tl;dr- This claim is false. While it's true that an investor who bought in at the start of each of the top-10 days, then sold at the end of them, could enjoy gains of 64% over a similar investor who didn't participate on those 10 days, it doesn't follow that these top-10 days "represent 63 per cent of the returns of the past 50 years".
The claim is incorrect due to a broken frame.
Imagine that someone tells you that the best strategy for your financial future is to use a certain strategy for playing poker at a Las Vegas casino. They might be correct about their strategy being optimal, in a frame that assumes that you're going to be betting your financial future at a casino. But, is it then true that the optimal strategy for your financial future involves playing poker in a Las Vegas casino?
It's not that the claim about how to play poker is necessarily wrong, but the frame around it is. Likewise, the title of this question asks:
Is it true that, “just ten trading days represent 63 per cent of the returns of the past 50 years”?
That "63 per cent" figure may come from correctly performed mathematical operations, but the frame suggests that someone who missed those 10 days could only hope to enjoy the remaining 37% of returns over the past 50 years.
The problem's that it's generally inappropriate to claim that members of some set constitute some portion of that set's total when that set isn't just adding over positive values. (More about the math at the end of this answer.)
For example, going by Wikipedia's numbers, it looks like an investor who skipped the worst-5 days would've applied a ~79.4% gain to their wealth. But does it make sense to attribute ~79.4% of wealth to skipping the worst-5 days while also attributing 63% of the wealth to being present for the best-10 days?
Analogy: Did one employee do 90% of a small operation's work?
Consider a small company with 25 employees. They do little contract jobs.
One employee, Bob, has a minor role as a specialist on 90% of the projects; the other 10% don't involve his area of expertise. Bob then writes on his resume:
My efforts represented 90% of the company's productivity during my tenure.
Which makes it sound like he did 90% of the work, and everyone else did 10%, right? Clearly he's a top-notch employee!
And he's not lying, in the sense of making numbers up. That figure's accurate, for the frame in which it was calculated.
The problem's that the frame's broken. This is, it doesn't make sense for Bob to describe his participation in 90% of the projects as having represented 90% of the company's work. It's not that he did the math wrong, it's that the logic behind the model itself is a tad silly.
Corrected claim: Investing on the top-10 best days would yield a ~64% return.
It seems that, if an investor invested at the start of a day and then pulled out at the end for each of the top-10 days, they'd have applied a factor of ~164%, for a gain of ~64%.
Without checking the math or figures, that looks plausible to me. But, assuming it's true, it becomes false if we misrepresent that claim through wording that makes it sound like this is a thing where it's meaningful to attribute 64% of gains to 10 specific days, such that anyone who missed those 10 specific days can only hope to enjoy 36% of gains, or something like that.
It's not that someone did the math wrong or messed up calculating the figure; rather, it's that the model frame in which the figure exists is itself broken. The correct solution is to exit that frame.
Discussion: Mathematical explanation.
When we say that something contributes x% to a total, we're typically working within a framework in which we're adding a bunch of positive values to arrive at a total.
We can make these investments additive. Specifically, we can take the natural-logarithm of each day's ratio to arrive at an additive parameter. Then, we can sum those additive parameters up, raise Euler's number to the power of their sum, and that's the total factor.
To grab the table from @MoneyAnn's answer, then augment it:
Date +% factor ln(factor)
1987-10-21 0.0910 1.0910 0.0870947068509337
1987-10-20 0.0533 1.0533 0.0519280928603591
1970-05-27 0.0502 1.0502 0.0489806222216219
1987-10-29 0.0493 1.0493 0.0481232751817282
1982-08-17 0.0476 1.0476 0.0465018336514199
1962-05-29 0.0465 1.0465 0.0454512629039174
1974-10-09 0.0460 1.0460 0.0449733656427312
1957-10-23 0.0449 1.0449 0.0439211870579281
1974-10-07 0.0419 1.0419 0.0410459694360010
1974-07-12 0.0408 1.0408 0.0399896482161584
1.645443519, which recovers that ~
1.64-factor, or ~64%, other answers were referring to.
However, the broken frame presented in the claim has 2 major problems:
The factors aren't totaled over all days.
If someone wanted to make a claim like this, then they should've
ln(factor)'d all of the days, then divided the sum of those parameters for the top-10 days by the sum of those parameters for all of the days.
The factors contain negatives.
Some days ended lower than they started. For example, apparently 1987-10-19 had a −20.47% hit, for a
ln(factor) ~= ln(1-0.247) ~= -0.231553819. In general, for any
factor of less than
ln(factor) should be negative.
We could try to fix up the numbers to account for Problem #1, but the frame doesn't lend itself to resolving Problem #2.