumed that the Annualized Return is related to the Average Return and Volatility according to:
Annualized Return = Average Return - (1/2) Volatility2

Note that (as Tom C. points out):
"Given two assets with the same average return, the asset with the lower volatility will result in a higher annualized return."
See CAGR

>I assume that's roughly true?
You got it

P.S.
Here are a few Volatilities
obtained by looking at the Standard Deviation of monthly returns
... then multiplying by SQRT(12)

... and some Vanguard Mutual Funds