The Magic(?) of Compound Interest |

For example, and I don't mean to single out anyone out in particular (though the title of the book sets itself up for overinflated expectations), I recently read

*I Will Teach You To Be Rich*by Ramit Sethi. In the book, he likes to use 8% as a portfolio growth rate assumption when providing examples about the power of compounding interest. Surely, if someone can earn 8%, getting rich becomes a lot easier. But let's try to break this assumption down a bit.

**Historical Data**

The 8% number is seemingly derived from US historical data. Using historical averages is pretty popular both for savings and for studying safe withdrawal rates in retirement. One key resource about the historical data is Morningstar and Ibbotson Associates SBBI database. From it, we can learn that the S&P 500 on average since 1926 earned an 11.8% return, while intermediate term government bonds earned 5.5% on average. However, these are not the numbers we should be using.

**Inflation**

For starters, when talking about wealth accumulation over a long period of time, we should be removing inflation from the numbers in order to make the results is more meaningful. We should be looking at wealth accumulation in today's dollars, not future dollars. Sethi, in particular, makes this mistake on page 170 when talking about historical returns. He provides nominal returns and seemingly gets it backwards by saying that the numbers do not include inflation, when in fact they do. Step #1 is to remove inflation from these numbers so that we can talk about them in terms that we can understand: today's dollars. Having $1 million in 40 years will not mean the same thing as it does today. I'm a multimillionaire in Japanese yen, but that won't get me so far when lunch costs 1,000 yen.

With inflation removed, the historical average stock return is 8.6%, and it is 2.6% for bonds.

We are not finished yet.

**Compounding Growth over Long Periods**

The next step is that we need to switch to compounded returns rather than arithmetic returns. These historical averages represent a possible best guess about what you can earn over the next year. But when talking about accumulated wealth over a long period of time, we cannot use these single period returns. We have to account for portfolio volatility. Sometimes the portfolio grows and sometimes it shrinks.

The way to understand this point is to consider what happens if your portfolio loses 50% of its value. How much does it need to gain in order to get back to its original starting point? The answer is not 50%. It is 100%. The portfolio needs to double to get back to where it started.

To make this more clear, suppose your portfolio is worth 100. Losing 50% means that the portfolio value drops to 50. The next year suppose your portfolio gains 50% in value. Well, 50% of 50 is 25 and so your portfolio would only grow to 75. The portfolio would need to grow by 50 to get back to 100 and that represents a 100% growth rate on top of its current value of 50.

This asymmetry must be incorporated into the analysis when talking about compounded returns over a long period of time. Stocks are volatile, and even though they earned 8.6% after inflation historically, with volatility one's wealth would've only grown at a rate of 6.5%. Bonds are less volatile and so their hair cut is smaller, but still the compounded returns for bonds falls from 2.6% down to 2.3%.

**Asset Allocation**

Another issue to consider is one's asset allocation. Back to that 8% assumption that is so popular to use, I don't know what the discussant has in mind for the asset allocation. Let's be charitable and say they are talking about it as a growth rate for stocks after removing inflation. Historically that was 8.6%, and perhaps the person is a bit conservative and reduces it down to 8%. Is this where the assumption comes from?

Nonetheless, this assumes that a person will hold 100% stocks over their entire working life and into retirement! Someone may start their career with a more aggressive asset allocation, but by the time they are approaching retirement and their wealth is hopefully grown to its largest value, where a given percentage return has the biggest effect in terms of dollars, the person is probably going to have an asset allocation far removed from 100% stocks.

In trying to choose one asset allocation to represent an entire lifetime, it's not exactly clear what to assume, but we do need to put more weight on what the asset allocation will be around the retirement date. That is when a given return will have the biggest overall impact on the portfolio. And that is when the asset allocation is likely to be more conservative and less weighted to stocks. Since bonds have a lower compounded return, this pulls the compounded return away from its loftiest values.

**Adjusting for Current Market Conditions**

There is one more adjustment we must make. It is that in today's current market environment, it borderlines on ridiculous to assume that the US historical averages will still apply in the future. Today bond yields are very low, and they are the best predictor of subsequent returns for bonds. That assumption of 2.3% inflation-adjusted compounded returns is really way too high, especially for those close to retirement who will be drawing down their portfolios.

As for stocks, even if they can provide the same risk premium over bonds as they have historically, the low starting position for bonds implies lower returns for stocks as well. Likewise, stocks are still considered overvalued by the cyclically adjusted price-earnings ratio, and that further implies lower future returns than average.

Joseph Tomlinson recently investigated these issues at Advisor Perspectives, and he found that popular software packages had assumed compounded inflation-adjusted returns for a 50/50 portfolio of 2.95%, while Tomlinson's own estimates for this portfolio are 1.13%.

Personally, I use a

**2% compounded and inflation-adjusted return assumption**in my own planning spreadsheet. I could always change the assumption to 8%, and this would let me imagine that I will be very rich, indeed, when I reach my 60s. But it would just be an illusion and I would need to prepare myself for becoming very disappointed.

I don't think that 8% assumption was all that well thought out. I do know that is not a good assumption.

Well gee I'd happily take a 2.5% 40yr mortgage from you Wade, that's a 25% increase over your assumption guaranteed by real estate. Sound good? I'll send you my SSN, address, and wire 20% to escrow and let's close on this!

ReplyDeleteIf you are willing to adjust those mortgage payments for inflation, let's talk. But mortgage payments are usually fixed, no?

DeleteOtherwise, 30-year Treasury bond yields are currently 3.2%, and I would find that more attractive that the deal you are offering.

But it has no impact on my inflation-adjusted assumption of 2%!

Just as a further follow-up, the current yield on 30-year TIPS is 0.61%.

DeleteThat is the relevant number for this discussion.

Nice article. I too use 2% for personal planning.

ReplyDeleteI question what I call the "volatility drain fallacy". If the PERCENTAGE changes average to zero, then indeed volatility drains the asset's value. But why should they average to zero? Why wouldn't, as likely, the ABSOLUTE value average to zero? Or something non-zero?

I think that volatility correlates with uncertainty, and in uncertain times, some investors pull out. That pushes down the price. But the "1% up is less than 1% down" is an irrelevant observation.

-jobjob

DeleteThanks.

I'm not saying that the percentage changes must average to zero. I was just trying to illustrate the point with a simple example. Changes will likely average to something positive.

But the issue that compounding returns over time are less than the simple average of past returns is a mathematical fact. Wealth will grow at a slower rate over time than the simple average.

The formula is:

compounded return = average return - 0.5 x (standard deviation)^2

Here's a great resource that gets deeper into the formula and Wade's explanation. http://moneychimp.com/features/market_cagr.htm (and even application of volatility drag http://moneychimp.com/articles/volatility/drag.htm ).

DeleteGood post Wade ... most consumers miss this and wonder why they keep coming up short of their projections. I use these links to help explain it to them. This post will help now too.

Larry,

DeleteThanks for sharing those additional resources.

Wade

ReplyDeleteIs your 2% personal assumption for a 50/50 portfolio?

In my simple spreadsheet, I was not really even thinking to connect the assumption to an asset allocation. I am currently about 70/30, but will probably be less later in life.

DeleteThanks

DeleteThis was an informative post. I do have one question: have the calculations on stock performance been adjusted with regard to survivorship bias?

Delete

DeleteThe data tries to track the S&P 500 and its predecessors. You have a good question about survivorship bias, and I don't know what they do exactly in that regard, but I suppose companies on a downward trajectory will be removed from the S&P 500 at some point and the bias will be partially accounted for.

Excellent post, Wade. I can't stand those 8%, 10%, even 12% compounding illustrations. There are more meaningful and accurate ways to demonstrate the advantages of saving early in life. I teach personal finance and try to convey this to my students (and to other instructors as well). I plan to refer folks to this post to help make the point.

ReplyDeleteWade

ReplyDeleteYou make some great points here again. I often see marketing just like the blue line on your first chart.

The point on the incorrect use of the arithmetic mean is something I often raise. Indeed I have recently written a short article on it here (http://cuffelinks.com.au/investment-strategies/why-you-should-know-the-difference-between-arithmetic-and-geometric-investment-returns).

Dylan and Aaron,

ReplyDeleteThank you both. Aaron, that is a good point about not confusing the compounded return with the risk premium.

Thanks! I know that what you are saying makes lots of sense, yet I still feel bad sometimes when I compare my wealth accumulation with the rosy scenarios books project. I have been saving for 20 years and am in a good place, but not where The Automatic Millionaire puts people after 20 years!

ReplyDeleteAnd this assumes "market returns". Most mutual funds underperform the market over time and the typical mutual fund investor underperforms his fund due to poor market timing. Except for devout buy-and-holders, I think your calculations are optimistic.

ReplyDelete

DeleteDirk,

You are right! I forgot! That should be a whole other category on my list: fees and most investors' inability to match market returns. I'll include that next time. Thanks.

Great Post Wade. A bit depressing, but a good post. I've used 5% real in my planning in the past, but tracking my returns since I started investing in 2004 shows I'm a little short of that, at around 4.4% real or so. That's also with a fairly aggressive asset allocation, 75% stocks and much of that in riskier stock asset allocations.

ReplyDeleteGiven current fixed income yields, it's pretty darn hard to average any reasonable amount of fixed income and come out with anything much higher than 2%. I don't know if it is fair to use the current historically low yields and project them for decades into the future, but I don't know of a better number to use.

DeleteThanks for contributing. I enjoy your website.

In the past I've tried calculating my own rate of return, but I lost the patience for separately inputting any inflows or outflows of funds, and I don't know any more. But that would certainly be a good tool to help someone gauge how they are doing relative to the broader market. I should get back into that habit.

Indeed, for someone far from retirement, I could understand upping the expected return a bit to account for what you are describing.

Actually, after re-running the numbers for a post coming up, I found I'm at 5.21% real. Of course, that's at the end of a 4 year bull market....

DeleteThe post I wrote (but won't run for about a month) isn't so much about arguing with your results, as what those results mean. I don't think it's so simple as just save more and retire later. If returns really are that low, it should logically push people out of the stock and bond markets all together in favor of alternative retirement strategies like working in retirement, real estate investing, investing in small businesses, taking on more portfolio risk, investing in insurance products, or just putting it all on red. Having your money double just once in your career simply isn't acceptable to most people planning for a traditional retirement.

Regardless of the growth assumption used, if retirees can avoid large drawdowns -- like the 50% you mentioned -- it will help preserve spending levels during retirement even if capital is withdrawn while out of the market. The challenge is to get back in when the time comes.

ReplyDeleteWade -

ReplyDeleteThis news is rather disconcerting- If my 12C app is functioning properly, it takes 71 years to accumulate $1.5M by saving $10K/yr at 2%.... and then hope that dog food has not outpaced inflation :(

What to do? If the 2% assumption is correct, then the only hope in the accumulation phase is highly leveraged securities or lottery tickets...

It's not correct. An internationally stock diversified fund, buy-and-hold and rebalanced, with ~ 30% bonds will get conservatively 7.5% nominal - .15% expense if you use vanguard index admiral - 3% inflation (which is also a high estimate). There is 0 loss in compounding over long period because he talked about share price exclusively completely disregarding the fact that US stocks also pay ~ 2% yield, and foreign is close to 3% yield. So that's 7.5 - .15 - 3% = 4.35% real, conservative.

DeleteYou have a point, of course, but you should also consider that the kind of volatility you are talking about - causing the divergence between arithmetic and geometric average returns - is actually a net positive for the investor during the asset accumulation phase of their investing life. (Assuming, of course, that the volatility in returns doesn't affect the availability of investable assets!)

ReplyDeleteIt is, of course, very bad for the investor during drawdown.

FWIW, I use a 3% assumed after-tax, after-expense, after-inflation rate-of-return assumption in my personal investing.

All money is debt.

ReplyDeleteAs such - the source of the compounding dividends are all inflationary and end up driving the cost of everything in the economy higher and at the same time the level of debt in the economy on a trajectory similar to that indicated in the post.

Given that the issuers of the debt (monetary assets) used to make the compounding interest payments can only be among the privileged wealthy, there can be no question that this entire process is both designed for, and results in, real wealth accumulation.

It's as easy as money is debt.

This is one the best debunking pieces I've seen. It really confirms to me that my "safe" dividends on the Cash Value in my Whole Life Life insurance is a much better way to accumulate wealth as apposed to the risk I take in the stock market.

ReplyDeleteBy-the-way, my dividend for 2012 was 6.5%

Keep in mind that the dividend rate on a whole life policy is NOT the same as the return on that policy.

DeleteWade, your math in the section "Compounding growth over Long Periods" couldn't be more wrong. In your examples where you talk about needing a much greater percent to get back to where you were, your discussion says only two things about the dividends you are earning and reinvesting while all of that is going on; jack and squat.

ReplyDeleteThe reality is, if an index stock fund earn 10% over, say, a specific 20 year period, and you buy-and-hold and reinvest all of the dividends, then you will earn exactly 10% nominal over that period. If its vanguard index, then you pay maybe 10 basis points. And if it's internationally diversified where valuations are absolutely outstanding, then you can expect to earn at least what we have earned historically, and more likely, something higher.

Its true the earlier that you start saving the more money you will have when you rtire. The greatest gains are made on the money that you put into a retirement fund when you are very young. Another factor at work here is the power of compounding. If I start a business and make thirty thousnad dollars in gross revenue the first year that I am in business and if I increase the annual sales of my business by just 15% every year for fortyfive years. My business will grow to 15 million dollars in annual sales in fortyfive years. Its one of the reasons why so many millionaires are business owners.

ReplyDeleteI've seen many times that a globally diversified portfolio of 60/40 rebalanced annually would return about 9.8% and that was from 1970-2008 not including the last 4 years of good returns

ReplyDeleteThat is probably a simple average of the returns and includes inflation. Those years were a great time for bonds and interest rates began dropping in the early 1980s. Also there was a major stock bull market in there as well.

DeleteYou can calculate your actual total annualized (geometric) return on one (or more) of your investments with the Internal Rate of Return function in Microsoft Excel (or another spreadsheet program). Just make two columns, one with the dates and the other with the amount invested/withdrawn on those dates. Investments are entered as negative numbers and withdrawals as positive. To get your return as of today add an entry with today's date and the amount you would receive (less commissions) if you sold at today's price. This should be entered as a negative number. The Excel XIRR function allows the dates to be irregular. Don't use IRR. Check Excel's Help on XIRR for more details.

ReplyDeleteJim,

DeleteThank you. Indeed, XIRR is the function to use when dealing with irregular inflows and outflows throughout the year.

Wade, again in this post you make great points. But I’d want to take one big step further – and I’m sure in a longer post you would too.

ReplyDeleteWhen we use a single after-inflation geometric rate, it gives us a single dollar result for each year, including the final year. But if our assumptions are right, we have only 50% probability of a result that high (or higher) – and 50% probability our actual result will be lower!

How much lower? From that single-rate approach, we don’t know.

We should want to explore what our assumptions mean as to what result we have 80% or 90% confidence of meeting-or-beating.

The way to explore that – as you know as well as anyone on the planet, you do it all the time – is to take into account the after-inflation return-rate’s standard deviation (with its arithmetic mean) to base the analysis on a real-return-rate probability distribution. Apply that with Monte Carlo, and see probability distributions for the dollar results of future years. From that, we can see what dollar results that according to our assumptions we have 80% or 90% confidence of meeting-or-beating.

That’s important – and the longer the investment time horizon, the more important, because as years go by, the probability distribution of possible results keeps getting wider, with the 80% and 90%-confidence results getting further and further below the 50% result produced by the single-rate approach.

Dick Purcell

Thanks Wade.

ReplyDeletehttp://wealthymindsetsuganda.blogspot.com/

Inflation is depressing. But this is a great article. I think you explained the entire scope of money as related to savings accounts.

ReplyDeleteIncome drawdown permit you to draw an income directly from your pension, although it is leaving the remainder invested. It is help the customers contracts and manage their pension affairs in a way that best suit them. Thanks for this excellent post. It will really help a lot of people.

ReplyDeleteIncome Drawdown

Wade, I'm interested in your formula

ReplyDeletecompound return = average return - 0.5 * stdev^2.

Is it exact? Can you point me to a (web) derivation. I'm assuming the returns are on an annual basis. I've been unable to replicate it in Excel (which might of course be my fault entirely) so I'm intrigued.

David,

ReplyDeleteThat is exact when we talk about a lognormal distribution for the underlying returns. If you are using numbers from a different distribution, then the formula does apply. But as it is exact, for any sample of numbers you use in Excel, your estimated compound returns for those numbers will not match the number from the formula. You can see the equation at Wikipedia. It's the 4th equation under the "Arithmetic Moments" section:

http://en.wikipedia.org/wiki/Log-normal_distribution

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