Having a young child is expensive, and most parents can't set aside much money toward the long-term future of their offspring.

But most young families (perhaps with help from grandma and grandpa) could find a way to save $1 a day. This extremely modest investment can yield amazing long-term results. Would you believe $4 million when your child reaches age 65?

Let me show you.

Obviously, you won't make money like that putting the daily dollar in the bank. It's got to be invested, and the results of such an investment are uncertain. But I will outline a scenario that can be followed by just about any parents (or grandparents).

Let's assume we're talking about a baby girl named Charlotte (an old-timey name that is reported to be one of the most popular choices for recently born girls). Let's assume also that Charlotte's parents save $365 a year for the first 18 years of her life.

I'll also assume that Charlotte (this is in her self-interest, after all) gives her folks enough time to figure out that an investment in small-cap value stocks is probably the lowest-risk way to invest with an honest probability of earning 12% a year.

Even with as little as $365 to start, Charlotte's parents can buy small positions in hundreds of small-cap value stocks through a low-cost ETF: Vanguard Russell 2000 Value Index . So let's assume they do that, and by the time Charlotte is 18 they have invested $6,570 (18 times $365).

Compounding at 12% a year, their annual investments grow to be worth $20,348 when she's 18. (Interestingly enough, the same result could be achieved if Charlotte's grandparents or other family members made a one-time-only investment of $2,700 in her first few weeks of life.)

Let's assume Charlotte, following her parents' lead, has the good sense to keep this money invested in the small-cap value asset class for the rest of her life. Such a sensible young woman probably will be able to earn at least $5,500 a year starting at age 19. Why does that matter? Because it will give her the opportunity to contribute that much of her growing nest egg every year to a Roth IRA that will never be taxed. She won't have to contribute her earnings, just the money her folks saved for her.

In her 19th through 23rd years, she doesn't have to add another penny as she funds an IRA and watches her money grow. On her 24th birthday, her investments are worth $35,857. (This is probably considerably more than the majority of her contemporaries have, and for this she can thank her parents.)

Now let's assume that Charlotte goes about living an interesting, busy life for the next 42 years until she is 66 (which is rapidly becoming the new "official" retirement age for Social Security purposes). Assuming that long-term growth rate of 12%, her nest egg is worth $4,185,342 on her 66th birthday.

Remember, she didn't add anything to this since she was 18.

Just for fun, as you think about the following numbers consider that they could all be twice as big if her grandparents had matched her parents' contributions dollar for dollar.

When Charlotte is 66 she can begin enjoying the payoff from this very long-term investment and all the patience it took to let it grow. Assume the money continues to grow at 12%, and once a year Charlotte withdraws 5% of it for her retirement income (letting the other 7% keep growing).

She may be startled (and I hope quite pleased) to learn that her first-year distribution will be $209,267. That turns out to be $573 a day — a huge increase from the $1 her parents initially saved.

If she keeps up this withdrawal plan and lives 30 more years, her cumulative retirement income will be $27 million. In addition, she can leave nearly $32 million to her heirs.

All this required was a modest initial commitment by someone who cared about this baby girl, an inexpensive investment vehicle, and a lot of faith and patience. Obviously the numbers won't be exactly what I've calculated, because the future is always unknown.

But based on 80-plus years of history, I believe this scenario is very possible. It's a testament to the power of long-term compounding coupled with lots of patience.

Does this scenario have "a catch?" Perhaps it does.

Here's one: Some people will scoff at assuming long-term future returns of 12%. So I made all the calculations as if the return were only 10%. In this case, Charlotte has $1,455,159 when she turns 66. The 30 years of assumed distributions would total $5.8 million, and she'd have about $7.2 million to leave to her heirs.

Is a 10% return reasonable to expect? Well, the Standard & Poor’s 500 Index achieved that for the past 88 years, despite a major depression, numerous recessions and bear markets, many wars, deflation, inflation and a host of political and economic challenges.

Here's another catch: Inflation will make these very impressive numbers much more modest far into the future. However, Charlotte probably will be able to add to her investments through her adult life.

Out of curiosity I calculated the result if she continued saving $1 a day from age 19 to age 65, an additional 47 years. This made a difference, but not as much as you probably think. At 12%, this would leave Charlotte with $4.8 million at age 66 instead of "only" about $4.2 million.

Even though these "adult" contributions totaled $17,155 (versus $6,570 for those in the first 18 years of her life), they accounted for only about 13% of the final return. That's because those very early parental investments had so many more years to grow.

However, that doesn't bother me much. Charlotte probably will have the capacity to add much more in her adult life than just $365 a year.

I have two additional suggestions for any parent, grandparent, aunt or uncle who decides to do this for a child:

Make the annual payment all at once at the start of the year instead of every day or at the end of the year. Because of the time value of money, this will boost your long-term returns. (My calculations assume Charlotte and her parents did it this way).

Give your Charlotte (or Harry or Trent or Imogene) a copy of this article for her 18th birthday so she will know your long-term intention for this investment.

With a little bit of money and a lot of foresight, you can help a child create a sound future. I hope you'll find a way to do it.

Richard Buck contributed to this article.

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