Stock Return Components
This section introduces the concept of stock valuations and then shows the relative contributions of company operating results and valuation changes to longterm stock returns.

Summary

Main

Details
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Stock Valuations
 To help determine if a stock’s price is appropriate, investors use various valuation measures.
 A stock’s valuation is usually calculated as the ratio of its current price to a measure of the company’s annual operating results.
 For example, an $80 stock whose company generates $4 of annual profit earnings per share has a Price/Earnings valuation ratio of $80/$4 = 20: a shareholder pays $20 to own a dollar of current annual earnings, along with future earnings and their growth or decline.
 A stock’s valuation is similar to the inverse of a bond’s yield. For example, a 4% bond’s price divided by its annual interest payment is $100/$4 = 25: a bondholder pays $25 to own a dollar of current annual interest payments, along with future interest payments until the bond’s maturity.
 The valuation’s denominator acts as a movable anchor that changes when the company’s operations significantly improve or decline.
 The valuation denominator we use in this section is the trailing tenyear average of real (inflationadjusted) operating earnings per share, where a tenyear average is used to smooth out the business cycle’s temporary effects on earnings.
Total Return Components
 A stock’s total return can be separated into four parts, the sum of its dividend or income return, its real (inflationadjusted) earnings growth, inflation, and its valuation change, where the first three represent the company's operating results independent of the stock market.
Results
 All results refer to the S&P 500 index of large US companies.
 Monthly stock price returns and their volatility dominate monthly dividend returns in importance.
 Over longer holding periods such as ten years, price returns remain the dominant effect but dividend returns play a significant role in each tenyear period.
 Separating the price return into nominal earnings growth and valuation changes has profound results: outside the Great Depression of the1930’s, tenyear nominal earnings growth is consistently positive and is often larger than valuation changes.
 Earnings growth and dividend returns, which respectively represent company profits and the portion thereof paid directly to shareholders, together are large enough to offset all but the worst impact of falling valuations.
 Tenyear total returns remain positive through the 1950’s and also from the mid1970’s to the mid1980’s, despite large valuation declines.
 Dividend returns and earnings growth together don’t fully offset falling valuations at the end of the 2000’s, the result of two 50% peaktotrough bear markets in less than eight years, but they keep a bad tenyear result from turning into a disaster.
 Inflation is positive over each tenyear period outside the Great Depression. Other than in the latter part of the Great Depression and briefly in the late 1980’s, real (inflationadjusted) earnings growth is also always positive over tenyear periods.
 Stocks’ tenyear dividend returns are very closely tied to their average dividend yield over the same period. Dividend yields are in the 2% range from the early 2000’s onward.
 Large valuation declines accompany sustained macroeconomic dislocations such as the Great Depression and the 1970’s stagflation. Valuations don’t have predetermined limits nor levels to which they’ll always gravitate, but with current US stock valuations having nearly tripled since 2008, their outsized contribution to the last decade’s returns is unlikely to be repeated.
Conclusion
 We claimed in earlier sections that investor emotion affects stock prices but that companies’ operating results ultimately matter much more. This section brings the proof!
 Over tenyear periods the lion’s share of returns stem from companies’ underlying operations and their growth, not from investor sentiment as reflected by valuation changes. The payment of some portion of corporate earnings as dividends, along with earnings growth that typically matches or exceeds inflation, are consistently positive and large in relation to valuation changes. Only rarely do valuation changes overwhelm companies’ operating results, and those instances are brief.
Introduction
The Bond Return Components section presented a security’s total return as the sum of its income return and its price return. The same applies to stock returns, whose income return consists of dividend payments but which are generally overshadowed by price returns for several reasons. First, as with bonds, shortterm stock price changes are volatile whereas dividend payments are small and stable.
Secondly, the growth of companies’ operating results such as earnings or the dividends paid from earnings ultimately translates into higher stock prices, as shown to the right for the S&P 500 stock index across three distinct periods. Stock prices don’t move in perfect lockstep with operating results but track them broadly. In contrast, bond coupons are fixed and so bondholders don’t participate in companies’ earnings growth. 
Stock Valuations
A stock’s value derives from its owners’ claim to the company’s current and future profits (earnings). To help determine if a stock’s price is appropriate, investors use various valuation measures.
A stock’s valuation is usually calculated as the ratio of its current price to a measure of the company’s annual operating results, i.e. what you pay for the ongoing annual amount of what you receive.
A stock’s valuation is usually calculated as the ratio of its current price to a measure of the company’s annual operating results, i.e. what you pay for the ongoing annual amount of what you receive.
For example, an $80 stock whose company generates $4 of annual profit or earnings per share (abbreviated EPS) has a Price/Earnings valuation ratio of $80/$4 = 20: a shareholder pays $20 to own a dollar of the company’s current annual earnings  the ultimate source of stock returns  along with its future earnings and their growth or decline.
A stock valuation is similar to a bond’s interest rate or yield. For example, a $4 annual coupon bond trading at its par value of $100 has an interest rate of $4/$100 = 4%. Inverting this formula, the bond’s price divided by its annual interest payment is $100/$4 = 25: a bondholder pays $25 to own a dollar of annual interest payments  the ultimate source of bond returns  along with future interest payments until the bond’s maturity.
The valuation ratio can also be viewed as a simpler method to relate stock prices to operating results than the chart shown in the introduction. The stock valuation’s denominator essentially acts as a “movable anchor” that changes when the company’s operations significantly improve or decline.
A stock valuation is similar to a bond’s interest rate or yield. For example, a $4 annual coupon bond trading at its par value of $100 has an interest rate of $4/$100 = 4%. Inverting this formula, the bond’s price divided by its annual interest payment is $100/$4 = 25: a bondholder pays $25 to own a dollar of annual interest payments  the ultimate source of bond returns  along with future interest payments until the bond’s maturity.
The valuation ratio can also be viewed as a simpler method to relate stock prices to operating results than the chart shown in the introduction. The stock valuation’s denominator essentially acts as a “movable anchor” that changes when the company’s operations significantly improve or decline.
For the rest of this section we use as a valuation denominator the trailing tenyear average of real (inflationadjusted) operating earnings per share, where a tenyear average is used to smooth out the business cycle’s temporary effects on earnings, as shown by the chart to the right. This measure is very similar to that popularized by Yale finance professor Robert Shiller in his book “Irrational Exuberance,” published the same month in 2000 that the tech/telecom bubble began to burst. Several other valuation measures are described in the Details tab. 
Total Return Components
In the Details tab we use a bit of basic algebra and some rearrangement to show that a stock’s total return as shown below,
can be further separated into four parts as below:
This relates a stock return to the company's dividend return, its real (inflationadjusted) earnings growth, inflation, and its valuation change, where the first three represent the company's operating results independent of the stock market. The rest of this section refers to the variables shown above for the S&P 500 index of the largest 500 US stocks, and focuses mainly on their annualized changes measured over tenyear periods.
Results
We start with shortterm returns of just one month. Stocks’ monthly price and income returns are displayed in the two charts to the right. Price returns’ volatility dominates the charts and the income return from dividends is barely visible. 

The next set of charts shows the annualized returns and components for tenyear returns. We start with price and dividend returns and then “peel back the layers” of price returns with each successive chart.
Extending the return period to ten years instead of one month drastically increases dividend returns’ importance. The price return (dark blue) is usually the larger of the two, but dividend returns (red) play a significant role in all tenyear periods, in contrast to their near invisibility in the onemonth return chart shown previously.
The second chart separates the price return into the growth rate of nominal earnings, in purple, and the change in stock valuations, in light blue. The result is profound: outside the Great Depression of the1930’s, tenyear nominal earnings growth is consistently positive and is often larger than valuation changes. Earnings growth and dividend returns together represent the portion of returns due to companies’ operating results, as opposed to valuation changes which largely represent changes in investor sentiment. Companies’ operating results are nearly always sufficient to offset periods of extended valuation decline. For instance, tenyear total returns remain positive through the 1950’s and also from the mid1970’s to the mid1980’s, despite respective annualized performance drags of 5% and 10% in each from declining valuations. 

The combined effect of earnings growth and dividend returns doesn’t fully offset falling valuations in the 2000’s. After two 50% peaktotrough bear markets in under eight years, the tenyear total return in black becomes negative by late 2008, but just barely and for only two years: the companies’ strong operating results keep the 2000’s from turning into a disaster, and are then augmented by flat or positive valuation contributions during the 2010’s.
The final chart above separates nominal earnings growth into its inflation and real components. Inflation (peach) is positive over each tenyear period outside the Great Depression. Other than in the latter part of the Great Depression and briefly in the late 1980’s, real earnings growth (burgundy) is positive.
The final chart above separates nominal earnings growth into its inflation and real components. Inflation (peach) is positive over each tenyear period outside the Great Depression. Other than in the latter part of the Great Depression and briefly in the late 1980’s, real earnings growth (burgundy) is positive.
The next two charts add more context to some return components. The first shows the dividend return along with the monthly dividend yield and its 10year average, where the yield is calculated as the annualized dividend divided by the monthly price. As with bonds’ income returns and yields, stocks’ dividend returns are very closely tied to their prior average yield, which averages about 2% over the last fifteen years. The second chart shows stocks’ valuation in grey along with the 10year annualized valuation change in blue. Even over nearly a century there’s a paucity of illustrative examples, other than the large valuation declines that accompany sustained macroeconomic dislocations such as the Great Depression and the 1970’s stagflation. Because valuations’ lower limit is realistically in the low singledigits for the broad market (but zero for an individual company that goes bankrupt), and because they have no legallybinding upper limit, assigning valuations a value to which they’ll always gravitate is impossible. 

However, with US largecompany valuations currently at their thirdhighest level in almost a century, having nearly tripled since their 2008 financial crisis lows, their outsized contribution to the last decade's returns is unlikely to be repeated.
Conclusion
We claimed in earlier sections that investor emotion or sentiment affects stock prices but that companies' operating results ultimately matter much more. This section puts that claim to the test and it comes through with flying colors.
Over tenyear periods the lion’s share of stock returns stem from companies’ underlying operations and their growth, not from investor sentiment as reflected in valuation changes. The portion of corporate profit or earnings paid as dividends, along with earnings growth that typically matches or exceeds inflation, is consistently positive and large relative to valuation changes. Only rarely do valuation changes overwhelm the companies’ operating results, and those instances are brief.
Over tenyear periods the lion’s share of stock returns stem from companies’ underlying operations and their growth, not from investor sentiment as reflected in valuation changes. The portion of corporate profit or earnings paid as dividends, along with earnings growth that typically matches or exceeds inflation, is consistently positive and large relative to valuation changes. Only rarely do valuation changes overwhelm the companies’ operating results, and those instances are brief.
1a) The inherent weakness of nearly all stock valuation denominators, particularly those that are flowbased, is that they’re supposed to “anchor” the valuation’s denominator so that the numerator’s price movements drive the vast majority of the valuation’s movements. But when the anchor itself experiences large yeartoyear swings, the valuation ratio is misleading.
Consider the example below. Company A’s earnings drop by half in the second period but fully recover in the third period. The market “shrugs off” its oneyear earnings stumble and its price remains constant. But its Price/Earnings valuation rises from 10 to 20, then declines to 10. Company B’s earnings are stable across the three years but its price doubles from $100 to $200 in the second year before falling back to $100 in the third. Its Price/Earnings valuation also rises from 10 to 20 then declines to 10.
The two companies’ Price/Earnings valuation ratios move identically over but their prices follow very different paths: Company A’s valuation changes are caused by movement in its earnings, the valuation's denominator, whereas Company B’s changes are caused by movement in its price, the valuation's numerator.
Consider the example below. Company A’s earnings drop by half in the second period but fully recover in the third period. The market “shrugs off” its oneyear earnings stumble and its price remains constant. But its Price/Earnings valuation rises from 10 to 20, then declines to 10. Company B’s earnings are stable across the three years but its price doubles from $100 to $200 in the second year before falling back to $100 in the third. Its Price/Earnings valuation also rises from 10 to 20 then declines to 10.
The two companies’ Price/Earnings valuation ratios move identically over but their prices follow very different paths: Company A’s valuation changes are caused by movement in its earnings, the valuation's denominator, whereas Company B’s changes are caused by movement in its price, the valuation's numerator.
The problem is exacerbated if the denominator becomes negative because the valuation ratio no longer has any interpretation.
1b) Various different measures can be used as stock valuation denominators. The most frequently used are shown in the table below, along with a brief description of the advantages and disadvantages of each. Among the flow measures, as one moves down the company’s income statement from Sales to Net Income the measure’s accuracy improves in describing shareholders’ claim to the company’s operations. The trade off is that moving down the income statement also increases the likelihood that in a poor year, the denominator falls and artificially boosts the valuation ratio, as described above, or even becomes negative and completely invalidates the valuation’s interpretation.
Suffice it to say that no single valuation measure for stocks is faultless or universally agreed upon.
1b) Various different measures can be used as stock valuation denominators. The most frequently used are shown in the table below, along with a brief description of the advantages and disadvantages of each. Among the flow measures, as one moves down the company’s income statement from Sales to Net Income the measure’s accuracy improves in describing shareholders’ claim to the company’s operations. The trade off is that moving down the income statement also increases the likelihood that in a poor year, the denominator falls and artificially boosts the valuation ratio, as described above, or even becomes negative and completely invalidates the valuation’s interpretation.
Suffice it to say that no single valuation measure for stocks is faultless or universally agreed upon.
1c) The 10year average of inflationadjusted earnings per share used as the S&P 500’s valuation denominator in this section draws from a similar concept used by Yale professor Robert Shiller, who calculates a CAPE (Cyclically Adjusted Price Earnings). The main difference between the Shiller CAPE and ours is that the Shiller CAPE’s denominator uses the companies’ As Reported net income (earnings), a GAAP measure defined by the US Financial Accounting Standards Board. Our measure also uses the As Reported earnings for its denominator prior to 1988 but from 1988 on uses the operating earnings figure published by S&P, which is comparable to analyst earnings forecasts that exclude unusual items.
The Shiller CAPE also uses the average of each month’s daily prices in its valuation’s numerator, whereas our numerator uses only the price of the last day of the month. We thank Professor Shiller for making the S&P 500 earnings and dividends history freely available on his website.
1d) A valuation is applicable to a single company but it can also be calculated for an index that contains multiple stocks. A stock index’s valuation is calculated as its member companies’ aggregate market value or capitalization, divided by the same companies’ aggregate valuation denominator. The formula for an index Price/Earnings ratio is shown below for an index with n members or constituents. The last term shows the calculation when each member’s total market value and total earnings must first be calculated from their per share amounts.
An index valuation measure facilitates comparison of its current valuation with its own historical values and also facilitates comparison of different countries and industries at any point in time.
1e) We’d be remiss if we didn’t take this opportunity to highlight the occasional oddity served up by doubleentry bookkeeping and accounting.
Most people assume that when company reports a loss for a year, the company had higher expenses than revenues; it spent more than it brought in and thereby lost cash in the process. But this assumption is sometimes very badly wrong.
USbased technology company Uniphase acquired Canada’s JDS Fitel in early 1999 and the combined company was renamed JDS Uniphase (hereafter JDS). In its fiscal years ending June 30 of 2000 and 2001, JDS reported revenues (sales) of $1.4 billion and $3.2 billion, respectively. But in 2001 it also reported a net loss of more than $56 billion. How did a company with barely $3 billion of annual sales lose $56 billion in the same year? Did it also lose $56 billion of cash?
The answers lie in several large acquisitions made by JDS between 1999 and 2001. JDS’ share price increased by a factor of 30 between June 1996 and June 1999 and rose another 6 times by August 2000, and the company used its highlyvalued shares to make acquisitions. The company did not issue shares to the public for cash and then using the resulting cash to purchase the acquired companies’ shares. Instead, JDS issued new shares directly to the acquired companies’ existing shareholders, who accepted those shares in exchange for their own companies’ shares. Each acquisition’s value was added to JDS’ balance sheet in the dollaramount of shares it had issued to finance the acquisition, along with any cash used, liabilities assumed, etc.
When an acquisition’s value exceeds the net fair value of the acquired assets, where fair value is estimated by the company’s accountants and auditors, the difference is deemed to be Goodwill and/or Intangible Assets and appears on the acquiring company’s balance sheet as an asset. After its acquisition of JDS Fitel, between June 1999 and June 2001 JDS issued stock worth over $62 billion in connection with acquisitions, of which over $59 billion represented Goodwill and other Intangible Assets.
After technology company business conditions took a pronounced downturn in late 2000 and 2001, JDS was forced to write down its Goodwill and Intangible Assets by over $55 billion in its June 2001 financial statements. Because doubleentry bookkeeping requires the writedown of assets on the balance sheet’s lefthand side to be balanced by samesized reduction on its righthand side, the $55.4 billion writedown of Goodwill and Intangible Assets also passed through JDS’ income statement, leading to its $56 billion Net Loss.
A company’s cashflow statement measures the impact on the company’s cash balances of its ongoing operating activities, its financing activities, and its reinvestment into its operations. The operating activities portion adds back noncash charges such as depreciation and amortization, and after adding back the $55.4 billion of writedowns, JDS’s Cashflow from Operations for the 2001 was slightly positive at $53 million. Not only was its $56 billion loss meaningless as a valuation denominator because it was negative, but it was also a grossly inappropriate measure of the company’s cashflows from its operations.
1e) We’d be remiss if we didn’t take this opportunity to highlight the occasional oddity served up by doubleentry bookkeeping and accounting.
Most people assume that when company reports a loss for a year, the company had higher expenses than revenues; it spent more than it brought in and thereby lost cash in the process. But this assumption is sometimes very badly wrong.
USbased technology company Uniphase acquired Canada’s JDS Fitel in early 1999 and the combined company was renamed JDS Uniphase (hereafter JDS). In its fiscal years ending June 30 of 2000 and 2001, JDS reported revenues (sales) of $1.4 billion and $3.2 billion, respectively. But in 2001 it also reported a net loss of more than $56 billion. How did a company with barely $3 billion of annual sales lose $56 billion in the same year? Did it also lose $56 billion of cash?
The answers lie in several large acquisitions made by JDS between 1999 and 2001. JDS’ share price increased by a factor of 30 between June 1996 and June 1999 and rose another 6 times by August 2000, and the company used its highlyvalued shares to make acquisitions. The company did not issue shares to the public for cash and then using the resulting cash to purchase the acquired companies’ shares. Instead, JDS issued new shares directly to the acquired companies’ existing shareholders, who accepted those shares in exchange for their own companies’ shares. Each acquisition’s value was added to JDS’ balance sheet in the dollaramount of shares it had issued to finance the acquisition, along with any cash used, liabilities assumed, etc.
When an acquisition’s value exceeds the net fair value of the acquired assets, where fair value is estimated by the company’s accountants and auditors, the difference is deemed to be Goodwill and/or Intangible Assets and appears on the acquiring company’s balance sheet as an asset. After its acquisition of JDS Fitel, between June 1999 and June 2001 JDS issued stock worth over $62 billion in connection with acquisitions, of which over $59 billion represented Goodwill and other Intangible Assets.
After technology company business conditions took a pronounced downturn in late 2000 and 2001, JDS was forced to write down its Goodwill and Intangible Assets by over $55 billion in its June 2001 financial statements. Because doubleentry bookkeeping requires the writedown of assets on the balance sheet’s lefthand side to be balanced by samesized reduction on its righthand side, the $55.4 billion writedown of Goodwill and Intangible Assets also passed through JDS’ income statement, leading to its $56 billion Net Loss.
A company’s cashflow statement measures the impact on the company’s cash balances of its ongoing operating activities, its financing activities, and its reinvestment into its operations. The operating activities portion adds back noncash charges such as depreciation and amortization, and after adding back the $55.4 billion of writedowns, JDS’s Cashflow from Operations for the 2001 was slightly positive at $53 million. Not only was its $56 billion loss meaningless as a valuation denominator because it was negative, but it was also a grossly inappropriate measure of the company’s cashflows from its operations.
2a) Stock Return Component Derivation
This section’s analysis uses a geometricallyderived dividend (income) return instead of the arithmetically derived dividend return. The geometric dividend return, denoted with the subscript g, is based on the following equation:
This section’s analysis uses a geometricallyderived dividend (income) return instead of the arithmetically derived dividend return. The geometric dividend return, denoted with the subscript g, is based on the following equation:
Multiplying the right side’s terms and subtracting 1 from each side gives:
Because of the crossproduct term on the right, the geometric dividend return differs slightly from its arithmetic equivalent, shown below with the subscript a:
Total return and price return are the same across equations A.ii and B and so the geometric dividend return is smaller than its arithmetic equivalent by the amount of the crossproduct.
Crossproducts can matter a lot; they’re the difference between simple and compound interest! But the crossproduct in equation A.ii is very small, less than 0.01% per year over 95 years for the S&P 500, and so for the rest of this section we use the geometricallyderived dividend return but without the g subscript.
The advantage of geometricallydetermined components is that equation A.i still holds when returns are expressed over longer periods. Each component’s subperiod return is linked to its prior subperiod return geometrically as shown below, but one plus each period’s total return is also the product of one plus its geometric components, for periods of any length.
For example, over subperiods 1 to n,
Crossproducts can matter a lot; they’re the difference between simple and compound interest! But the crossproduct in equation A.ii is very small, less than 0.01% per year over 95 years for the S&P 500, and so for the rest of this section we use the geometricallyderived dividend return but without the g subscript.
The advantage of geometricallydetermined components is that equation A.i still holds when returns are expressed over longer periods. Each component’s subperiod return is linked to its prior subperiod return geometrically as shown below, but one plus each period’s total return is also the product of one plus its geometric components, for periods of any length.
For example, over subperiods 1 to n,
or, more formally,
Each of the three terms in equation C.i is very closely related to its compound average. For example, the compound average total return, denoted by a double bar above it, is calculated as:
which is just the nth root of equation C.i’s first term, less 1.
Taking the nth root of each term in equation C.i converts its components into their respective compound averages as below, where the double bar above represents a compound average:
Taking the nth root of each term in equation C.i converts its components into their respective compound averages as below, where the double bar above represents a compound average:
Or we can leave it as in equation C.i, which shows the total growth instead of a compound average, by taking each term in C.ii to its nth power as below:
The compounded price return in both equations C.i and C.iii is the ratio of the nth subperiod’s ending price to the first period’s starting price, shown below and denoted by subscripts n and 0 respectively.
Next, we expand the economic content of the price ratio’s numerator and denominator by dividing each by its corresponding earnings per share, denoted En and E0 for periods n and 0 respectively, thereby turning each into a valuation. Then we multiply each valuation by the same earnings per share:
The price change is now viewed as the joint effect of the change in the company’s valuation (its P/E ratios) and the change in its earnings.
To remove the distorting effect of inflation from valuations or from earnings growth, equation E uses real prices Pn,r and P0,r and real earnings En,r and E0,r. However, this slightly alters its output to Pn,r/P0,r , the ratio of real stock prices as shown below:
Equation G below shows how this ratio differs from the ratio of nominal stock prices, where CPI refers to the Consumer Price Index:
Equation G also shows how to convert equation F back to the nominal price ratio Pn/P0 : just multiply it by CPIn/CPI0 , the inverse of the rightmost part of equation G:
We can now express the compounded totals in equation C.iii as the product of four different components:
which consists of the compounded dividend return, the change in valuations, the growth in real earnings, and the change in the consumer price index.
We then annualize each term by taking its nth root:
We then annualize each term by taking its nth root:
If we multiply out the terms on the equation’s right and ignore the crossproducts, this simplifies to:
The compound annual total return is roughly the sum of the compound annual dividend return, annualized valuation change, annual real earnings growth, and annual inflation.
For example, the S&P’s annualized 10year total return to the end of 2020 is 13.89% and its annualized dividend return is 2.09%. Its real price (in 2020 dollars) is $3756.1 at the end of 2020 and $1495.1 at the end of 2010. Its 10year average real operating EPS is $126.07 at the end of 2020 and $83.97 at the end of 2010. The US Consumer Price Index, CPIU, is 260.47 at the end of 2020 and 219.18 at the end of 2010. Entering these values into equation H gives:
For example, the S&P’s annualized 10year total return to the end of 2020 is 13.89% and its annualized dividend return is 2.09%. Its real price (in 2020 dollars) is $3756.1 at the end of 2020 and $1495.1 at the end of 2010. Its 10year average real operating EPS is $126.07 at the end of 2020 and $83.97 at the end of 2010. The US Consumer Price Index, CPIU, is 260.47 at the end of 2020 and 219.18 at the end of 2010. Entering these values into equation H gives:
Annualizing all the terms by taking each to the power of 1/10 (i.e. taking their 10th root) gives:
The 13.89% compound annual total return over this tenyear period consists of about 2.09% annual dividend return, 5.28% annual increase in valuation, 4.15% annual real earningspershare growth and 1.74% annual inflation.
The components’ sum is only 13.26%, 0.63% less than the 13.89% total return, because it excludes their crossproduct terms. Over the sample’s 85 years’ worth of tenyear returns the components’ sum is an average of 0.31% lower than their corresponding total return, which doesn’t affect our conclusions.
The components’ sum is only 13.26%, 0.63% less than the 13.89% total return, because it excludes their crossproduct terms. Over the sample’s 85 years’ worth of tenyear returns the components’ sum is an average of 0.31% lower than their corresponding total return, which doesn’t affect our conclusions.
2b) The extremely mathinclined reader will recognize that taking natural logs of both sides of equation J turns it into an exact additive equation, because the natural log of a product equals the sum of its natural logs, as below:
Given the crossproducts' modest size and their irrelevance to this section’s results, the logarithmic form’s benefit is outweighed by its additional complexity. Compound annual returns are far more intuitive and understandable than the natural log of one plus the compound annual return.