William J. Polley

Did Big Picture tell us what he was assuming for the nominal rate of interest? I'd be curious if anyone worked his story out using a 2% real interest rate.

OK, I answered my own two questions. The example assumes a 10% interest rate and concludes that the late investor gets only 3.1% more than the early investor even though the late investors saves for his last 40 years and the early investor only saves for his first 7 years. If one assumes a more realistic 2% interest rate, the late investor gets 268% more in his retirement account than the early investor. So the title should be "The Power of Compound Interest Assuming Absurdly High Interest Rates"!

2%? I agree 10% is too high, but let's split the difference at around 4 or 5% and not even get into how a person's portfolio might be more aggressive when young and more conservative close to retirement. (Such a consideration would also make a big difference.)

What it all boils down to is the time it takes to double. At a lower interest rate, doubling is so slow that you'd have to speed things up by contributing all along the way. At a higher interest rate, an initial investment doubles faster and can achieve the same result as an investor who started later but put in more over time.

OK - I'll agree to the 4% to 5% if you agree to factor in the risk-taking factor. As far as being young and aggressive - Paul Samuelson's 1963 Scientia paper needs to be considered here.

I'm familiar with the question posed by Samuelson. Much has been written about it (of which I have read only a fraction), and I understand why you're bringing it up here.

I still think that on-balance you want to have a more aggressive portfolio when young and more conservative when old. But I would also say that the precise allocation would not simply be dictated by time, but also by past performance.

Is this what you meant back in March when you said, "And if he [Don Luskin] bothers to read Kritzman’s excellent discussion of Paul Samuelson’s Scientia 1963 paper, he might finally understand that it is the variability of terminal wealth and not the variability of average returns that matter."

I can certainly buy the idea that variability of terminal wealth matters a lot. That makes it a more complicated optimal control problem as opposed to a simple formula based on age, but I'm ok with that.

Is this what you meant?

The idea that one can make a constant "real" contribution that is significantly high is, well, unrealistic, given the financial conditions most young workers must overcome.

Also, the theory that 5-10 years of aggressive investing can beat 30 years of conservative investing works sometimes, but not all the time. The aggressive investor who started in 1929 would end up with less money overall at retirement than the conservative investor in this scenario.

Change 1929 to 1925 in the last comment - I was thinking crash and put the wrong number in.

Well, just for fun, why not run the Lemony Snicket vs King Midas (Link: http://politicalcalculations.blogspot.com/2005/04/lemony-snicket-vs-king-midas.html) scenarios? Not that either is actually achievable....

M1EK:

I agree that it is hard for young workers to save a constant real amount. A better way to have illustrated the problem is to show that starting a year or two earlier makes a difference--even if that first year or two of contributions is small. That is the advice I would give young people.

I address your second point in my first comment. It depends on how both the interest rate and the additional contributions contribute to the time it takes to double.

Ironman:

That's great! The Economist magazine did something like that a few years ago. I think it was for the turn-of-the-millenium issue where they did it for the 20th century. Maybe I'll find it someday.

William,

But the problem with showing that starting with a little bit a little earlier makes a difference is that it only makes a small difference. No free lunch and all that. That's why nobody shows it with a more realistic contribution schedule - because it just doesn't sell.

Suppose you plan to start saving "t" years before retirement (earning interest for t years). And suppose that you are evaluating whether or not to save an amount "x" one year prior to that. (In other words, you are planning to start your retirement fund next year but you are wondering what difference it would make if you started this year with "x"... all other things equal.) Assume also that the interest rate is "r".

Your retirement fund will increase by x(1+r)^(t+1). That is the mathematics. Whether you think it is small or not is up to you. As a rough estimate, if t=30 and r=.05, every dollar invested at the beginning (one year earlier) becomes about $4.50. If t=40, it's closer to $7. Saving (and compounding) monthly bumps it up slightly. In the limit, the formula is exp((t+1)r)x.

I could also point out that a dollar invested today is equivalent to a few cents (how few depends on r and t) invested per year every year. But the longer you wait, the more cents you have to commit to put in every year to make up the difference. Everyone has to make up his/her own mind about how long to wait and how many cents to put in.

So the real issue is this. When you're young, the impact of a dollar of savings on your terminal portfolio value is larger, but due to lower current income, a dollar saving represents a larger current sacrifice. As time goes on, the impact of a dollar of savings decreases slowly, but saving that dollar becomes less of a sacrifice. Somewhere in that continuum, it is rational to begin to save, and rational to increase savings over time. The precise timing depends on your personal situation. You can make all the tables you want, but the math always comes down to exp(rt). The rest is commentary.

"The Economist magazine did something like that a few years ago. I think it was for the turn-of-the-millenium issue where they did it for the 20th century. Maybe I'll find it someday."

I was unaware that they had done it - I'll have to try to track it down myself....

Also, since we're talking about investment math, I can vouch that the math is very much as you describe, at least for the case of continuous compounding. For more common compounding options, the relevant tools I've developed are:

Investing: Future Value
(Link: http://politicalcalculations.blogspot.com/2005/02/investing-future-value.html)

for a single investment, and for a portfolio of up to five investments:

Your Investment Portfolio
(Link: http://politicalcalculations.blogspot.com/2005/04/your-investment-portfolio.html)

It was the December 16, 1999 issue of The Economist. Not exactly the same as what we're talking about, but sufficiently close as to be relevant.

William,

Again, all I argue that people who push this theory without mentioning inflation are vastly overselling the benefits. That's all. In your formula, for instance, r needs to be replaced with (r-i). And for most people, it really isn't that easy to drastically exceed the rate of inflation on investments. Certainly not for the lower-income young worker in your example, who is unlikely to have access to enough money to earn a good rate of return from an investment institution; less likely to work somewhere with a good 401(k); etc.

http://www1.jsc.nasa.gov/bu2/inflateCPI.html

(first link in google to CPI deflator is NASA. How cool is that?)

I don't buy that. You can open an IRA with a low initial deposit, no loads, and low expenses if you shop around just a little. Those accounts are earning high single digit rates of return depending on the funds. That beats inflation. I will admit that going the IRA route requires some commitment since there are penalties for early withdrawal. Not a good idea if you need ready access to that money to fix your car or something. But if you can spare $50 a month (and commit to it), it might be worth it.

In my formulas, "r" means real. When I think about these things, I just automatically take out inflation in my head. I also believe strongly that real returns to long term savings are generally positive. I believe that even young low-income people have access to instruments with positive real returns. I also believe that while saving is generally good, the best time to start depends on your expected lifetime income stream. Some people should optimally start saving later than others. I'm pretty sure that we agree on at least this last point.

But you keep coming back to inflation as the reason why it's not optimal for young or low income people to save. I disagree. If you can spare $50 a month toward an IRA, you can beat inflation (on average). Will you drastically exceed the rate of inflation? Maybe... maybe not. Returns are uncertain--but that's true for older, higher income folks too. In the long run, your odds are quite good.

If anything, inflation is more of a worry for those close to retirement who optimally shift their portfolio towards fixed-income assets. There's where inflation rears its ugly head--not when you're 25 with an growth oriented IRA.

William,

I question whether a young person sticking 50 bucks a month in an IRA is doing significantly better than somebody who starts 5 or 10 years later, but puts 100 bucks in, or 200 bucks in, or whatever. Yes, you can 'beat' inflation. That might mean a 1% rate of return, while accepting some risk. I haven't seen a lot of people beating inflation by a lot - the popular conception about stocks' rate of return seems to be driven by a combination of outliers and of ignorance of the inflation factor.

The thrust of these arguments seem to be that starting early is, by far, the biggest factor, since compound interest has such a dramatic effect. Certainly some people buy it to their own detriment - such as an acquaintance of mine who came into a lawsuit windfall, and promptly invested it instead of paying off a high-interest car loan.

I'd almost go as far as to say that the better investment for a young person, in today's financial environment, would be paying off even a home mortgage first, rather than investing in stocks. Depends on the mortgage, of course, and where you are on it for tax purposes.

In short: the original chart looks like cheerleading to me. Having lived through the dot-bomb and at least having friends-of-friends who lost everything in it, I have a hard time not admiring my current cow orker who owns his house outright after 5 years in on his mortgage, even though he doesn't own any stock of consequence.

You said, "I question whether a young person sticking 50 bucks a month in an IRA is doing significantly better than somebody who starts 5 or 10 years later, but puts 100 bucks in, or 200 bucks in, or whatever."

I've already given you that point when I said, "I also believe that while saving is generally good, the best time to start depends on your expected lifetime income stream. Some people should optimally start saving later than others. I'm pretty sure that we agree on at least this last point." Don't we?

And the thrust of my argument is not that starting early is the biggest factor. Again we agree. The whole point is how quickly the money doubles. That depends on r, t, and the size of the periodic payment. There are many paths up the mountain. Preferences matter. Which path do you prefer?

You do make a very good point about paying off debt. Let us never forget that paying off debt, by reducing future interest payments, is analogous to compound interest from saving. Paying down a mortgage is somewhat questionable if you have a low fixed rate and deduct the interest on your taxes. However, paying off a high interest car or credit card is almost always the better option. Good point and well said.