and the 4% Rule

The S&P 500 calculator is on the previous page, and its calculations include past fund survivability of withdrawals. My interest was about how appropriate is the 4% Rule for a 100% stock fund like the S&P 500? (as opposed to a balanced fund with bonds?) This page is the general introductory info describing the 4% Rule, the S&P 500, the general market, bonds, dividends, gain, Bear markets, etc. This is some market stuff that you should know. You might find a few things of interest.

This is basically Page 2 of that calculator page. It's a large page, but a Menu to several subjects below is:

A few important facts to know about stock Dividends

A few important facts to know about Bonds

Computing compounded gains, and Annualized results

Three Annualized Compounded Gain calculators

How to make a Million dollars for retirement

Bad times in the market with a recent graph of S&P 500

This was originally about **balanced funds**, meaning stocks balanced with bonds, perhaps containing 60% stocks and 40% bonds. The idea was that bonds can shield some portion from market volatility, and at least in the past, bonds would also provide some earnings when the market is down. My own notion is that a 50-50 balanced fund might indeed drop half as much during a market crash, but it also only earns half as much in recovery and also in the good times (and there are many more good times than bad). The 4% withdrawal rate has been promoted as safe, determined by testing past market history with 4% withdrawals lasting through 30 years of withdrawals in retirement from any starting date. However it did not specify any specific fund investments, nor any value of fund growth, but which seems a major factor to me. And it was from history when bonds paid more than now, but 2022 has to be the worst year for balanced funds, because both stocks and bonds are down big now.

But instead of balanced funds, the point here is that I also wondered about 100% stocks (like the S&P 500 Index funds for example). Any X% percentage withdrawal rate might seem safe if the fund average earning gain was X% to support it, except years vary in gains, at an irregular rate. A 50% loss (say $100 down to $50) requires a 100% gain to recover, which might take a few years. The bonds in balanced funds used to pay more to aid that, but times change, and with interest rates increasing now, bonds are also losing money now (see below). The 4% concept specifically means the withdrawal dollars are **adjusted each year** to not exceed withdrawing more than 4% of the then current fund. And market years do vary erratically, when a couple or three seriously bad years in a row can make a serious departure from the average. So the rule examines market history verifying survival of all starting dates enduring all known bad year periods. This sites S&P 500 calculator (link at top above) has the Test that does the same thing, with variable withdrawal rates. The future is unknown of course, but knowing the history should help know what possibilities could happen (the 2000 decade was particularly poor).

**Origin of the 4% Rule:** Interest rates of bonds were higher in older years, and the purpose of a balanced fund (*Balanced meaning equities mixed with some degree of bonds, often 40% composed of bonds called 60/40 stocks/bonds*) was market safety, because bonds are not affected by the market, and the bonds contributed to help tide it over in bad market times.

A good description of the 4% Rule was that it comes from a 1994 investigation of historical market data that tested for a reliable safe withdrawal amount, specifically for a balanced fund. Its conclusion was that a 4% withdrawal would survive 30 years of retirement withdrawals in past situations if invested anytime since 1928. However, it was done earlier than the worst times in the 2000s. Bonds are a different factor. The bonds did provide some income in those days for a degree of safety in bear markets. Here's a chart of the Federal Reserve Bank's interest rate history, and I'm thinking the 4% Rule look in 1994 could not foresee today's zero interest rate. A good recent article about the 4% withdrawal number 4 is at Morningstar.

IMO, a downside of the 4% Rule is that it does not consider the gain of the fund, nor how much money it has accumulated before withdrawal begins. These seem serious factors in predicting how long the fund can survive the withdrawals.

The S&P 500 is widely considered to be one of the best market investment choices for most people (those who are not market professionals following the market closely every minute). An Index fund keeps its holdings exactly matched with the index it is tracking to match the same performance. The S&P 500 is the collection of the 500 largest US publicly-held companies, (all are the largest large-cap stocks, including both growth and value stocks), all well established, and widely including most industry types. Might say it's where the money is, since the S&P 500 includes about 80% of the total US market capitalization.
Capitalization is a companies total dollar market equity value, **equal to the companies (number of public stock shares x current price per share)**. The S&P 500 index is weighted by capitalization, including a stock amount proportional to the companies total dollars, so that the largest companies count proportionally more in the index, according to their overall capitalization dollar value.

All of the many S&P 500 Index funds try to exactly match the S&P 500 Index performance, but these funds do have different expense fees. There are actually 503 listed tickers in the S&P 500 (today, but it varies), because three of the companies have two major classes of public common stock included (Google, Fox Corp and Discovery Communication). Google's company name is actually Alphabet, with two public stock classes A and C, with two tickers GOOG and GOOGL, which are added together here in the table here. The S&P rules today prohibit including more than one class of public stock, but these three are grandfathered.

S&P 500 Weighting Top 10 as of 27 Jan 2023 |
||
---|---|---|

Apple | AAPL | 6.40% |

Microsoft | MSFT | 5.40% |

Alphabet (Google) | GOOG | 3.31% |

Class A & C | GOOGL | |

Amazon | AMZN | 2.66% |

Berkshire Class B | BRK.B | 1.63% |

Nvidia | NVDA | 1.49% |

Tesla | TSLA | 1.40% |

Exxon Mobile | XOM | 1.40% |

United Health | UNH | 1.33% |

Johnson & Johnson | JNJ | 1.29% |

The weighting method is that since Apples capitalization is several hundred times more dollars than the smallest member company (Apple's worth is about $2 Trillion, which is about 6% of the total S&P 500 capitalization), so the weighting is per each invested dollar instead of per company. Some do fault it because a few companies dominate the total, but it seems very proper to me. So the weighting numbers vary with the daily prices, computed each trading day (to compute the S&P 500 Index). All of the 500 "Weights" add to 100, so these numbers are the actual percentages of the total Index value.

See the current weighting of all of the S&P 500 companies. The numbers change slightly every market day.

To be eligible for S&P 500 inclusion, each company's stock must be publicly held, and to be in the top 500, currently each is at least $14.6 Billion capitalization. However each company is selected by a committee with additional performance concerns. Companies can also be similarly removed from the S&P 500 (one must be removed for every company added). These S&P 500 are the Big Boys, the largest and most financially successful. Then all of the S&P 500 index funds simply plan to exactly track and match the performance of the S&P 500 index (less the fund's expense fee). The S&P 500 is near 80% of the total U.S. market capitalization.

Probably about any question you might have about the S&P 500 Index would be answered here: S&P U.S. Indices Methodology.

See the largest company stocks NOT in the S&P 500, but in some cases, these are just different classes of stock, those typically being non-public Class A. For example, this list shows Berkshire Hathaway private class A as the largest entry Not in the S&P 500, however Berkshire Hathaway public class B was added to S&P 500 in 2010.

Similarly weighted by capitalization, there is also the **Total Stock Market Index Fund** (Vanguard VTSAX) with stock of 3992 companies (blend of selected large-cap, mid-cap, and small-cap U.S. companies). To me, it seems mostly a conceptual idea, because in practice, it is weighted by capitalization like the S&P 500, which boils down to be that much of the S&P 500 are still at the top of it, with the smaller ones still weighted much less strongly, with much smaller contributions. So it still has similar performance as the S&P 500 (usually around ± a percent or so from the S&P 500 index in individual years).

There also are various S&P Equally Weighted funds, either for all of the S&P 500 or just of specific industries or concerns, but equally weighted, for example ticker RSP. It seems to have a slight advantage in the current bad times, but my notion is that the largest leaders are ahead in good times.

A company's publicly held capitalized value is their number of existing stock shares multiplied by one share's price value. Apple's value this way computes to be $2 Trillion. The idea of the dividend is to share company profit with the owners. The dividend money paid out (as a few dollars per share) was removed from the company's value, as a payment to you. This reduces the company's value by millions, so the stock price (which reflects the remaining value of the company) **is automatically and equally reduced by the same dollars per share when the dividend is paid**. Due to that corresponding stock price drop, your value of the paid dividend and your stock's remaining value **remains exactly the same sum value as before the dividend**, i.e., **there is no gain on that day**. Yes, the stock price did drop, and the company value dropped, but **you did not lose anything either** — You already owned that stock distributing the dividend, and you do have that cash difference now. It was just a withdrawal. There is no gain and no loss, not from the dividend on that day. The distribution was just from the company's stock value that you already owned before, but now is instead transferred to your cash. See more here.

**The best and most optimal thing to do is to reinvest that dividend, which retains today's fund value, and adds a few more shares each time, and makes a huge difference long term.** You already owned the dividend money, so it was just a withdrawal, NOT new income. The only advantage of dividends (but it sure is a big one) is that a stock paying 2% dividend a year is (only if reinvested) contributing 2% more shares **every year**, which seriously increases your earnings, it really adds up.

Reinvesting dividends is a very major part of long term earnings. Long term, the cost of withdrawing dividends is too high to consider, much more than you might ever imagine (over 40 years costing half of the total gain potential). There is a computed chart on the previous page showing a typical cost of withdrawn dividends over the years. It is a S&P 500 Index fund paying dividends of perhaps about 2% a year, more or less.

Near 80% of the S&P 500 companies pay dividends in some degree. Dividends are dollars per share, percentage is annual, and is typically paid each quarter. See a list of those companies in the S&P 500 **ranked by dividend**. The S&P 500 dividend itself is the dividend paid by all 500 companies (but about 20% of those don't pay dividends. Many growth companies instead invest profit into creating more growth.)

This next table is the **Top 20 in that largest dividend list**. It contains the Morningstar year end **Total Return** numbers (annual gain percentages, which includes both price gains and dividends). Year 2023 is omitted because it is far from complete yet. The **10 year Gain** column is the Total Gain over the ten years (3 years for DOW). The **Annuliz(ed) Return** column is the hypothetical comparison AS IF this Same ten year gain number had instead been at that fixed constant interest rate (like a bank), the same every year of the ten years. The overall gain is the same either way, but the mixed up and down years of stock are very difficult to compare long term results, so this Annualized Return is a good way to compare the stock gains.

A bug in this chart (in the 10 year Gain) was corrected 1/18/2023

Tickr | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 10 year Gain | Annuliz Return | Company |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

NLY | -18.30 | 20.46 | -2.13 | 19.08 | 31.29 | -7.32 | 6.62 | -0.64 | 2.96 | -21.36 | Annaly Capital | ||

LUMN | -13.06 | 31.05 | -30.98 | 3.10 | -20.77 | 3.78 | -6.20 | -18.62 | 38.97 | -52.43 | Lumen Tech | ||

PXD | 72.76 | -19.09 | -15.71 | 43.68 | -3.97 | -23.73 | 16.01 | -23.31 | 65.69 | 39.56 | Pioneer Natural | ||

CTRA | 72.76 | -19.09 | -15.71 | 43.68 | -3.97 | -23.73 | 16.01 | -23.31 | 65.69 | 39.56 | Coterra Energy | ||

MO | 27.96 | 33.55 | 22.55 | 20.20 | 9.36 | -26.63 | 7.69 | -11.04 | 24.17 | 4.22 | Altria Group | ||

DVN | 20.54 | 0.45 | -46.15 | 44.03 | -8.82 | -44.83 | 16.77 | -36.50 | 191.08 | 51.37 | Devon Energy | ||

VFC | 67.60 | 21.92 | -15.11 | -11.84 | 41.93 | -1.04 | 51.15 | -12.36 | -11.97 | -59.55 | VF Corp | ||

VZ | 18.36 | -0.45 | 3.54 | 20.41 | 3.51 | 10.70 | 13.52 | -0.29 | -7.26 | -19.22 | VerizonComms | ||

SPG | -0.81 | 30.07 | 10.09 | -5.28 | 0.69 | 2.42 | -6.39 | -38.72 | 94.21 | -22.15 | SimonProperty | ||

OKE | 48.91 | 0.49 | -45.59 | 142.78 | -2.16 | 7.01 | 46.80 | -44.34 | 62.85 | 18.18 | OneOK Inc | ||

LYB | 44.12 | 2.25 | 13.29 | 2.54 | 32.75 | -20.99 | 18.60 | 1.46 | 5.47 | 0.76 | LyondellBasell | ||

KMI | 6.31 | 22.25 | -60.17 | 42.16 | -10.33 | -10.87 | 43.82 | -30.53 | 23.87 | 20.95 | Kinder Morgan | ||

T | 9.64 | 0.77 | 8.04 | 29.18 | -3.97 | -21.45 | 44.08 | -21.08 | -7.23 | -1.33 | AT&T | ||

DOW | 6.52 | 7.24 | -6.22 | Dow Inc | |||||||||

STX | 88.53 | 21.67 | -41.49 | 10.99 | 16.22 | -1.74 | 60.77 | 8.87 | 86.12 | -50.96 | Seagate Tech | ||

WPC | 24.35 | 20.27 | -10.38 | 6.81 | 23.39 | -0.72 | 28.83 | -6.61 | 22.21 | 0.42 | WP Carey Inc | ||

PM | 8.45 | -2.07 | 12.89 | 8.76 | 20.09 | -32.56 | 34.38 | 2.87 | 20.67 | 11.84 | PhilipMorris Intl | ||

IP | 26.20 | 14.62 | -26.58 | 45.47 | 12.71 | -27.02 | 19.08 | 12.42 | 3.86 | -22.35 | Internat'l Paper | ||

BXP | -0.56 | 35.29 | 2.10 | 0.74 | 5.80 | -10.75 | 25.89 | -28.59 | 25.99 | -37.92 | Boston Propert | ||

WBA | 58.39 | 34.93 | 13.58 | -1.09 | -10.38 | -3.59 | -11.09 | -29.22 | 35.53 | -24.70 | WalgreenBoots | ||

AAPL | 7.64 | 40.03 | -2.80 | 12.15 | 48.24 | -5.12 | 88.09 | 81.85 | 34.48 | -26.32 | Apple | ||

MSFT | 43.69 | 27.24 | 22.22 | 14.65 | 40.22 | 20.75 | 57.12 | 42.37 | 52.24 | -27.94 | Microsoft | ||

AMZN | 58.96 | -22.18 | 117.78 | 10.95 | 55.96 | 28.43 | 23.03 | 76.26 | 2.38 | -49.62 | Amazon | ||

NVDA | 33.20 | 27.28 | 66.36 | 225.32 | 81.82 | -30.69 | 76.73 | 122.20 | 125.41 | -50.26 | Nvidia |

Many of these companies had a low ten year Total Return (or a loss), so notice that (for stocks and funds) the highest dividend rate is Not the main criteria to be considered — not always a good criteria. There is more to value than dividends. Overall gain seems more important. For example, the last four rows add Apple and Microsoft and Amazon, which are #1 #2 #3 in the S&P 500, and Nvidia currently #10, which pay "only" 0.6%, 1.07%, 0% and 0.10% dividends, but they have added much more value. They can be volatile, but these 20 appear volatile too. You can compute this 10 year Gain and Annualized Return for any stock or fund in Morningstar at the 3rd calculator below. And the 50 largest 50 S&P 500 stocks (and several others) are shown here.

**Indexed funds vs. Actively Managed funds:** Index funds simply try to match performance of the fund to the actual daily index of the group of companies by using computers to maintain the index match by automatically buying the matching shares of each company (passive investing, computers instead of managers, low fee cost). Whereas actively managed funds instead have a human manager trying to pick the selected best paying investments (at larger fee cost). And managers might accomplish that now and then, but next year may be rather different, and it is commonly said that indexed S&P 500 earnings beat managers about 90% of the time (the lower fee is some part of it).

Another category is **ETF funds** (Exchange Traded Funds), relatively new. Standard mutual funds can only be traded after the market closes, for the close price, and then Only if ordered before the market closes. ETF funds are traded like stocks, any time the market is open, at the market price. That could be important if you trade frequently, but won't be important if you buy and hold. One example is the NASDAQ 100 ETF, ticker QQQ, which is 100 of the leading S&P 500 stocks.

**Fees**: Brokers typically charge commissions on buying or selling stocks, however some brokers now are free (and some almost so). Market exchanges have a slight difference between buy and sell prices of stocks (bid and ask), which is a fee on the buyer. Some brokers sell only funds with a "sales load" commission which has been quite high, however there are also very many no-load funds with no charge (you must look those up yourself). Those are one time charges. However, funds also have annual fees, charged every year for the management, so while maybe a fee may sound low, it can add up big every year. Some fees are near zero on Index funds. The large brokers with no commission will handle your trade, probably only online, but don't expect advice on what to buy. Those earning commission may offer more advice, but they might be more interested in their commission (they have lists of what to sell today). You'd be advised to shop around a bit to be aware about the fees and commissions. It's your choice and there are good choices, and less fee is more final profit for you.
I'm not familiar with most brokers, but regarding fees, offhand I'd start at Vanguard or Fidelity (however all funds there won't have the same fee, so still pay attention). Brokers can also use slightly different algorithms in trading stocks daily to maintain the same S&P weightings, so compare Morningstar's Performance annual gain numbers too, in the case of some of the S&P 500 Index funds, the lowest fee does not always match the maximum gain. Morningstar Total Return results do include annual fund expense fees, but do not include any commissions or sales charges.

**So yes, there are other funds and stocks** that might sometimes earn more than the S&P 500 Index (the largest of those companies are also probably in the S&P 500, contributing their share, and the S&P 500 is 500 stocks which is some degree of diversification). However their downside is these currently hot stocks are more volatile, their prices can swing widely, which is great when the market is good, but is the worst when the market crashes. And it does seem that the faster they grow, the faster they can fall. The current year 2022 was not a good year, the S&P 500 bottom was -25%, but -18% to -20% was more common. with the biggest usual leaders (like Apple, Microsoft, Amazon) way down, values from -30% to 40%, and a few special case down even more (for an awakening, see current Total Returns of 90+ stocks). But the market always recovers, eventually. The market is usually good overall, but there certainly can be big downside surprises. If investing in individual stocks, you may want to watch closely, and know when and why to switch stocks (preferably before it changes, but that is extremely difficult to know, it happens before we know about it). The S&P 500 can be more comfortable long term without close watching, but it does go negative with the market, like this year. It has always recovered to continues growing, but that is not very comforting while waiting in the bad times.

**Diversification — Don't put all your eggs in one basket.** The S&P 500 mix of 500 companies is a diversification in the various industries (tech, energy, financial, consumer, health, industrial, materials, etc). **However all of the 500 are US large cap stocks** (which includes No small caps, mid caps, emerging markets, foreign markets, bonds, etc.). A S&P 500 Index fund earns more than **balanced funds** ("balanced" means majorly mixed with bonds for diversification), **but bonds can be very volatile too**, because bond value varies with interest rates, which goes up and down too (see Bond **Duration** in the Bond box below). But overall, the S&P 500 trend line is quite appealing. The nature of investing is that some risk is necessary to earn higher gains, a low risk investment doesn't earn much. The S&P 500 does have the normal daily market ups and downs, including the rare economy crashes, but the overall S&P 500 averaged gain has historically always been of about 10% a year, compounded long term. *Which is NOT a guarantee — years vary, a few years are negative, but there are many more good years than not. However a bad crash with a few bad years in a row will have a large effect*. The entire 2000 decade was down -9.4% with 2001 and 2008 crashes that were pretty bad. But the long term picture is very appealing, with only a few dips, which have always recovered of course.

**Nothing ventured, nothing gained.** Ben Franklin said that too, but the thought is centuries older. Some people do fear anything in the market is too much risk for them (yes, the market can crash in bad times, but then it always has recovered, after a while). At least it does if it was a good investment, and the S&P 500 are the largest and most successful Blue Chip companies in the US, which is a good strong bet.

**Compounding gains:** The greater number of years of **long-term** investments makes the compounding of gains be a huge exponential function (with years as the exponent power), which is an astounding big deal. A fixed 10% gain in each of 40 years becomes a final value of 1.10^{40} = 45x the initial value and 4400% overall gain. The market gains do vary up or down each year (often large variations, but sometimes negative), see below for computing the comparable overall gain (called Annualization). In addition to the price gains, another major factor is that reinvested S&P 500 dividends add very roughly about 2% to total returns each year, and then those additions see gains too, and those gains see more gains, compounding every following year. Even if a few random years are negative, compounded earnings are a Real Big Deal, so think of a long term investment always with reinvested dividends. Start young, and then it will be waiting at retirement. The S&P 500 calculator on previous page shows this with 50 years of past S&P 500 history. The math of compounding is shown below.

**The Average of the fund's annual gain is Not the measure of stock performance**. Because the average is just the **sum** of all years gain divided by the number of years. But the long term compounding result is the **multiplication product** of the years of gains. The best plan is to use Annualized Return instead, which is described below.

The overall summary can hide a few adjacent down years during which withdrawals could deplete the fund. The S&P 500 will recover, it always has, but if all your money was depleted earlier, it ends there. The first early years are the higher risk, when value is small before it has earned much, since more money will of course always last longer in a crisis. So first building more money in the fund (before the retirement withdrawals) is the insurance to last longer when down, and to make recovery easier. Reason would suggest that first allowing maybe 20 or better 40 years for the fund to build and grow without any withdrawals would make all the difference of survivability, and would of course also provide much greater income during retirement. The market years do vary erratically, but continually withdrawing 10% also with average earnings near 10% might (on average) usually keep it drained down to always about the same level, more or less. It can't grow more then, but its value won't vary so greatly through a long retirement. Except there are variations outside of average, and limiting withdrawals to about half of the average fund earnings rate significantly improves odds against going the fund going bust (and would also leave something for future inheritance to your heirs).

Never withdrawing anything will not go to zero, also unlikely if withdrawal is a small percentage, but fixed withdrawals can become relatively huge when the fund is small, so recompute the percentage every year. Fixed dollar withdrawal become very large when the stock value drops. Even an extremely bad rare crash probably leaves at least 50%, which is certainly no fun then, but it has in fact always recovered. Here is a chart of a few years of S&P 500 record highs. But when and if it is down low, but then percentage withdrawals become fewer dollars of withdrawals when the fund is low. Instead, the biggest danger is fixed dollar value withdrawals, which if blind to current situations and not limited to a reasonable current percentage, of which an example is shown in the Test section on previous calculator page. Your planning for that should have occurred decades earlier. Withdrawals are the desired and necessary goal in retirement, but are very counterproductive during the growth phase. In every case, withdrawals should be reconsidered if the fund value gets low. We don't know the future but we can look at the effect of "typical" past periods, regarding our withdrawal feasibility.

**One issue of a 4% Rule is that it does not specify any specific fund contents, nor any specific fund value.** However a fund containing more money can obviously survive withdrawing in a crash longer than would the same fund with less money. Meaning, a large million dollar fund and a small $10K fund both withdrawing after a bad crash might both fall to 50%, but 50% of a $Million is much more survivable (and with greater gains in recovery) than 50% of $10K. **The survivability of investing for 20 or 30 years to build before starting withdrawals is a huge factor of retirement withdrawal success.**

The survivability of a reasonable **percentage withdrawal** not hitting zero seems relatively independent of value — only meaning a fixed withdrawal percentage rate (*if the withdrawal dollars are readjusted every year to hold that percentage rate*), it withdraws much less when fund is low, near zero withdrawal when near zero value, and worst case still always leaves a tiny value instead of zero. Maybe only a few cents left, but not exactly zero, so hitting zero can take a very long time. Which is the reason an adjustable $100 minimum limit was added to the S&P calculator to more clearly define the end of Survival due to depletion. Possibly this limit should be higher for a stronger recovery, and you can change it, because the fund does need some money left to be able to earn a faster recovery. But in the real world of fixed withdrawals in dollars, hitting zero is certainly about the fund value, since a higher value fund will always last longer through any crisis. The important thing is to maintain a fund value that can recover and survive. Withdrawals make remaining Fund Value be a very major survivability factor (and many years/decades of growth with no withdrawals until retirement is the obvious way to easily increase retirement fund value). If you had $1 Million in a fund, a bad crash might drop to 50%, but half a million would still last a very long time, and then the larger value will also recover with more dramatic gains than a tiny value could.

**Investing in bonds is a very different game than investing in stock.** I am not encouraging bonds now, just offering some facts you need to know first. **This is all true both of direct bond purchases, and of bond funds too.**

But you should be aware of the difference of buying bonds vs. bond funds.

Bonds may have paid higher dividends in 1994 (for the 4% Rule study), helping to support withdrawals, but interest rate has bottomed out near zero today, so IMO, now bonds seem an outdated investment idea. However, bonds do protect savings from market volatility, and in a market crash, a 50/50 balanced fund may drop half as much as a 100% stock fund, and the bonds still could provide some earnings. But markets always do recover, and bonds don't earn what the 100% stock fund can, and don't earn today what bonds have historically earned, and don't even match inflation today. **Most of all, you must realize that bonds are also quite volatile too, maybe safe from the market, however bond value is very seriously affected by current interest rate changes, which are increasing this year, lowering resell value of existing bonds.** Also, inflation is a serious concern, still at a 40 year high at 6.5% for Dec 2022.

Government bonds are different yet, sold at auction to bidders. U.S. Treasury bonds of 1 to 10 year maturity are called Notes, and those of 20 or 30 years are called Bonds (why I don't know, but "bonds" here refer to both). The Treasury bonds are safely backed by the government, and have face values and a fixed interest based on face value, and are redeemed at maturity at full face value. However initially, buyers buy by bidding at treasure auctions on new treasury notes and bonds, so the actual final yields vary with the selling prices (basically what buyers will pay). The variable auction prices are why interest rates of new government notes vary every day. The bid price drops if buyers won't pay more, and then the yield goes higher (the yield to maturity), and of course vice versa too. **But until redeemed at maturity (at full face value then, for the expected income), the bond resell value varies every day, due to current interest rates.** The bond pays a fixed initial dividend in dollars, but resell value varies, making investment gain percent vary too, if sold early. The current **SEC Yield** reported is the previous 30 day interest result annualized to be a hypothetical 12 month rate, which is a standard way to compare the current earning rate of varying gains.

Bond interest is not compounded. It's rate is computed from the face value, which does not change.

**Definition of very important bond "Duration":** Bond resell value varies with current interest rates. The term Duration computes that **for Each 1% change in current interest rates, the resell value of existing bonds is expected to change in the opposite direction by "Duration" percent**. If Duration is 5 years, and interest rates increase 1%, the bonds decrease value by 5%. Interest rates dropped the couple of prior years, so existing bond resell values increased then, and bond funds showed better results. But vice versa, today existing bonds lose resell value when interest rates increase (and now last one year return is negative, and interest rates are near zero now with only one direction they can move). The U.S. Federal Bank has announced plans for more interest rate increases in 2022 to fight inflation.

Again, regardless of current interest rates, **bonds do pay full face value when redeemed at maturity**. However, holding until maturity may be difficult to achieve unless you directly own the bonds to make it your choice, independent of mutual funds.

Morningstar shows the bond **Duration** (in balanced funds at Portfolio tab, Bond sub-tab, if any). The meaning of a **Duration** of say 3 means the expected bond value will decrease 3% with each 1% interest rate increase, and vice versa, existing bond values also increase when interest rates fall. But either way, **when and if held and redeemed at maturity they do still pay face value**. Short term bonds will have lower duration with lower risk from interest rates, but they pay even less. Speaking of all bonds, including commercial, municipal or treasury (but excepting federal Savings Bonds, which are not sold at auction, and can be cashed in, but cannot be resold to a third party). Bonds have a face value and pay a fixed interest rate, of dollars based on their face value. However, when bought new at treasury action, or any bonds resold later in the market may pay a different price than face value, but they still redeem face value at maturity. Bond resell value varies with current interest rates and with Duration, and then the actual purchase price computes the new effective interest rate (when redeemed at face value). If interest rates increase, existing bonds earning lower effective interest can only sell at a lower price to attract any interest in them (and also vice versa, price goes higher when interest rates decrease). This fact is Extremely Important, especially with interest rates currently increasing from a rock bottom low. The bond value decreases and also inflation exceeds the bond interest.

Bond resale value becomes volatile when interest rates change. I say "resale" because if held, bonds are still eventually redeemed at full face value (if directly purchased and in your own control). So short term bonds, and bonds nearing maturity date, will have low durations. Long term bonds will have higher duration (more volatile due to interest rates). Morningstar.com shows the Duration of bond funds (on the Portfolio tab, computed each quarter, I think).

However, as individual bonds approach their maturity date, their duration drops towards zero, because **directly purchased bonds do still repay full face value when redeemed at maturity or recall**. And existing bonds do continue to pay their same fixed face value dividends, but their resell value varies with current interest rate (existing bond resell value drops with higher current interest rate). **But if you buy bonds directly yourself, and hold until recalled or redeemed at maturity, the face value return then will be as expected.** But if sold early, they will have current market value.

**But if in a bond fund, or if planning resell, there's much more to know about the volatility. Resell value of existing bonds varies inversely with current interest rates**. And bond funds must buy and sell bonds continually as investors buy and sell shares, and bond values are computed daily, not necessarily held until redeemed, but the fund values its bonds at the resell price. So if you buy or resell in the fund, you get whatever the fund value is paying that day (due to interest rate changes). That could be a plus if interest rates fall, or costly if interest rates rise. Right now, interest rates were at near zero but rate is increasing due to the Fed planning to fight inflation with multiple interest rate increases during 2022.

*When inflation increases interest rates of new bonds, it lowers old bond resell values. That's because no one would pay full price for old 1% bonds if they can buy new 2% bonds at same face value price. So then buying two existing 1% bonds at half price is required to match earnings of one new 2% bond.* So existing old bond resale value can drop to half each time interest rate doubles (because then it takes two old bonds to pay what one new bond pays). And also vice versa, lower interest rates will increase the value of existing bonds that still pay more. (But see Duration too).

Some bonds are "callable" (most municipal bonds and some corporate bonds), with a callable date when the issuer can redeem the bond early (at full face value but which terminates dividend income), perhaps planning to issue new bonds paying lower interest rate (if rates are falling).

Note that **Junk bonds exist too (politely called high-yield bonds)**, from companies with lower credit rating, paying more dividend to attract buyers, but with higher possible risk of default failure (meaning 100% loss of your investment). For bond funds, Morningstar.com Portfolio tab shows bond ratings too. AAA is most secure rating, including government bonds. Bonds rated down to A are considered investment grade. Generally bonds rated B or less are considered speculative and non-investment grade (speculative meaning offering greater dividends if paid, but with greater risk they could fail and default and stop, and not redeem at all). See bond credit rating.

The significant fact to know is that Interest rates dropped in 2019 and 2020 (to near zero), significantly increasing existing bond resell values, so bond funds showed good results then. The results may look real good, but **it is important to realize why you see that value increase in the history**, because the interest rate situation has changed. Since then. Existing bonds lose resell value when interest rates on new bonds increase, and then bond return went negative. When interest rates are near zero, there is only one direction they can move. The U.S. Fed has done several interest rate increases in 2022 due to inflation, and more are expected.

Bond price changes when reselling are taxed as Capttai Gains if held one year or more. However *bond dividends* are taxed like interest, at regular income tax rates (except municipal bond dividends are tax free of federal tax, and sometimes free of state tax in same state, at least in some states).

Repeating the important stuff: **Bonds do still pay full face value when redeemed at maturity** (or when recalled earlier). And until then, they continue paying face value interest rate. **So if you buy bonds directly yourself, Not in a fund, but in your name as owner, and hold them until redeemed at maturity, the return will be as expected. That's the good news.**

However **bond funds currently own existing bonds, which will have dropping value as the U.S. Federal Bank interest rate increases (which the Fed is currently doing in a big way, due to record high inflation). The bond funds must buy and sell bonds continually, as investors buy and sell shares**. Bond funds value their bonds at the current sale price. And bond resell value varies daily with current interest rates. Bond funds recompute bond values with every days interest rate change. However, bond value does not vary much when close to maturity, so a complication is that bond value sensitivity to interest rates depends on how close it is to maturity, in the special calculation called Duration.

Again, a big point is to not be confused by market return statistics showing bond and balanced funds had good performance history in recent past years. They in fact did, but that is history, when interest rates fell more than 2% over 2019 and 2020, causing the existing bonds to increase in resell value. **But it's very wise to also check their current YTD (Year To Date) results too. The current SEC 30 day yield is shown annualized to represent one total year, which is the accurate current annualized rate value today, but it of course changes all during the rest of the year.** The Federal Reserve Bank interest rate was near zero at the start of 2022, so the only way it could go is up. And the high inflation continues to increase it now, and increased interest rates means resell values of existing bonds are going down. However, bonds held until redemption at maturity do retain full face value then, except bond funds might not be able to hold them to maturity when their shareholders are ordering withdrawals because the value is going down. Since interest rates are near zero now, the danger for existing bonds is that rising rates (and lower values) are the only change possible now. We are expecting inflation to cause increasing interest rates. Bond dividend income is taxed with regular income tax rates, but the price changes are Capttai Gains or losses if held a year or more.

We already covered above that **Stock Dividends are valuable, but are NOT new income, so withdrawing them is just a withdrawal** (and a limit on future gain). There are about 65 stocks referred to as Aristocrat Stocks, which to be included in their list, the companies must be in the S&P 500 (largest US companies), and **must have increased their dividend every year for 25 years**, a point of pride indicating a stable business. See that Aristocrat Stocks list. Their annual increases can be small, and about half of them still pay less than 2.5% dividends, and some of those pay less than 1%. A favorite of Warren Buffet at Berkshire Hathaway is Coca Cola (KO) paying about 3% plus decent earnings. Whereas most of the best growth stocks pay no dividends, and also can have prices varying widely.

Inflation has historically averaged about 3%, and after being 1.2% in 2020, now inflation in 2022 is high but down some, but is still currently at 6.5% (December 2022) which is still a 40 year high, so the times have become very different. But the S&P 500 Total Returns (includes dividends) has averaged 11.77% gain for the last 50 years (including the 13 negative years during that period).

Since very low earnings today from bonds is less appealing, my interest was about something like the 4% Rule, but for 100% stocks, such as the very popular S&P 500 index funds. A good stock fund earns a lot when the market is good, but market value can drop significantly when market times are bad. But which is more just a delay, since long term, even a 50% drop is not the end of the world, since the bad market crashes have always fully recovered if you can hang in there and wait it out. **This is definitely NOT speaking of bad investments recovering**, but is instead speaking of good investments in bad times, which have always recovered. However there is always risk that withdrawals at suffering prices can deplete a small fund. More money in the fund can survive fixed amount withdrawals longer. But if no withdrawals, it should recover and last indefinitely.

So instead of 60/40 balanced funds, what about a "4% rule" for a 100% S&P 500 Index fund? The S&P 500 calculator and Survival Test on the first page is about that.

My emphatic opinion is that all withdrawals ought to be planned to occur after retirement, not before. Repeated withdrawals drastically limit long term performance. The big issue is that retirement generally means having no job or salary, requiring living off of savings for maybe 30 years, perhaps even with major health care expenses, which will require some planning. The time to realize this is when young with still many years of great opportunity.

Compounding is the largest gain effect of **long term market investments**. One year has earnings, which (if positive) then that greater working total increases the greater total amount producing earnings the next year, continually repeating every year. But market gain is variably different every year, making it hard to judge actual performance numbers, but there is a mathematical way. Here's a math example with six years of example Total Return statistics. For it to be Total Return, don't forget to add any reinvested dividends and readjust cost basis.

The same percentage numbers are computed regardless of any initial value, $10 or $10,000,000. These are just made-up example gain numbers, accurate math but NOT representative of any actual real result.

Example of the Manual Calculation Method of Annualized Return | ||||||
---|---|---|---|---|---|---|

Year | Initial Value | Gain | Annual Gain | Final Result | Compound Gain | Annualized Return |

1 | $10000.00 | 15% | $1500.00 | $11500.00 | 15% | 15% |

2 | $11500.00 | 23.5% | $2702.50 | $14202.50 | 42.025% | 19.174242% |

3 | $14202.50 | 10.4% | $1477.06 | $15679.56 | 56.7956% | 16.174628% |

4 | $15679.56 | -5.2% | $-815.34 | $14864.22 | 48.64223% | 10.416901% |

5 | $14864.22 | 12.1% | $1798.57 | $16662.79 | 66.62793% | 10.751486% |

$16662.79 | 20% | $3332.56 | $19995.35 | 99.95351% |

**Method A:**

Percentage gain =

New value - Old value

Old value

× 100Old value

From this computed table, this overall example six year gain (initial to final amount) is

**((19995.35 - 10000) / 10000) × 100 = 99.9535% gain**.

If you already know the final and initial values, then use this gain formula, or see the 1st Gain calculator below. Stock price does Not include dividends, but dollar amount does include your reinvested dividends. Any withdrawals are received value which should be added back into the final total, but withdrawals will drastically reduce gains.

**Method B:**

This first year gain of 15% (in the table above) means the first year result was 1.15 × (the initial 1x amount). That becomes the initial amount for the second year. These same six individual yearly gains (as 1 + percent/100) will also compute total compounding result as

**(1.15 × 1.235 × 1.104 × 0.948 × 1.121 × 1.20 - 1) × 100 = 99.9535% gain**.

Or see the 3rd Compounding of Yearly Total Return Percent calculator below. Morningstar.com shows ten years of these annual **Total Return** numbers (including reinvested dividends, which for stocks are at the *Price vs Fair Value* tab, and for funds are at the *Performance* tab).

The order of the years makes no final difference. Compounding is simply repeated multiplication of gains. These methods do use the annual gain numbers with only 3 or 4 significant digits, but it does fairly well. The initial dollar value does not affect the gain percentage. And then, the final Total Result **value** is (1 + gain/100) × initial invested value. **For an initial $10000, then 1 + 99.9535% is 1.999535 × $10000 = $19995.35 result value.**

**Important Details: The ±1 in these equations represents the initial 1x investment value.** That 1 becomes 100% when multiplied by 100 for percentage. If the gain is 12%, the final value will be (1 + 0.12) = 1.12x (as done in the six yearly terms at B above, and with current dollar value of 1.12 × initial value. Whatever the initial amount was, it is 1x here). Subtracting the initial 1 from final multiplied result leaves just the gain portion. *The gain computes the same numbers regardless of whatever the value of initial dollars.* The 99.9535% gain in this example is 1.999535x value, which is essentially 2x total value. So the final value would be 200%, but subtracting 1x initial value, the gain was 2×100% - (2-1)×100 = 100% (1x less than value, for any gain). The 1st Gain calculator below tries to differentiate gain and value.

Each year's gain is an individual multiplier of the initial value. Each factor is **(1 + gain/100**, with 15% gain becoming a 1.15x multiplier of value. **Negative gains** use the same method, for example **-5.2% is 1 + (-5.2/100)** resulting in 0.948x value that year. The -1 is subtracting the 1x initial value, to see just the gain portion. Or the 1 is added to gain (the 1 is actually 1 x 100, which is 100% of the original initial value), to see the final total value result. The amount of "gain" does not include the initial value, but the total value result does.

Note that a **price** result does not include dividends or compounding, but a formal **Total Return** does, including reinvested dividends. However, if there were no withdrawals, the final **value** resulting from the investment is a clear total answer including everything that happened. So $50,000 value result from $25,000 invested is a 50000/25000 = 2x gain, pure and simple. And it is a (2x - 1x)×100 = 100% gain. If a market result that took 10 years, it is (2^{1/10} - 1) x 100 = 7.1773% Annualized Return. Meaning, (1 + 0.071773)^{10} = 2x value would be the same result if it had come from a fixed rate interest source. Annualized Return seems a useful way to compare varying market results. Annualized Return is the same result, but of a fixed gain rate.

The overall gain number is of course important, but it's largely about the total number of years. Annualizing it to an equivalent annual gain shows the rate of gain, allowing valid comparisons.

These Annualized procedures are based on the standard compounding formula of *(1 + fixed interest rate/100) ^{years}*

You might imagine that the total gain percent divided by the number of years would be a reasonable annual number, but it is a noticeably wrong number, far from reasonable. The correct number is smaller, because it does compound into the same actual final gain. Over just a year or two won't add up much, but long term, compounding is very impressive.

What is the resulting gain of the example six years shown as: 15%, 23.5%, 10.4%, -5.2%, 12.1% and 20%? The answer is shown above to be 99.9535% total gain, and 12.24185% Annualized Return, which is more understandable. **Annualized Return** (the same total gain viewed as the equivalent return each year if each year were equal) computes **the hypothetical fixed interest rate AS IF ** it were that same interest rate every year. It still gives the same final total result, still an accurate result number, it just didn't actually happen that way. But it would be the same final result if it did, so that's a good way to understand the gain. Annualization can be helpful because otherwise it is very difficult to visualize the result of a string of variable years, maybe both large and small gains, some even negative. The Average of the years is NOT the correct answer. Average is a sum (not compounded), but compounded gain is a multiplied product.

**To compute the Annualized Return rate** for the example in the above table with the above gain formula is:

**Total gain**is ($19995.35 - $10000) / $10000 = 0.999535 x 100 =**99.9535% gain in 6 years**(Method A). The final total value in dollars would be (1 + 0.999535) × initial investment. The 1 is the initial 1x investment included in the final total value.Or using Method B, (1.15 × 1.235 × 1.104 × 0.948 × 1.121 × 1.20 - 1) × 100 =

**99.9535% gain in the six years**.Note that Percentage Gain formula above (and calculator below) can enter units of either this method (like a 1.15 multiplier for 15% gain) or can use actual values like dollars (for example, if the initial value was $1, the 1.15x gain is $1.15 value). Percentage comparisons compute correctly regardless of the actual value, just meaning, $8 vs $1 or $800 vs $100 is exactly the same percentage as $8,000,000 vs $1,000,000. These are 8x or (8-1)×100 = 700% gain. The -1 subtracts out the 1x initial investment to show only the gain. The gain percentage is not affected by the initial investment value. Percent only reflects the degree of difference between Start and Final value.

**Annualized Return**of this overall gain of 99.9535% is**((1 + 0.999535)**^{(1/6 years)}- 1) × 100 = 12.24185%Annualized meaning AS IF this result were from the same fixed gain every year that would produce the same result. The first +1 adds the initial amount to compute gain, and the final -1 subtracts it to show just the gain, instead of the final total money amount. Again, if the final gain period is negative, like -8.2% over 6 years, still use

**1 + (-8.2/100)**which is (0.918^{1/6}- 1) × 100 = -1.4158% annualized.**Tricky math part again:**In all these formulas, there is the initial and the final amounts.**The ± 1's represent the initial amount, separate from the gain**(it distinguishes between gain and resulting value, so 1x value + gain x = x of resulting value). It doesn't matter if that initial was $10,000 or whatever, 1x represents it, and we are working with percentages to come up with a final result. As a percentage, the 1 is multiplied by 100 to be 100%. So in the above Annualized Return formula, the first +1 adds the "initial x" to the gain (to be the 1.999535x final value result), applies the reversed exponent of 1/years to compute one year, and then subtracts the initial amount (the -1x again) to be the x percentage of only the gain (less the initial amount).The Annualized Return is of course Not what actually happened each specific year, however it starts with the actual real final gain result and works backwards. The point is if an investment did actually achieve 1.999535x value (99.9535% gain) in 6 years (again, this is speaking of the Total Return result including reinvested dividends), the resulting annualized 12.24185% interest in six years would be that same 1.999535x value result. It is a very good way to visualize performance of investments that vary so much (+ and -) each year. It could still matter if comparing with a published fixed rate compounded every day or month, but using each year is what Annualized Result means. Stocks are basically compounded on the date of every dividend and also with every market day's price, but annualization computes accurately using whatever final dollars are actually present at the end of the year (instead of how they got there). That is also how annual Total Return stock numbers are computed. So whenever Morningstar says the 10 year Total Return of a stock was say 12.5%, that means annualized, as if every year. That didn't literally happen every year, but the final result was the same. You may want to think of the fixed rate gain as the resulting dollars annualized too (should be the same, unless fees or other costs). The purpose of Annualization is for comparing the return of a varying rate as if it was a fixed interest rate of results, useful to better visualize a number for the varying market gain.

The Withdrawal Depletion Test on the S&P 500 calculator page shows the S&P 500 Annualized Return is usually around 10 to 12% (a few times are a bit more or less) if starting in any of the last 50 years. Or 26% for the previous three years (read at the 2019-2021 start line), before 2022 drops it to 16% (2019-2022). But that is only 4 years, and it always has recovered. Always recovering is the only way we got to today. Any withdrawal does drastically reduce the total returns. You do get the withdrawals, but then the future earnings are seriously reduced, which does not compute well.

- Reversing the computing direction to verify that the Annualized number is works accurately (with same example numbers).
**Some relationships of this 1 (which here the 1 is always the 1x initial amount (and 1 x 100 = 100%):**($19995.35 - $10000) / $10000 = 0.999535 x 100 = 99.9535% total**gain**in 6 years

(1 + 0.999535) final gain x initial $10,000 = $19995.35**final amount**

((1 + 0.999535)^{(1/6 years)}- 1) × 100 = 12.24185**% Annualized Return**(with the -1 subtraction)

((1 + 0.1224185)^{6 years}- 1) × 100 = 99.9535**%**(with the -1 subtraction)**Verified**Overall gain percent

((1 + 0.1224185)^{6 years}) × 100 = 1.999535**x**(**x multiple**computed without the - 1 subtraction)

It gets easier after the first few tries. It is all based on the standard annual compounding formula of**(1 + fixed interest rate/100)**= x gain^{years}This examples 1.999535x gain is very nearly 2x. And the Annualized Return 12.24% is near 12%. And the doubling in 6 years x 12% is the Rule of 72. Which is an approximation, but it does indicate the math is working OK.

The large number of digits are shown in the calculators in case someone wants the precision to reverse calculate for verification. Values like $1,000,000.01 are 9 significant digits. If reverse calculations don't quite reach the same exact number, you need more decimal digits for better precision (which are shown in the calculators here). (Only 8 significant digits here, but the final value is only 7 digits.) It is possible for a calculator or computer to use FULL precision of all the numbers, especially for exponent calculations. Meaning, don't round off the data until time to show it. For example, the initial default values for calculator 1 & 2. Computing 9 digit values (like $1,000,000.01) needs at least that many significant digits all along for full precision. You would round off final results to show them, but while in the computer, compute with the full available precision, without any rounding. Rounding during calculation (of dollars and cents or of percent) limits result precision to that limit, however even a rough approximation might still be adequate as a ballpark comparison.

**The Average of a fund's annual gains is Not the measure of stock performance**. Because the average is just the**sum**of all years gain divided by the number of years (with no compounding). But the long term compounding result is the**multiplication product**of the years of gains. It's difficult to judge a final result just looking at several mixed year results (with maybe both + and -), but the Annualized Total Return is a very good way to realize the actual performance of a stock.

**1st calculator, Gain:** Hopefully it is both self-explanatory and maybe the most useful. Technically, any units work (like price or distance or weight or time). Or Dollars or Euros or Yen, but gain calculations need Not be about money.

**2nd calculator, Future Value:** This one is perhaps less used, but if no withdrawals or additions, it might estimate expectations of final value, however a future variable market gain rate is not predictable.

Or it can be used to reverse compute to verify an Annualized calculation is correct, and in that use, the apparently exorbitant number of digits shown is because large values like $1,000,000.00 have 9 digits, which needs at least that many significant digits in the interest rate to accurately match the precision desired (years is an exponent of interest). For example, in the initial default shown, using instead 12.2% (3 significant digits) computes $999,342.31, also accurate to three digits. For example, the initial default value could instead be the 12.2% and still compute final value within 0.07% ($658 difference in a million). So approximations might be an adequately useful estimate, but full precision does require the necessary number of significant digits. The precision of future market result estimates is unknown anyway, but exactly matching a reverse verification of Annualized rate needs about all the digits you can manage (compute it first, before any rounding).

**3rd calculator, Total Return from yearly gains, Two methods:**

- It accepts Copy and Paste data
**direct from Morningstar screens**

(or other web tables of Total Return %). - Or you can simply enter each year's gain separated by a space.

To be the Total Return, each year's data percentage should be the years **Total Return %**, which includes reinvested dividends. Morningstar.com shows ten years plus YTD of these annual **Total Return %** numbers (which for stocks are at their *Price vs Fair Value* tab, and for funds are at their *Performance* tab). Morningstar also shows Annualized Return numbers (called Trailing Returns) for various year periods which are from the *current date instead of year end values as here*. Enter the yearly percentage, like as 15% instead of 1.15x. Then this method makes getting the compounded gain numbers be easy.
The data is shown interpreted in case you tried editing the data and messed up somehow, it may help see the trouble. If any data trouble editing the Morningstar dara, just start over with the simple Copy and Paste. That works.

The Average gain is shown, to make the point that the average is not useful. Instead, the Annualized Return computes correctly.

**Extra Calculator 3 Details:** The Copy and Paste from Morningstar uses **tabs** for year separators (between each year value). And that tab works fine here, you can leave it just as copied.
Actually, this tab thing likely also works from any HTML screen table showing **Total Return** percentage, such as the dividend stock table above (but of course only columns like the 2013-2022 Total Return values there, meaning, NOT the first or last three columns there.)

But the concern was that if you edit the data, you cannot enter a tab in the browser's field. So the calculator always first replaces all tabs in the data with the space character, and it uses that space between numbers to separate them, and shows it that way. So if desired, you can instead just type the data directly, or use a text editor with a space between each year, and then copy/paste it here and that will work too. The calculator doesn't care which is present, it works both ways, with either space or tab, or even mixed (but the separators must be one of them). You will see the space in the displayed data, but the tab still works too (but a tab will become a space). Again, a Copy of the Morningstar data will have tabs in it, which works as is, but which will then become a space. Multiple embedded spaces in the data are OK, they will be condensed to be single spaces.

The calculator's initial Apple data default result exactly agrees with Option 5 of the Funds Calculator (if it is extended to include 2013, for Span 2013-2022. The final year data has surely changed since then.)

**Significant digits:** The extreme number of decimal places shown for the Annualized Return percentage are actually intentional here. It could have been rounded, but the Full precision doesn't hurt anything this way that allows reasonable precision to be possible to calculate backwards from the result back to the original data. To validate the first calculation for example. For example, $1,000,000.01 has 9 significant digits of precision, and to compute that exact value requires the data values to have at least that precision. I've shown the Annualized Return with 10 significant digits, which may be excessive to view, but when the math has an exponent of 40 years for a result in the millions, it is more accurate to provide adequate accurate precision.

Any computed result will not have any more precision than the precision in any number used to compute it. Some numbers are **Exact Numbers**, and are fully precise, like 3 apples or 10 people or a $20 bill. Or perhaps 2 times investment or 5 years if actually exact. And sometimes, especially with exponents and large numbers, even one more digit will help. One issue though is that the Morningstar annual Total Return percentage is shown with two decimal places, so typically the multiplier is limited in significant digits of precision. So we have to think of it as ballpark accuracy, but it should be close enough.

**All three calculators above:** The Annualized Return is defined as the gain for an entire year. If the final year is as yet a partial incomplete year, there is a tricky issue to know about. Data for 2023 is a partial year too.
For example, say we only had data for last year, which was a 10% gain, and for January of this year, also 10% in that 0.085 of a year.

The data gain is 1.10 x 1.10 = 1.21x or 21% so far.

1. Annualized 1.21^{1/2 years} = 1.1x or 10%. (2 full years, each is 10%. The remaining 11 months are zero gain, which they are so far)

2. Annualized 1.21^{1/1.085 years} = 1.19x or 19%. (19% does not match anything I see)

But 10% in a month is actually a rate of 120% if it continued a year, which it has not yet.

3. (1.10 x (1 + 1.20))^{1/1.085} = 2.27x or 127% (annualized return seems right, but it does not match the 1.1 x 1.1 earnings so far, because the full year has not occurred yet)

I like the first one. It seems correct to me. Two 10% gains appear to match the money then. The 0.085 year does not seem to do anything useful here.

Calculators 1 and 2 will allow numerically entering a partial year, like 6.4125 years. If you do enter the final partial year as less than 100% of the year, the gain values will still correctly show the SAME value and gain percent numbers (which is what actually happened so far either way), but the annualized rate may be absurd, or at least won't be correct.

Calculate the Fixed Interest Rate

and actual Rule Number that will

compound to x value

and actual Rule Number that will

compound to x value

An Enter key in these fields will

recompute this table

Extend range to years

Off-topic a little, but if checking that these numbers are reasonable, a simple rule of thumb check is the Rule of 72 that says an **investment value about doubles if the years × fixed percentage gain = 72**. So 6 years × 12% = 72 would approximately double to be 2x value (1.12^{6} = 1.9738, almost 2x).

The Rule of 72 is an approximation dating back to at least the first known mention in year 1494 (in the time of Columbus), when the calculation was difficult. The Rule of 72 is most accurate for 8 to 10 years, and technically 6 years should be Rule 73.4772 for 12.2462% doubling in 6 years (1.122462^{6} = 1.99999948). The 12.2462% Annualized Return is an impressive rate of gain when compounded, awesome over many long term years.

This calculator purpose was to look at accuracy of doubling with the Rule of 72, however you can enter different multipliers here (other than 2 for double, like 1.5x or 3x or 10x).

Or the accurate "Rule" can also be determined by the first calculator just above (by specifying two simple values that will double, like 2 and 1, or that will triple, like 3 and 1) and the years.

In exploring the Rule of 72 (in this table), it became clear it is only a simple rough approximation. Speaking only of doubling, the worst accuracy with 72 is if 5 or less years. The best case for Rule 72 doubling is for 8 to 10 years. But a Rule of 70 works better for doubling long term 15 or more years, but that much time to double would not seem a great investment 😊 .

So I suggest that in specific situations, the first Gain calculator above will be more useful and versatile and certainly more precise than the Rule of 72. In that 1st calculator, New = 2, Old = 1 over 6 years is 2.0x Value, 100% gain at 12.2462% Annualized Return, which agrees with this table. The percentage numbers are the same for any money value.

Retirement is the income issue of course, salary typically stops then, but we may live longer, possibly even 30 years more. So we need a plan for income then.

Based on past S&P 500 performance history, earning a million dollars has been relatively easy, if given the sufficient span of years to let it grow. See the 2nd Future Value calculator above.

Maybe it does take either a lot of years or a lot of investment, but the years are automatically doable if starting early enough (meaning if you will just do it). One million dollars is roughly 40 years from $10K, or 30 years from $35K, or 25 years from $60K, or 20 years from $100K. **The long range of years is a magic opportunity.** The S&P 500 calculator on previous calculator page can also show that only $10,000 invested in 1980 for 40 years (which sounds like an extremely long time, but it is like age 25 to 65) and left untouched until today would have been worth about $1 Million now. That result assumes 12.2% Annualized Return which has been a valid average (NOT meaning an average annual gain, but instead an average Annualized annual gain), with dividends reinvested and **compounded for 40 years, despite including the few very bad market years** (the 2000s decade did not help, but it still got there). And much of it remains to continue growing 20 or 30 years after retirement too. Results of starting in any year, and/or with any other starting value, is shown in the Survival Test Mode chart on the previous S&P 500 calculator page. The $10,000 doesn't earn so much initially in the beginning, but after it's grown to more digits in the last several years, the growth seems amazing. And that growth keeps earning more too, which is the concept of compounding.

12.25% Annualized Return for 40 years is 1.125^{40} = 101.73x gain from $10K is $1,017,311.58 result.
Or say it was less, call it 8% Annualized. Then an initial $50K would be needed to make it to $1,000,000. And of course, you can start with less, and continue adding more as you go along. It is very doable if you do it. Calculator 2 will compute these cases. Or the S&P 500 calculator will show results of starting in any past year with any amount, and adding to it each year in any amount, to show the result today according to the past performance.

Compounding is easy, all you have to do is start early and then just wait long term. And think what adding even more investment to that now and then could have done. **Starting or adding when the market is down (certainly including today, 2022) is a really good time (to buy low for maximum growth opportunity). A drop in the market is Not the end of the world, and recovery provides opportunity for much additional gain.** The young probably think other things are more important now, but I promise that your priorities will change near retirement time, after it is too late (trying to get your attention if you need it). That growth will become quite important at retirement time, and the best tool is an early start. It also continues earning and compounding after retirement, during 20 or 30 years of retirement withdrawals. If looking for magic, this comes pretty close, and seems a mighty big deal.

$10,000 might have seemed impossible for me at age 25, but starting with $1200, and adding $1200 a year ($100/month) to it for 40 years all along (without fail, adding $51.6K overall) also creates $1 Million. Think of it as supporting yourself in your old age.

Or one approach is you can create a self-directed IRA that invests in a S&P 500 fund. A S&P 500 Roth or IRA that adds the $6000 maximum every year could reach $1 Million in just 25 years (example 5 on previous calculator page). A 401K plan has a much higher maximum contribution, and a possible company addition. And of course, if possible, a Roth instead of IRA or 401K would eliminate the taxes on the million, which would be a real big deal then.

Age 65 will come for all of us, when salary stops and we will need replacement income, which will become extremely important then. It is too late then, but planning makes that possible if you start early. Then thereafter, 4% withdrawals from $1,000,000 is $40,000 a year to add to Social Security. The fund would continue making its gains then, but if $1 Million, then withdrawing $40K a year would last 25 years even if zero additional gain. However taxes will be due on it, making any large lump sum withdrawal seem unwise. But spread out into smaller withdrawals over more years, taxes on high income will be the best problem you could have.

**The easy and best solution** is simply to start a good investment early, without fail, as early as possible, today. The 4% Rule was concerned with market bad times surviving 30 years of retirement withdrawals, after building substantial value with years of investment without withdrawals. From my own experience, my notion is that it takes many young people many years to realize that the many years of opportunity available to them would have been their very best and easiest and greatest tool BY FAR, but then there is no going back for a redo. Wasting that most valuable opportunity would be a tragic shame.

Again, these results are computed from the past years in history, **and future results are not known. The standard obligatory investing advice is that past success does not guarantee future performance**. There have been bad times (including today, 2022), but it always has recovered. Past success of long term investing in the S&P 500 seems clear enough (the 500 largest and most successful companies in the USA).

Compounding is certainly a real big deal in investments, making many **long term** years be the most profitable part. Only a year or two is not so dramatic, but compounding is exponential with time, becoming huge over many years. Long term can be exceptionably good. The S&P 500 (gain and reinvested dividends) has averaged an annual return around 12% for the last 50 years. The future is not known, but it sure seems a good bet if you consider "long term"). It is true that the S&P 500 is down now. Two facts though, this or worse has happened several times over the years before, and it always recovers and continues. The S&P 500 was down 25% at $3585.62 on Oct 3 2022, but 40 years ago it started with only about $122, which is an increase of about 30x so far, not even counting the dividends.

Plug in your own numbers, but if your age is 40 years or less, then you still have at least 25 years before retirement at 65 (when you will certainly be needing a source of income). Today is the latest time to be considering that. And the investment can continue earning during 30 years of retirement withdrawals too. The years will be your largest growth multiplier, so wake up, and get with it, now (the term *Buy Low* means, the market is currently very low to making buying right now be the very best and most profitable time, very wise). I've just shown how $10K now can grow to $1 million in 40 years, so don't foolishly waste the years. (25 years will need about $60K.) The market always recovers, but lost years cannot be recovered.

The market goes up and down a little every day. It can make you crazy to watch it every day. But don't sweat the small stuff, it will be different tomorrow. Do understand that it is very normal to go up and down every day. Another page shows four years of this daily S&P 500 activity highlighting the peaks and valleys.

There are some bad times, and some people are scared off and will cash in and get out of the bad market, but that simply locks in their losses and makes it permanent, not recovered. Others grit their teeth and bear it, and hang on and wait for the correction, and then continue on recovering and happily earning more money. I recommend this latter course. It happens now and then, and waiting it out is no fun, but it pays off. The alternative is accepting the loss. But the world continues on, it does not end.

A Brief History of U.S. Bear Markets provides a very clear and informative view and details of our bear market history, that you ought to see. That one does not show the good times, but for that, also see its second green graph just below it (click it to enlarge it slightly). Certainly you should realize that crashes do happen now and then, but also, that they do recover. A Bear Market is defined by at least a 20% decline, which can seem mighty uncomfortable at the time. The worst ones have hit -50%. Many investors panic and sell and end their fund then, which just makes their loss permanent and very real. But instead hang in there, and it will eventually recover into happy times again with continued gains. Most years are good, and the long term gains are hard to ignore. Politics and taxes do need watching, and bad times do happen every once in a while, but then recovery also happens too.

The market is usually good, with many more good years than not, and long term wins. But starting the calculator data at 1970 was deliberately chosen here to include actual real data for some seriously bad times. The crashes of 1974 and 1982 and 2001 and 2008 were exceptionally bad economic and market times. In contrast, the 2020 pandemic crash, -34% was tough on the economy and market, but its cause was not economic or political, and the market recovered quickly to current all time record highs. There were other smaller dips, but the 1970s were poor (one crash) and the 2000s were worse (two crashes), all near 50%. The recovery from 2008 took the longest in modern history (until 2012), and the entire 2000s decade was down 9.4% (a "lost decade"). So 2000 was the worst year to start the fund in the last 50 years of history. The price of the actual S&P 500 was under $1000 in 1997, again in 2002, and again in 2008, but even so, reached $4700 in 2021. That is just the price (less dividends), but the compounded gains have been exponential in the many years of gains. Investing for long term is the way to bet.

The current version of this Google chart is here.

The 2001 and 2008 dips made the entire 2000-2009 decade lose 9.4%. It fuly recovered in 2013. The 2020 pandemic dip was deep (-34%) but relatively short duration. The S&P 500 was down 25% at $3585.62 recently (Oct 3 2022). The current inflation (2022 is the highest in 40+ years) is a factor, but if the elections go right, it will recover. See a current status of Total Returns of 70+ stocks.

**Corrections:** Market drops of more then 10% are called Corrections. These are fairly routine, and happen more often then you might think, but they typically don't last long before the correction recovers. Again, we learn to take it in stride, and in fact, the low times are often welcomed as great times to buy more at the lower price. That is the meaning of "Buy low, sell high".

**Bear Markets:** Drops of more than 20% are called Bear Markets, occurring less often but much more severe. These might reach 50% down in truly bad economic times, but they have always finally recovered (could take a year or two then, or even more). The worst action would be to cash in by selling during the low times, **which simply locks in your loss permanently with no opportunity for recovery**. Buying more then is the better choice, the recovery will be profitable, but timing the exact bottom of the market is impossible (the bottom likely will not be in the first few weeks though).

One accounting of this says "Most declines are quickly erased but the deeper the stock market decline, the longer the recovery." They make this report about history (I am unsure how precise the numbers could be in the future):

- 5% or greater pullbacks occur about every 7 months
- 10% or greater pullbacks occur about every 2 years
- 20% or greater pullbacks occur about every 7 years

And the few worst past ones have reached 50% down. But it happens, and then it recovers, always has. The 2020 pandemic dropped the market 34% in March, quite bad but short. It recovered quickly 100% by August, and the year ended up at a new record high with 18% annual gain despite the lost months. In the following March the S&P had achieved a 76% gain (a year after the low). Recovery of bad economic situations can take a couple of years though, until the economy is corrected. 1974, 2001 and 2008 crashes were spectacularly bad, and each took a few years to recover. But they do recover.

Currently, most companies are down and negative for the year, but the leading growth stocks (Apple, Microsoft, Amazon, Google, Nvidia. Tesla, etc) are down big time, -25 to -45%. It's just market fears due to all the current problems. There is nothing wrong with the companies, their earnings are doing great. The Russian invasion of Ukraine is of course a big worry, but the painful self-inflicted inflation is another of the current big concerns about the US economy. The government's massive spending of Trillions is a large factor, and their own self-imposed policies last year limits our own U.S. oil production, which has had very strong effect increasing inflation. The U.S. oil production was self-sufficient before, but now we must import oil again, and pay the price. Oil affects the price of about everything (transportation, plastics, etc), and the doubled oil price has increased U.S. inflation, up from 1.2% in 2020 to 7.7% in Oct 2022 (2022 is by far the worst inflation in 41 years). But the cavalry will come and the market has always recovered.

**Recessions:** The definition of a recession is about the decline of national GDP growth and the rise of unemployment. Recessions are NOT about the stock market. Some imagine a recession is just when there is two consecutive declining GDP quarters, and we do have that now, but a recession is also additionally about unemployment statistics, which are still rather low now, so there has been no recession called. Technically, the National Bureau of Economic Research (NBER) decides if and when it is actually a recession. The unemployment rate is still quite low right now, so it is not yet a recession. It's bad though anyway.

Predictions about the market future are only guesses, and at any given time, many "expert" guesses heard will always be rosy and bright, and many others are always gloom and doom. It doesn't take long to understand that no one actually knows the future. I am certainly no expert, and I don't know either, but it is easy to see that the long term S&P 500 graph (meaning a few decades) sure always looks great, but with some dips. The market goes up and down every day of course, with many more good years than bad years (but yes, expect a few bad years as a matter of course). Withdrawing everything when it is down in bad years is the worst plan, which simply guarantees the loss is real and permanent, with no recovery possible. It is scary, and it takes some patience, but it will recover if no withdrawals. Market crashes do happen every few years, and they are survivable. The S&P 500 does recover.

**But there is no one safe magic percentage withdrawal such as 4%.** Because how long a fund can survive retirement withdrawals in bad times actually depends on how much money it makes available. **This 4% number does assume it is recalculated every year** (same 4% percentage, which calculates different withdrawal dollars each year, depending on the different current investment total).

- Before retirement, just hang in there. Make no withdrawals, preserve the funds value. Assuming it is a good investment, adding additional investment is good, and doing it while prices are low is the right time to see even greater gains during recovery.
- At retirement, salary stops and withdrawals likely begin. There might still be Social Security then, but your fund can add to it. How large is the market drop? How much had the fund previously grown? How much of the fund value was previously withdrawn?
- Specifically, how much value is left in the fund to support recovery and withdrawals during retirement? Was the fund newly started or had it grown large?

4% withdrawal from $1 million is $40K a year and will last 25 years even if no further growth, but 4% of $10K is $400 a year ($33 a month) which doesn't seem much help. - How long the recovery will take is unknowable. Perhaps a year, or maybe two, or maybe less, or perhaps even more. Jan 2 2023 is one year since the last record high of the S&P 500 Index (and the S&P 500 low on 30 Sept 2022 reached -25%, and still nearly -20% at year end). But the 2020 Covid pandemic crash (-34%) was only six months between record highs. The 2001 and 2008 crashes were each about 50% and made the entire 2000 decade negative. But it always has recovered.

Building a larger fund by retirement time is the best help, and it should be seriously planned. The more years you have to do it will make it much easier. Pulling the money out of the market after it falls low simply ensures the loss is both real and permanent (with no chance of recovery then). But it always has recovered.

We don't know those things about the future, but we can see such instances in the past, to suspect what we might expect sometime in the future. We can see that it has always recovered. If the fund value drops 50%, then from there, it must recover 100% to reach the same original value again. Our own withdrawals also during the low times are dangerous to the survival of our fund. Even innocent looking fixed amount withdrawals can become drastic in bad times (see the $200/month fixed withdrawal in the 50 year S&P 500 calculator which fails soon, but a 10% withdrawal finishes 50 years, without growth, but no failure). The advantage of a percentage withdrawal is that (if the withdrawal rate is then adjusted every year to the same percentage of the funds then current value), the withdrawal becomes very low when the fund value is low. Except actual withdrawals are usually set up as fixed dollar amounts every month. But a percentage withdrawal definitely implies the withdrawal is recomputed every year from current fund value, which becomes less withdrawal when the fund value is lower.

Withdrawals of course depend on money still remaining available in the fund. If no withdrawals, the fund will survive and continue growing, but withdrawals will drop the fund value fast, especially when low in bad times. The S&P calculator program cannot predict future gains, but its purpose is to see the result of some typical actual bad times from recent history, and also to see the results of withdrawals, to help know the best future plan.

**Again, this is definitely NOT speaking of bad investments recovering**, but is instead speaking of good investments in bad times.

25 years ago, the original 4% Rule data looked at the market back to include the Great Crash of 1929, but times and laws and market rules have since changed so much, and IMO the last 50 years seem typical enough of today's world. The calculator Test on the previous page is ONLY about actual S&P 500 Index history. It has no historical data for any other funds except S&P 500 Index funds (which are a very popular class). All of those will show the same S&P result, except they do vary in the fee they charge (the fund fee is withdrawn every year, and a fund with a low fee is a big plus).

**How much withdrawal can survive bad crashes is a vague question though.** Situations vary. A market crash just when you need the withdrawals is the fear. Another danger is an early crash before the fund has grown to be able to survive it. Do realize if a fund loses 50%, the low price then has to regain 100% to recover.

The market goes up and down every day, but fund survival depends on how much value is in the fund, and specifically, how much value is also being withdrawn from the fund.

- A really bad crash can drop value to 50%, and has occurred a few times, but its date is not predictable. So then, how much other money is also being withdrawn? If no withdrawals, a crash will recover in time. Sitting tight long term might be uncomfortable, but it is has always recovered to continue growing, whereas significant withdrawals during the crash could endanger emptying the fund.
Withdrawing from a fund of low value won't last long in worst situations, but a larger fund value certainly helps. If for example, a $2 Million fund crashes to 50%, it still has $1 Million, which is $40K a year for 25 years, even without any recovery or future gains. That case seems a lesser problem, and recovery will come. Just saying, starting early to accumulate a larger fund is a great plan.

- A 4% Rule sort of assumes the fund should survive if its average return is at least 4%. The S&P 500 has always earned about 10%, however in practice, the fund might often gain more, but then suffer a 50% drop. So it is uneven, and unknown, so there can be some difficult spots. Withdrawing only about half of the funds annual gain seems safer. The S&P 500 does gain more than a balanced fund can, but it also crashes deeper.
Realize that 4% of not much money is even less money, but 4% of a lot of money is much more satisfying. The S&P 500 has always averaged about 10% all along, meaning withdrawing 10% should stay at more or less the same fund value (with risky variations), but withdrawing the average gain cannot grow much, so will 10% not pay much either. The S&P calculator shows (if starting with $25K) the S&P 500 would have survived 10% withdrawals every year of the last 50 years, but 10% of say $25000 is only about $200 month, and is even less withdrawal when fund value is low.

- The most reasonable plan is to first let the fund grow to a larger value, for say 20 or 30 years before starting any withdrawals. That would then allow withdrawals greater than 4% during retirement. This is shown in Example 4 on that previous calculator page (10% withdrawals survives more then 30 years then, and still grows to 10x ending value). This seems very worthwhile.
And for example, 10% withdrawal even from the first year did survive if starting with $25K in any of the past 50 years, but it could not grow or pay much if 10% withdrawals. The point is, more fund money does last longer, but withdrawals approaching the fund gain rate cannot grow. But the main goal in retirement is that it just needs to last 30 or 40 years.

The commonly seen market advice about risk is **"Past success does not guarantee future performance."** Meaning, we don't know the future, and unexpected bad times do happen. But IMO, that is speaking of short term events (up to a few years). I get my encouragement by looking at a graph of the S&P 500 history. Market gains certainly offset inflation, however *do unclick the Inflation-Adjusted box there* to show the actual S&P data. The world might someday end, but the graph long term trend does look very promising. 😊 The notches in the rising curve are the bad times, and there's been many of them, but they get forgotten as the curve goes up. It does show that the 1970s and the 2000s decades were serious bad times (a mouse-over there shows the dates). The bad times will seem drastically bad at the time, but they always recover (might take a year or two, but retirement is a long term goal, right?)

**The actual risk** is that if the fund is saving for a specific time, like for retirement or a child's college expense, a 100% recovery might not be fully available at the time needed. But college is a four year duration, not all needed at once on the first day, so it has more time. And retirement is possibly a 30 year duration, and growth continues all during that time. We don't know about the future, but the program can show the effects of some past bad time drops.

Fund values seriously suffer from any withdrawal, both by reducing the remaining balance, which also limits the future gains. IRA RMD (Required Minimum Distribution) is required after age 72, but otherwise withdrawals are a choice, but if the withdrawn money had remained invested, that money would have earned more money itself, repeated every year, compounded. **It is certainly wise to cut back on withdrawals in really bad times, to avoid depleting the fund. And it is always best to reinvest the dividends**, and you can see the tremendous difference that makes here (of compounded growth in time). Bad times are the worst possible time to sell out and close the fund since that absolutely locks in and guarantees maximum loss, with no recovery possible. The market will drop in value now and then, maybe to around 50% in the very worst times, which will seem catastrophic and unbearable at the time. But if you can hang in there, it will recover and will then be forgotten (eventually, which could be fast, or could take one or more years). It no withdrawals, the S&P 500 has always recovered to hit new highs, and will resume and continue earning more. Currently, the last ten years have had good results, but the market behavior before 2010 might be considered expected now and then, however it always recovers.

Also see these pages:

Previous page with the S&P calculator.

Next page, Stock Dividends are valuable, but withdrawing them is Not New income.