Chapter 2 of 22 · Unit 1 · Foundations · 11 min read
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Every investment decision trades one thing against another: what you hope to get back against what could go wrong on the way. This chapter names both sides. We start with plain definitions, sort the many kinds of return and risk into families, then measure them with the tools that the Series 65 and CFA syllabuses use. Level 4 tests those tools on two of the worst stretches US stocks have had.
For every reader: the idea in plain words, no formulas.
Return is what you get back, measured against what you put in. If you buy a share for $100, collect $2 of dividends and sell it a year later for $108, you made $10 on $100, a return of 10 %. Part of that came from the price rising and part from the income it paid. Investors who quote a return without saying which parts they counted can make a fund look better than it was.
Return also has a time dimension. Ten percent over one year differs a lot from ten percent over five years, so we usually convert returns to a yearly rate before comparing them.
Risk is what can go wrong. For most savers that has two faces. One is the swing in value from year to year, which can force a sale at a bad moment. The other is falling short of a goal, which can happen even with the safest assets if inflation eats more than they earn. A third worry, the deep and long loss from which you cannot recover in time, sits between the two.
Markets pay people to bear risk. A Treasury bill pays little because you face almost no chance of losing money over its three months. A stock pays more on average because its owner takes the first hit when profits fall. No investment offers high expected return with no risk for long, because buyers would bid its price up until the extra return disappeared. When someone shows you one, ask which risk you have not been told about.
For readers who want the mechanism, with the diagrams.
Beginner path stops here
Price return counts only the change in price. Income return counts the dividends or interest. Together they make the total return, which is the figure to compare across investments, since a stock fund paying 2 % in dividends and a growth fund paying none would otherwise look alike.
Nominal and real. A nominal return is in dollars as printed. A real return takes out inflation and measures what the money buys. In a year when a bond pays 5 % and prices rise 6 %, the holder is poorer in what matters, even though the statement shows a gain.
Arithmetic and geometric. An arithmetic average adds the yearly returns and divides by the number of years. The geometric average, also called the compound annual growth rate, asks what single steady rate would have turned the starting sum into the ending sum. The two can differ by a lot. Take a fund that gains 50 % one year and loses a third the next. It has an arithmetic average of +8.3 % and a geometric average of 0.0 %: $100 became $150 and then $100 again. Your wealth follows the geometric figure.
Annualised. We turn a return over several years, or over a few months, into a yearly rate so it can sit beside others. Over two years, 21 % annualises to 10.0 %, because 1.10 times 1.10 is 1.21.
We group the common risks into six families. Most investments carry several at once.
Market risk, also called systematic risk, is the part of a loss that hits almost everything at once: a recession, a financial panic, a pandemic. Owning more stocks does not remove it. Concentration risk is the opposite: the loss from owning too much of one company, industry or country. Diversification reduces concentration risk sharply and market risk hardly at all.
Interest-rate risk is the fall in a bond's price when market rates rise, because new bonds pay more and the old one must get cheaper to compete. Reinvestment risk is its mirror: when rates fall, the coupons and maturing principal earn less when you put them back to work. Inflation risk is the loss of purchasing power, the reason cash and long fixed-rate bonds can fail a long-term saver.
Credit risk is the chance a borrower pays late, pays less or never pays. It shows up first as a wider gap between the borrower's yield and a Treasury yield. Liquidity risk is the cost of selling in a hurry: a wide gap between bid and ask, or no buyer at a fair price for days.
Currency risk affects anyone holding assets priced in another money. A European stock that rises 8 % in euros while the euro falls 8 % against the dollar gives a US holder roughly nothing. Political or country risk covers capital controls, seizures, sanctions and sudden tax changes.
Model risk is trusting a formula that fits the past and breaks in a new regime. Operational risk is the loss from errors, fraud or failed systems at the firms that hold and trade your assets. Both are easy to forget because they rarely show up in a return series until the day they do.
Sequence-of-returns risk matters once money flows in or out. Two retirees can earn the same returns in a different order and end with very different sums; Level 3 works one case. Shortfall risk is the chance of ending below a target, such as the sum needed to retire at 65. For a long-horizon saver it often matters more than any yearly swing.
Across long periods, the classes line up roughly as theory says: more swing has come with more return. Treasury bills sit at the low end on both counts, long government bonds in the middle, large company stocks above them and small company stocks at the top. The SBBI yearbook puts large US stocks at about 10 % a year compounded from 1926 through 2023 with a standard deviation near 20 %.
| Asset class | Average yearly return (%) | SD of yearly returns (%) | Years |
|---|---|---|---|
| US stocks (S&P 500) | 11.85 | 19.4 | 98 |
| Baa corporate bonds | 6.9 | 7.65 | 98 |
| 10-year Treasury | 4.82 | 7.9 | 98 |
| 3-month T-bills | 3.42 | 3.04 | 98 |
| US homes | 4.38 | 6.18 | 98 |
| Gold | 7.36 | 21.51 | 98 |
Source: Aswath Damodaran, NYU Stern, historical returns on stocks, bonds and bills (annual total returns in percent: S&P 500 with dividends, 3-month T-bill, 10-year T-bond, Baa corporate, real estate, gold), as of 2025-12-31.
Arithmetic means of calendar-year total returns. The plan names the SBBI yearbook; its series are licensed, so the chart uses Damodaran's free annual series instead.
A drawdown measures the fall from a previous peak to a later low. It answers the question a saver asks in a bad year: how much of what I had is gone? A standard deviation averages good and bad years together; a drawdown looks only at the worst stretch and how long it took to climb back.
| Month | Value ($) | Below the peak (%) |
|---|---|---|
| 0 | 100 | 0 |
| 6 | 112 | 0 |
| 12 | 126 | 0 |
| 18 | 140 | 0 |
| 24 | 118 | -15.7 |
| 30 | 96 | -31.4 |
| 36 | 84 | -40 |
| 42 | 92 | -34.3 |
| 48 | 104 | -25.7 |
| 54 | 116 | -17.1 |
| 60 | 128 | -8.6 |
| 66 | 140 | 0 |
| 72 | 150 | 0 |
Source: computed: an illustrative path; drawdown = value / running peak - 1; recovery = peak / trough - 1, as of 2026-10-28.
Each risk measure answers a different question, and none answers all of them. We use this table when choosing what to quote.
| Question a saver asks | Measure | Where it misleads |
|---|---|---|
| How bumpy is the ride? | standard deviation | treats a big gain as risk; assumes a bell curve |
| How bad have bad years been? | downside deviation | needs a threshold, and the answer moves with it |
| How deep was the worst fall? | maximum drawdown | one episode, so it depends on the window chosen |
| Was the return worth the risk? | Sharpe or Sortino ratio | noisy over short windows; a few years prove little |
| How much does it move with the market? | beta | changes over time; says nothing about other risks |
| Did a manager add value? | alpha, information ratio | depends on picking the right benchmark |
Losses also compound against you. A fall of 50 % needs a gain of +100 % to get back to even, and a fall of 33 % needs +50 %. That asymmetry explains why drawdowns weigh so heavily on people close to spending their money. Someone with thirty years to go can wait for a recovery. Someone retiring next year may have to sell into the fall. For that reason we quote a drawdown beside every standard deviation in this series.
For Series 65 and CFA readers: formulas and worked examples.
Series 65 path stops here
For yearly returns we write the two averages as follows.
In words, the arithmetic mean adds the yearly returns and divides by how many there are.
In words, the geometric mean multiplies the yearly growth factors, takes the n-th root and subtracts one.
Worked example Plus 50 % then minus a third
Take and . The arithmetic mean is , about +8.3 %. The geometric mean is , which comes to 0.0 %. A $100 stake goes to $150 and back to $100. Losses weigh more than gains of the same percentage, and the geometric mean records that.
A total return earned over years becomes a yearly rate by taking the -th root of the growth factor.
In words, the annualised return is one plus the total return raised to one over the number of years, minus one.
For a 21 % gain over two years that gives 10.0 %; for a 6 % gain over six months, with , it gives 12.4 %. Annualising a short gain can make it look large, which is why fund rules restrict quoting it for periods under a year.
The standard deviation measures how far the yearly returns scatter around their mean. We use the sample version, dividing by .
In words, the standard deviation squares each year's distance from the mean, averages those squares and takes the square root.
The downside deviation counts only the years that fall short of a threshold, here the risk-free rate . Years above it count as zero.
In words, the downside deviation squares only the shortfalls below the threshold, averages them over all years and takes the square root.
The maximum drawdown tracks the value and its running peak.
In words, the maximum drawdown is the largest percentage fall from any earlier high to a later value.
Both ratios ask how much extra return a portfolio earned for each unit of risk it took. The Sharpe ratio uses the standard deviation; the Sortino ratio uses the downside deviation, so it does not punish a fund for big up years.
In words, the Sharpe ratio is the average return above the risk-free rate divided by the standard deviation.
In words, the Sortino ratio is the same excess return divided by the downside deviation.
Worked example A five-year fund
A fund returns +12 %, −8 %, +20 %, +5 %, −3 % over five years, and cash pays 2 % a year. Its arithmetic mean is 5.2 % and its geometric mean 4.7 %. The sample standard deviation is 11.3 % and the downside deviation against 2 % is 5.0 %. That gives a Sharpe ratio of 0.28 and a Sortino ratio of 0.64. The maximum drawdown is −8.0 %, the second-year loss. The Sortino figure is more than twice the Sharpe figure because two of the five years were large gains, which inflate the standard deviation without hurting anyone.
Worked example Sharpe ratio of US stocks on SBBI's round figures
With a 10 % return, a 20 % standard deviation and Treasury bills at 3.3 %, the Sharpe ratio is , about 0.34. Readers should treat this as a rough figure: SBBI's 10 % is a compound rate, and the textbook ratio uses the arithmetic mean, which runs higher.
Beta measures how much a stock tends to move with the market. With the correlation between them, beta is the correlation scaled by their relative swings.
In words, beta is the covariance of the stock with the market divided by the market's variance, which equals the correlation times the stock's standard deviation over the market's.
Alpha is the return left over after paying for that market exposure at the rate the capital asset pricing model sets.
In words, alpha is the actual return minus the risk-free rate and minus beta times the market's excess return.
Worked example A stock with beta above one
A stock has a 30 % standard deviation, the market 20 %, and their correlation is 0.8. Beta is , or 1.20. With cash at 3 % and the market returning 10 %, the model expects 11.4 %. If the stock returned 14 %, its alpha is +2.6 %.
The information ratio judges an active manager against a benchmark: the average return above the benchmark divided by the tracking error, the standard deviation of that difference. A manager who beats the index by 1.5 % a year with a 4 % tracking error scores 0.38.
Order does not matter to a buy-and-hold investor, since multiplication gives the same product in any order. Once money comes out each year, order matters a lot.
Worked example Two retirees, the same ten returns
Each retiree starts with $1,000,000 and withdraws $40,000 at the start of every year. Retiree A earns −20 %, −12 %, +5 %, +10 %, +12 %, +15 %, +8 %, +10 %, +14 %, +18 %, a compound rate of 5.3 %. Retiree B earns the same ten returns in reverse order. Without withdrawals both would end with $1,673,685. With them, A ends with $957,423 and B with $1,254,419, a gap of $296,996. A sold shares after the losses, so fewer shares were left to catch the recovery.
For readers who want past episodes run through the Level 3 math.
2008 The S&P 500 loses 37 % in a calendar year
The S&P 500 returned about −37 % in 2008, dividends included, and fell −56.8 % from its close of 1,565.15 on 9 October 2007 to 676.53 on 9 March 2009 (S&P Dow Jones Indices). We set those losses against what a normal curve with SBBI's long-run figures would have expected.
Suppose an investor in 2007 had used the long-run figures, a 10 % average and a 20 % standard deviation, and treated yearly returns as normal. A −37 % year sits −2.35 standard deviations below the mean. A normal curve gives a year that bad a 0.94 % chance, about once in 107 years. That sounds rare, and SBBI's own data already held one worse year, 1931.
The peak-to-trough fall is the more painful number, and the standard deviation says nothing about it. Treat the −56.8 % fall as if it were one year's return and it sits −3.34 standard deviations down, an event the normal curve expects about once in 2,387 years. Two things went wrong with that expectation. Losses ran across calendar years, so the yearly numbers split one fall into two smaller ones. And real returns have fatter tails than the bell curve, which Chapter 3 takes up. Our view: quote the standard deviation, and always quote the worst drawdown next to it.
1974 Stocks and inflation hit savers in the same two years
From 11 January 1973 to 3 October 1974 the S&P 500 price index fell from 120.24 to 62.28, a drop of −48.2 % (S&P Dow Jones Indices). In the same two years consumer prices rose 22.1 %.
SBBI records large-company total returns of −14.7 % for 1973 and −26.5 % for 1974. Compounding them gives a two-year nominal loss of −37.2 %. The consumer price index rose 8.7 % in 1973 and 12.3 % in 1974, December to December. Dividing one plus the nominal return by one plus inflation gives the real result.
In words, one plus the real return equals one plus the nominal return divided by one plus inflation.
The real two-year loss comes to −48.6 %. A portfolio statement in December 1974 showed a loss of about a third; the loss in buying power was close to half. Market risk and inflation risk arrived together, which is the case that the 60/40 portfolio handles worst, since bonds also lose real value when inflation jumps. We return to that pairing with 2022 in Chapter 7.
Both bear markets ran longer than a calendar year, so yearly statistics understate them. Both arrived with something else: a banking panic in 2008, an oil shock and inflation in 1973. In each, the realised loss was far larger than a normal curve with long-run inputs allowed, and the drawdown took years to recover: an investor who lost −56.8 % needed a gain of +131 % to get even. For planning, we take three lessons. Quote risk in more than one measure. Plan spending so that a forced sale in a drawdown is unlikely. And count inflation as a risk on the same footing as a price fall, since in 1974 it took about as much from savers as the market did.
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