Buy for Less ... and increase your return

Recently I was asked whether buying a stock for a few percent less would make much difference in your annual return and I said I didn't think so because ...

>Buy for less? What does that mean?
I mean instead of paying, say, $10 and 11:30 AM you manage to buy it for $9.50 at 3:30 PM. Would that make much of a difference at year's end?

>From $10 to $9.50? That's a 5% decrease so I'd say the difference is peanuts.
Yeah, that's what I said.
In fact, if the stock went from $P to $Q in one year, your Annual Return is Q/P - 1.
For example: from P = $10 to Q = $11 the return is 11/10 - 1 = 0.10 or 10%.

Now, if you bought it at $9.50 your return would be: 11/9.5 - 1 = 0.158 or 15.8%.

>That's not peanuts, eh?
No, it isn't. In fact ...

>So you just add the 5% to your return, right?
In fact, if the stock went from $P to $Q then the return is Q/P-1.

However, if you managed to buy the stock at less than $P, say at P(1 - R),
where R = 0.05 means 5% less, then your return becomes
Q/{P(1-R)}-1 which is roughly Q/P-1 + R.
For different price reductions (R running from 2% to 8%) the increased annual return is shown in Figure 1.

The plot is the increased return: Q/{P(1-R)}-1 versus the stock return: Q/P -1.
For example, if the stock return from $P to $Q is 15% and you managed to buy the stock for 8% less than $P, you'd get a 25% return.

>The dot?
The red dot.

>Red? Aah, but there's a grey dot.

Figure 1
That's the example we started with: P=$10, Q=$11 hence a 10% return ... compared with the return when we buy at 5% less than P.

>From now on I'm buying at 3:30 PM.
Good luck.