Chapter 18 of 22 · Unit 7 · Portfolios and behaviour · 17 min read
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Earlier chapters looked at one asset class at a time. Here we put them together. A portfolio is more than a list of good investments, because the way its parts move against each other decides how rough the ride feels. Harry Markowitz turned that idea into mathematics in 1952, William Sharpe built a pricing model on top of it in 1964, and Eugene Fama and Kenneth French added factors in 1992 and 1993. We follow the same path, then spend the second half of the chapter on the plain work of keeping a mix on target, which matters more to most investors than any optimiser.
We use one set of round inputs throughout so the numbers can be checked by hand: stocks return 10 % a year with a standard deviation of 20 % (SBBI's large-company figures from the overview article), bonds return 5 % (SBBI's long-term government bonds), Treasury bills pay 3.3 %, and we assume a bond standard deviation of 8 % and a stock-bond correlation of 0.1. The last two are our assumptions, chosen to match Chapter 3; the Gaps section says so.
For every reader: the idea in plain words, no formulas.
A portfolio is the set of investments that together pay for a goal. Retirement in thirty years, a house in five, a cushion for next year: each goal sets how much can be lost along the way and how much growth it needs. Choosing a portfolio means choosing how much of each asset class to hold, a decision called asset allocation. Study after study of pension funds finds that this choice, more than which single stocks or funds a person picks, explains most of how much the value moves from month to month.
Suppose one investment zigs when another zags, even a little. On a day the first falls, the second often holds up, and the loss on the whole pot is smaller than the loss on the worst part. Over many days this cancelling makes the pot steadier than any single holding. We call it diversification, and its strength depends on correlation, a number from minus one to one that says how closely two things move together. At one they move in lockstep and mixing them gains nothing. Below one, the mix wobbles less than the average of its parts.
With our inputs, a 60/40 mix of stocks and bonds has a standard deviation of 12.7 %. Average the two parts' swings by their weights and you would expect 15.2 %. The difference is what diversification gives you without giving up any expected return.
Some risk disappears when you hold many investments, and some does not. A drug trial that fails, a factory fire or a chief executive caught in a scandal can halve one company's shares, but in a fund of five hundred companies each of those is a small dent. A recession, a jump in interest rates or a war hits nearly every company at once, and no amount of spreading removes it. Finance calls the first kind specific risk and the second market risk. Markets pay investors, on average, for bearing market risk, since somebody has to hold it. They pay little for specific risk, because anyone can get rid of it by owning a broad fund. Holding a few stocks therefore means taking extra swings with no extra expected return to show for them.
Markets move the mix. A strong year for stocks leaves you holding more stocks than you planned, and more risk than you signed up for. Rebalancing means selling some of what has grown and buying what has lagged, to return to the target. People do it on a calendar (once a year, say) or when a weight strays outside a band around its target. It feels wrong in the moment, since it sells the winner, and that discomfort is one reason a written rule helps.
| Point | Stocks ($) | Bonds ($) | Stock share |
|---|---|---|---|
| Start | 600,000 | 400,000 | 60.0 % |
| After year 1 | 750,000 | 420,000 | 64.1 % |
| After year 2 | 937,500 | 441,000 | 68.0 % |
| After rebalancing | 827,100 | 551,400 | 60.0 % |
Source: computed: stocks +25 % and bonds +5 % in each of two years, from $600,000 / $400,000, as of 2026-10-28.
For readers who want the mechanism, with the diagrams.
Beginner path stops here
Take two assets and try every mix from all bonds to all stocks. Each mix has an expected return and a standard deviation, and plotting them traces a curve that bulges to the left: as you move from all bonds toward stocks, risk first falls, because a little stock diversifies a bond portfolio, then rises. The point furthest left is the minimum-variance portfolio. With our inputs it holds 11.1 % in stocks, has a standard deviation of 7.7 %, below the bonds' own 8 %, and earns 5.6 %.
Only the upper half of the curve interests a sensible investor. A mix below the minimum-variance point has a twin above it with the same risk and a higher return. That upper half is the efficient frontier: for each level of risk, the most return the assets allow. With many assets the same idea holds, though the frontier now comes from an optimiser rather than a sketch.
| Share in stocks | SD at ρ 1 (%) | SD at ρ 0.5 (%) | SD at ρ 0 (%) | SD at ρ -0.5 (%) | Return (%) |
|---|---|---|---|---|---|
| 0 % | 8.00 | 8.00 | 8.00 | 8.00 | 5.00 |
| 10 % | 9.20 | 8.38 | 7.47 | 6.44 | 5.50 |
| 20 % | 10.40 | 9.09 | 7.55 | 5.60 | 6.00 |
| 30 % | 11.60 | 10.05 | 8.21 | 5.81 | 6.50 |
| 40 % | 12.80 | 11.20 | 9.33 | 6.97 | 7.00 |
| 50 % | 14.00 | 12.49 | 10.77 | 8.72 | 7.50 |
| 60 % | 15.20 | 13.88 | 12.42 | 10.76 | 8.00 |
| 70 % | 16.40 | 15.34 | 14.20 | 12.97 | 8.50 |
| 80 % | 17.60 | 16.86 | 16.08 | 15.26 | 9.00 |
| 90 % | 18.80 | 18.41 | 18.02 | 17.61 | 9.50 |
| 100 % | 20.00 | 20.00 | 20.00 | 20.00 | 10.00 |
Source: computed: stocks 10 % / 20 %, bonds 5 % / 8 %; σp from the two-asset formula, as of 2026-10-28.
The bend depends on correlation. Holding the 60/40 weights fixed, the mix's standard deviation is 15.2 % at a correlation of one, 13.9 % at 0.5, 12.7 % at our 0.1 and 10.8 % at minus 0.5. Expected return stays at 8.0 % in every case.
Now add Treasury bills, which we treat as riskless over a year. Any mix of bills and a risky portfolio lies on a straight line between them. Of all those lines, one is steepest: it touches the frontier at a single point, the tangency portfolio, and we call it the capital market line. Every investor, whatever their nerve, does best on it. A cautious investor holds the tangency portfolio plus bills; a bold one holds it with borrowed money. Picking the risky mix and picking how much risk to take become two separate decisions, an idea James Tobin set out in 1958.
With our inputs the tangency portfolio holds 41.2 % stocks and 58.8 % bonds. Its Sharpe ratio is 0.38, above stocks alone (0.34) and bonds alone (0.21).
| Item | Value |
|---|---|
| Tangency weight in stocks | 41.2 % |
| Expected return | 7.06 % |
| Standard deviation | 9.89 % |
| Sharpe ratio | 0.380 |
| Bill rate | 3.3 % |
Source: computed: max-Sharpe weights for stocks 10 % / 20 %, bonds 5 % / 8 %, ρ 0.1, bills 3.3 %, as of 2026-10-28.
Sharpe asked what happens if every investor thinks this way. They all hold the same tangency portfolio, so it must be the whole market. An asset's risk then matters only through what it adds to the market portfolio, measured by beta: how much the asset tends to move when the market moves one percent. The capital asset pricing model (CAPM) says expected return rises in a straight line with beta, starting at the bill rate. That line is the security market line. Risk that diversification removes, such as a single firm losing a lawsuit, earns no reward, because nobody who holds the market has to bear it.
CAPM is a model with heroic assumptions (everyone agrees on the inputs, borrows at the bill rate and pays no tax), and tests since the 1970s find that low-beta stocks have earned more than it predicts and high-beta stocks less. It survives in practice as the default way to set a cost of equity, which is how Chapter 10 uses it, and as the yardstick behind most performance measures.
Fama and French (1992, 1993) found that two more traits explained returns that beta missed: small companies beat large ones, and cheap companies (high book value per dollar of price) beat expensive ones. Their three-factor model adds a size factor and a value factor to the market. Mark Carhart (1997) added momentum, the tendency of last year's winners to keep winning for a while. Later models added profitability and investment. A factor is a long-short portfolio, and a fund's exposure to it is a beta like any other. Whether factor premia are pay for risk or the residue of investor mistakes is still argued, and Chapter 11 shows how long the value premium can vanish.
A fund that earned 12 % beat one that earned 9 % only if it did not take far more risk to get there. Four measures adjust for risk, each in a different way.
| Measure | Formula in words | Risk it charges for | Use it when |
|---|---|---|---|
| Sharpe ratio | excess return over standard deviation | total risk | the fund is most of your money |
| Treynor ratio | excess return over beta | market risk only | the fund is one slice of a diversified whole |
| Jensen's alpha | return above the CAPM line | market risk only | you want the result in percent a year |
| M-squared | Sharpe ratio rescaled to the market's volatility | total risk | you want Sharpe's ranking in percent |
| Information ratio | active return over tracking error | risk relative to a benchmark | you judge an active manager against an index |
Level 3 runs two funds through all of them. Chapter 2 covers the Sortino ratio, which charges only for downside swings.
Markowitz's optimiser is one way to choose weights. Several simpler rules compete with it. Market-cap weights, the index fund's rule, hold each asset in proportion to its market value and need no forecast at all. Equal weights give each holding the same share. A minimum-variance portfolio ignores expected returns and minimises risk alone, which makes it less sensitive to the input most prone to error. Risk parity sets weights so that each asset adds the same amount to the portfolio's risk; because bonds swing less than stocks, it holds far more bonds, and funds that follow it often borrow to bring the expected return back up. Each rule answers the same worry: optimised weights depend on forecasts that nobody makes well.
Seen through risk rather than dollars, a 60/40 portfolio is mostly a stock portfolio. With our inputs, stocks are 60 % of the money and 91 % of the variance. Level 3 shows the split.
A calendar rule rebalances on a fixed date. A band rule waits until a weight leaves a range, such as five points either side of target, and acts then. Bands trade less in quiet years and act faster in violent ones. Both rules sell what has risen, so in a long trend they give up some return, and in a market that swings back and forth they gain. Costs and taxes argue for wide bands in a taxable account, and for using new savings, dividends and withdrawals to steer the mix before selling anything.
A glide path shifts a portfolio from stocks toward bonds as a goal approaches, the design behind target-date funds. The reason is the shrinking time to recover from a loss, and for a worker, the fall in future earnings that act like a bond holding. Where each asset sits also matters: a taxable bond's interest costs more tax in a taxable account than a stock's long-term gains do, so many investors keep bonds in tax-deferred accounts. Chapter 22 puts a number on that.
| Years before the goal | Stocks (%) |
|---|---|
| 40 | 90 |
| 30 | 90 |
| 25 | 90 |
| 20 | 82 |
| 10 | 66 |
| 5 | 58 |
| 0 | 50 |
| -5 | 40 |
| -10 | 30 |
Source: computed: an illustrative path: 90 % until 25 years out, 50 % at the goal, 30 % ten years after, as of 2026-10-28.
For Series 65 and CFA readers: formulas and worked examples.
Series 65 path stops here
A mix's expected return is the weighted average of its parts. Its variance is not, because the cross term carries the correlation.
In words, the portfolio's expected return is each asset's expected return times its weight, added up.
In words, the portfolio's variance is each asset's variance times its weight squared, plus twice the product of the weights, the correlation and the two standard deviations.
Worked example A 60/40 mix with our inputs
We set , , , and . The three terms are , and , which add to . Its square root gives a standard deviation of 12.7 %, and the expected return is 8.0 %. At the cross term grows to and the standard deviation becomes 15.2 %, the weighted average; any correlation below one pulls it under that.
Setting the derivative of the variance with respect to to zero gives the weight with the least risk.
In words, the least-risk stock weight is the bond variance less the covariance, over the sum of both variances less twice the covariance.
Worked example The least-risk mix
The numerator is and the denominator , so we hold 11.1 % in stocks. That mix has a standard deviation of 7.7 % and an expected return of 5.6 %: less risk and more return than all bonds.
With a riskless rate , the best risky mix maximises the Sharpe ratio. For two assets it has a closed form.
In words, the tangency stock weight trades each asset's excess return against its variance and the covariance between the two.
Worked example The tangency mix with bills at 3.3 percent
Excess returns are 6.7 % for stocks and 1.7 % for bonds. The formula gives 41.2 % in stocks, with an expected return of 7.1 %, a standard deviation of 9.9 % and a Sharpe ratio of 0.38.
In words, on the capital market line each unit of standard deviation earns the tangency portfolio's Sharpe ratio on top of the bill rate.
Worked example A cautious investor on the line
An investor who wants a standard deviation of 6 % holds 60.7 % in the tangency portfolio and 39.3 % in bills, which works out to 25.0 % in stocks overall. Expected return is 5.6 %. A bolder investor who wants the 80/20 mix's risk would earn +0.47 pp a year more on the line, but only by borrowing at the bill rate to hold more than all of the tangency portfolio, which few individuals can do.
Inputs are estimates, and the optimiser treats them as facts. Raise the stock return assumption from 10 % to 11 % and the tangency stock weight moves to 45.5 %; lower it to 9 % and the weight falls to 36.4 %. One point of error in one input, well inside the uncertainty of any forecast, moves the answer by four to five points of weight here and by far more with many assets. This is why practitioners shrink their estimates toward a neutral value, cap the weights, or start from market weights as Fischer Black and Robert Litterman proposed in 1992. DeMiguel, Garlappi and Uppal (2009) found that an equal-weighted portfolio beat most optimised ones out of sample, for the same reason.
The capital market line shows the choices; risk aversion picks one. A mean-variance investor scores a portfolio by its expected return less a penalty for variance, and the best share in the tangency portfolio follows from that score.
In words, the best share in the risky portfolio is its excess return divided by the investor's risk aversion times the risky portfolio's variance.
Worked example Three investors
With our tangency portfolio, an investor with a risk aversion of 8 holds 48 % of the money in it and 52 % in bills, which comes to 20 % in stocks overall. At the share rises to 96 %, about 40 % in stocks. At the formula asks for 192 %, borrowing to hold nearly twice the tangency portfolio. Nobody measures directly; questionnaires and the investor's capacity to absorb loss stand in for it, which is the subject of Chapter 22.
For assets held in equal weights, each with standard deviation and the same pairwise correlation , the variance has a short form.
In words, the variance is one asset's variance divided by the number of assets, plus a share of the common covariance that does not shrink as assets are added.
Worked example Single stocks with a 35 percent standard deviation
Take stocks with and . One stock has a standard deviation of 35.0 %. Ten stocks bring it to 20.0 %, thirty to 18.4 % and a hundred to 17.8 %. The floor, with infinitely many, is = 17.5 %. Diversification removed 75 % of one stock's variance, the firm-specific part; the rest is market risk, the part CAPM pays for.
Each holding's contribution to variance is its weight times its covariance with the whole portfolio. The contributions add up to the total, so they split the risk the way weights split the money.
In words, the stock share of risk is the stock weight times the stock's covariance with the portfolio, divided by the portfolio's variance.
Worked example Where a 60/40 mix's risk comes from
Stocks contribute of the total, so 91 % of the risk comes from stocks and 9 % from bonds. For two assets, equal contributions require , so a risk-parity mix holds 28.6 % in stocks. It earns 6.4 % with a standard deviation of 8.5 %, a Sharpe ratio of 0.37. Levered 1.50 times to match the 60/40 mix's risk, it would earn 8.0 %, the same as the 60/40 mix, because with our inputs the two mixes have nearly the same Sharpe ratio. Risk parity gains only when bonds offer more return per unit of risk than our inputs give them, which is what Clifford Asness, Andrea Frazzini and Lasse Pedersen (2012) argued: investors who will not borrow bid up risky assets and leave safe ones cheap.
A portfolio rebalanced to fixed weights grows at about its arithmetic mean less half its variance. Its parts, each with a larger variance of its own, grow more slowly on average than the whole. David Booth and Eugene Fama called the difference the diversification return in 1992.
In words, the extra growth from rebalancing is half the gap between the weighted average of the assets' variances and the portfolio's own variance.
Worked example The 60/40 mix
The weighted variances are and the portfolio variance is . Half the gap is 0.52 pp a year of extra compound growth, before trading costs, with returns that wander around stable means. In a strong trend the effect can turn negative.
After years with stock return and bond return , the stock weight becomes the stock value over the total.
In words, the new stock weight is the grown stock value divided by the grown value of everything.
Worked example Two strong years
Stocks rise 25 % in each of two years and bonds 5 %. A 60/40 mix becomes 68/32, a stock weight of 68.0 %. On a starting $1,000,000 the portfolio is worth $1,378,500, with $937,500 in stocks and $441,000 in bonds. Back at 60 % the stock holding should be $827,100, so the trade sells $110,400 of stocks and buys the same amount of bonds.
The math When a band triggers
With bonds flat, a 60 % stock weight reaches the top of a five-point band at 65 % after stocks rise +23.8 %, and the bottom at 55 % after they fall −18.5 %. Solving the drift formula for the stock return gives the trigger.
In words, the stock move that hits a band edge is the edge weight times the bond share over the stock share times one less the edge weight, less one.
In words, an asset's required return is the bill rate plus its beta times the market's premium over bills.
In words, beta is the asset's covariance with the market over the market's variance, which equals the correlation times the ratio of the two standard deviations.
Worked example Two betas
With bills at 3.3 % and a market premium of 6.7 %, a stock with a beta of 1.3 needs 12.0 % and one with a beta of 0.6 needs 7.3 %. A stock with a standard deviation of 30 % and a market correlation of 0.6, against a market at 20 %, has a beta of 0.90: high total risk does not mean high beta.
Worked example Ranking two funds
Fund A returned 12 % with a standard deviation of 24 % and a beta of 1.2. Fund B returned 9 % with 12 % and 0.6. The market returned 10 % with 20 %, and bills 3.3 %. These are illustrative funds.
| Measure | Fund A | Fund B | Market |
|---|---|---|---|
| Sharpe ratio | 0.36 | 0.47 | 0.34 |
| Treynor ratio | 7.25 % | 9.50 % | 6.70 % |
| Jensen's alpha | +0.66 % | +1.68 % | 0.00 % |
| M-squared | 10.55 % | 12.80 % | 10.00 % |
Fund A earned more and has a positive alpha, yet Fund B ranks first on every measure. M-squared says that B, levered up to the market's volatility, would have returned +2.80 % above the market, against +0.55 % for A.
In words, M-squared is the return the fund would have earned if scaled up or down to the market's standard deviation.
An active manager who beats the index by 1.5 % a year with a tracking error of 4 % has an information ratio of 0.38. Richard Grinold and Ronald Kahn's textbook calls 0.5 good and 1.0 exceptional, and few managers sustain either.
For readers who want past episodes run through the Level 3 math.
2022 Stocks and bonds fell together
In 2022 the S&P 500 returned −18.1 % with dividends (S&P Dow Jones Indices) and the Bloomberg US Aggregate bond index −13.0 % (Bloomberg). A 60/40 mix of the two, rebalanced in January, lost −16.1 %, one of its worst years since the 1930s.
Our inputs give the mix a mean of 8.0 % and a standard deviation of 12.7 %. The 2022 loss sits −1.9 standard deviations below that mean. Under a normal distribution a year this bad or worse has a chance of 2.9 %, about 1 year in 34. That alone does not make 2022 freakish. What broke was the input we called fixed: the stock-bond correlation. Inflation drove both the discount rate on stocks and the yield on bonds up at once, so the cross term in the variance formula, which had helped for twenty years, turned against the mix. Put into the formula and the standard deviation rises to 13.9 %. In the 1970s, when inflation also ran the show, stock-bond correlations in US data were positive too. A 60/40 investor's diversification depends on whether growth or inflation is the main shock, and Chapter 8 gives the bond side of 2022 through duration.
2008 Rebalancing into the crash
The S&P 500 returned −37.0 % in 2008 and the Bloomberg US Aggregate 5.2 %; in 2009 they returned 26.5 % and 5.9 %. A $100 60/40 mix fell to $79.90 by the end of 2008, with stocks now 47.3 % of it.
Here the two halves moved apart, the case diversification is built for: the mix lost −20.1 % while stocks lost −37.0 %. Rebalancing on 31 December 2008 meant selling $10.14 of bonds per $100 and buying stocks after that fall, with the news at its worst and Lehman Brothers three months gone. By the end of 2009 the rebalanced mix was worth $94.48 and the one left to drift $92.39, a gap of $2.08 per $100. It looks small, and it came from one trade that most investors found hard to make. The lesson has a second half: had stocks kept falling in 2009, as they did after 1930, the same trade would have lost money for a while. Rebalancing is a bet that the asset mix you chose still makes sense, which a written plan settles ahead of time. Chapter 19 covers why the trade feels so hard.
2009 A lost decade for US stocks
Over the ten years 2000 to 2009 the S&P 500 returned −9.1 % in total with dividends (−0.95 % a year), so $100 became $90.88. The Bloomberg US Aggregate returned 84.8 % over the same years (6.33 % a year).
A 60/40 mix of the two, rebalanced each January, turned $100 into $129.24, 2.60 % a year. Stocks' calendar returns over those ten years had a standard deviation of 21.1 %, close to the long-run 20 % we assumed, while the Agg's was 2.9 %. Their correlation came to −0.47, well below our 0.1, so bonds rose in the years stocks fell (2000 to 2002 and 2008). Our two-asset formula with that correlation would have predicted a much calmer mix than our inputs suggest, and it was.
Could the equity risk premium have vanished? With a 20 % standard deviation, the standard error of a ten-year average return is 6.3 percentage points (Chapter 3). A decade near zero sits well within what a long-run 10 % allows, so one bad decade tells us little about the premium. We find this the most useful number in the chapter for a nervous investor: ten years is a short sample for an asset this volatile.
1952 Markowitz publishes Portfolio Selection
Harry Markowitz's paper in the Journal of Finance in March 1952 gave the mean-variance model: judge each investment by what it adds to the risk and return of the whole. He shared the 1990 Nobel prize in economics for it with Merton Miller and William Sharpe.
1964 Sharpe publishes the capital asset pricing model
William Sharpe's 1964 paper in the Journal of Finance derived the security market line; John Lintner (1965) and Jan Mossin (1966) reached the same model on their own.
Each idea arrived as a simplification that practitioners then had to soften. Markowitz needs estimates nobody has with precision, so the optimiser's output is shrunk or constrained. CAPM's single beta misses size, value and momentum, so factor models took over performance measurement in the 1990s. What has lasted best is the plainest part: diversify across assets that respond to different shocks, choose a mix that fits the goal, and keep it there with a rule.
We used round inputs in place of a fitted data set: the 8 % bond standard deviation and the 0.1 stock-bond correlation are our assumptions, chosen to match Chapter 3, and not SBBI's published figures. We typed the calendar total returns of the S&P 500 and the Bloomberg US Aggregate for 2000 to 2009 and 2022 from the publishers' year-end figures as we know them; each needs a check against S&P Dow Jones Indices and Bloomberg before the strict build, and so do the decade totals, standard deviations and correlation computed from them. The claim that 2022 was among the 60/40 mix's worst years since the 1930s, and that stock-bond correlations were positive in the 1970s, stay in words until we chart a sourced series. The statement that allocation explains most of a pension fund's month-to-month variation refers to Brinson, Hood and Beebower (1986) and later work, which we cite in words without a figure. The Grinold and Kahn benchmarks for the information ratio are quoted from memory of the textbook and need a page check.
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