In an article which came to my attention a week or so ago for some reason, Rob Gardner, who was described as a ‘financial expert,’ wrote in an article unequivocally titled ‘How parents can make their child a MILLIONAIRE’ in the Daily Mail dated 25/02/2018, that if you open a bank account for your child on the day he (she) is born,
 …the daily saving [of £5·50 a day] can grow into a £1m pension pot for child [sic]
 Contributing £5·50 a day equates to £6·88 a day due to tax breaks
 By the time they reach 10 years old, the pot will reach a total of £25,000 [misrelated pronoun sic]
 If money has been invested wisely, it could have grown to £35,000 — £40,000 [incorrect sequence of tenses sic]
 Compound interest means the pot will roughly double every decade
Since I did something similar for my children back in the days when I had a job and interest rates were higher than they are now, I was most curious to find out what I had done all wrong. My children had ended up only with what I had paid into their accounts and a light spattering of interest.
I tried Mr Gardner’s schedule of payments on an Excel™ work sheet. I assumed a bank interest rate paid on savings accounts of 2% pa and an annual inflation rate of 3½%. In order that you may judge how realistic a projection that is, here is the Bank of England’s own diagram of recent interest rates. If anything, 2% appears a bit higher than you might realistically expect.
I did not consider tax on the interest payment, quite likely 20%, or management fees. Since the child has an income tax allowance, currently £12,570 in his own right, interest payments would have to be very large before you had to think about hiding your wealth off shore, like Members of Parliament do.
Firstly, the annual contribution that the parent has to make into the child’s account, £5·50 per day, evens out at £167·41 per month, taking leap years into account. In addition to the contribution to the account that the parents pay in, the bank also contributes interest on the capital that’s in the account already, as well as on payments that were made during the year. If you pay money into the account in January, it will have earned some interest by the time December comes round.
So, if the parents don't increase their daily contribution to account for inflation, this is how the account for the first ten years will look:
Year

Paid in

Amount at year end 

0  £0·00  £0·00 
1  £2,008·88  £2,028·96 
2  £2,008·88  £4,098·51 
⋮  ⋮  ⋮ 
9  £2,008·88  £19,791·79 
10  £2,008·88  £22,216·59 
At that point, the child’s tenth birthday, Mr Gardner tells us to stop making payments into the account.
 When [the child turns] 10 years old, stop contributing. By this point the pot will have grown by almost £50 a week, or £2,500 a year, to a total of £25,000.
Actually it hasn’t, but at £22,216·59 it isn’t too far off target. Then Mr Gardner asserts that,
 Because of the miracle of compound interest, if left alone this pot will grow (roughly doubling every decade) to over £1million by the time your child turns 65.
Ha! So that’s the secret of feeding a crowd of five thousand from a current account with nothing in it but five loaves and two fishes. That’s how to make bricks without straw. That’s how to catch a Woozle. The rabbit was in the hat already. It’s all done by smoke, mirrors and compound interest.
Well, no, it isn’t.
With a bit of mathematical reverse engineering, we find that in order to double every ten years, the loaves and fishes have to be invested at a minimum of 7·17734% pa interest. (Solve the equation (1 + r)^{10} = 2 if you don’t believe me.) In 2018, when the article was originally published, bank rate was ½% pa and even now (July 2022) bank rate is only 1¼% pa, which is well short of what you would need for the child to find a million pound legacy in his current account on his sixtyfifth birthday.
Here is the result of stopping payments. I allowed for the likelihood of the government increasing pensionable age to 70 years, but it does not make the child's position much easier, even though by now he walks with a stick, has grey hair and a tweed jacket, listens to Radio Three and can't work any technological device more complicated than the thermostat on the central heating.
Year

Paid in

Amount at year end 

10  £2,008·88  £22,216·59 
11  None  £22,660·92 
⋮  ⋮  ⋮ 
65  None  £66,021·71 
⋮  ⋮  ⋮ 
70  None  £72,893·31 
At this point, I ask myself, is there anything we could have done to rescue this poor soul from spending the rest of his life in grinding poverty? Is Mr Gardner’s vision actually practicable? Can you really endow a millionaire for only £5·50 per day?
I suppose the first thing would be for the parents to index their contribution for inflation. Let’s assume an inflation rate of 3½% pa. Now those first ten years look a little better, and we’ve achieved Mr Gardner's waymark of £25,000 in the account by the child’s tenth birthday, but the total at age 65 or 70 is still nowhere near a million pounds.
The daily contribution will rise from £5·50 in year 1 to £7·50 in year 10. After that, the parents don't contribute any more.
Year

Paid in

Amount at year end 
Daily contribution 

0  None  None  None 
1  £2,008·88  £2,028·96  £5·50 
2  £2,079·19  £4,169·52  £5·69 
⋮  ⋮  ⋮  ⋮ 
10  £2,737·89  £25,917·22  £7·50 
⋮  ⋮  ⋮  ⋮ 
65  None  £69,758·47  None 
⋮  ⋮  ⋮  ⋮ 
70  None  £85,035·19  None 
The result is a bit closer to a million pounds than before, but it is still a long way off target.
The parents are going to have to spend more months making contributions. For how long might they continue?
My first child arrived when I was 25 years old, and I drew my old age pension at the age of 65. Let’s assume for the sake of argument that everyone else, including the parents of this hypothetical name on a passbook, does the same. Then the parent will retire after forty years of making payments.
Things look more favourable if the parents continue to make regular payments, indexed for inflation, for forty years after the birth of their child. Then they stop making payments and leave their child’s retirement income to the mercy of time and chance.
Year

Paid in

Amount at year end 
Daily contribution 

0  None  None  None 
1  £2,008·88  £2,028·96  £5·50 
2  £2,079·19  £4,169·52  £5·69 
⋮  ⋮  ⋮  ⋮ 
10  £2,737·89  £25,917·22  £7·50 
⋮  ⋮  ⋮  
40  £7,684·69  £236,877·47  £21·04 
41  None  £236,877·47  None 
⋮  ⋮  ⋮  
65  None  £388,622·59  None 
70  None  £429,070·75  None 
We are beginning to see some encouraging results. By age 40 the child will have £¼m and by age 70, getting on for £½m.
We need to get more money from somewhere, so let's make the assumption that by age 43, the child's own eighteen year old children are gainfully employed, so that between the age of 43 and pensionable age, the child can contribute something out of his own pocket every day. Assume he pays the same as his parents did, indexes it for inflation and makes his first contribution on his 44th birthday.
We now look close to success. If the child makes his own contributions at the same rate that the parents did, then by age 65 the pension fund looks quite reasonable:
Year

Paid in

Amount at year end 
Daily contribution 

65  £18,160·81  £736,293·73  £48·04 
70  £21,569·35  £918,741·16  £57·06 
Which is close enough to a million pounds, but no cigar.
Going back to Mr Gardner’s original scheme, what size contribution would you need to make it work at 2% pa interest? Again, we can try some reverse engineering and see.
At 2% pa interest, accumulating one million pounds by age 65 by making deposits every day for ten years and then leaving compound interest to do the hard graft requires contributions starting at
£83·31 per day
and indexed for inflation.
It would also be possible to accumulate one million pounds by age 65 by making deposits of £5·50 every day for ten years and contributing nothing after that if the interest rate were higher than
6·70% pa.
If you made contributions for forty years, at 2% interest your child would have a million pounds in his bank account by his 65th birthday if you
contributed £14·15 a day instead of £5·50.
Conclusion: In the financial environment at the time when Mr Gardner’s suggested savings plan appeared in the Daily Mail, it was possible to start from nothing and pay daily contributions into a bank savings account, but it is considerably more expensive than Mr Gardner says it is. His payment plan only works if the interest paid on the account is guaranteed to be 7% pa or more throughout the period in which the savings are being amassed.
By that time, of course, with our guesstimated 3½% inflation, one million pounds in Year 65 would be worth a mere £109,012·79 in Year Zero money,