Investments & Portfolio Theory: Building a Bundle, Not Picking a Winner

A friend says her fund gained 50% one year and lost 50% the next. "Average return zero," she says, "so I broke even." She did not. $10,000 became $15,000 and then $7,500. The average of the two numbers is zero; the money she holds is a quarter smaller. Nothing was lost to fees or bad luck in the arithmetic. It is how percentages work, and it is the first thing a graduate of this course never forgets.

Investments is the study of holding many uncertain things at once. How should you measure what they earn? Which risks are you paid to carry? How much of a record is skill? What do costs do over decades? The answers are less about finding the winner than about the shape of the bundle.

Plain statement: this piece describes how portfolio theory works. It is not investment advice, and it does not tell anyone what to hold, add or reduce. Every fund, asset and number in the examples is invented for teaching; none is a forecast.

A return is a spread: why +50% then −50% loses money

A single year's return is one draw from a spread of possible outcomes. Two numbers describe the spread well enough to start: its centre and its width. The centre is the expected (arithmetic average) return. The width is the volatility, the standard deviation of returns. What you actually live through, though, is the geometric return: the constant yearly rate that would produce the same final wealth. Because gains and losses multiply, the geometric return is always below the arithmetic one when there is any volatility.

Take the simplest spread: each year the return is the average plus or minus the volatility, with equal chance. With an average m and volatility s, one up year and one down year multiply to (1+m+s)(1+m−s) = (1+m)² − s². So the yearly growth you keep is:

With m = 8% and s = 20%, g is 6.13%, about two points lower, and $10,000 after 30 years becomes roughly $59,600 along a path with equal ups and downs, against $100,600 if you wrongly compounded the 8% average. The expected value is still $100,600, because rare very good paths pull the average up; the typical path is lower. Slide the volatility down and watch the gap close. The same arithmetic is why a loss is harder to repair than it looks: after a fall of x, you need a gain of x ÷ (1 − x) to get back, which is +100% after −50%.

Two assets: correlation bends the curve

Now hold two things. Their average return is simply the weighted average of the two. Their risk is not. It depends on how they move together, measured by the correlation ρ, from −1 (opposite) through 0 (unrelated) to +1 (in lockstep). Harry Markowitz's 1952 paper "Portfolio Selection" turned this into a method, and he shared the 1990 Nobel prize in economics for it, with Merton Miller and William Sharpe.

Our two invented funds: Alder Equities (expected 8%, volatility 18%) and Birch Bonds (3.5%, 5%). At ρ = 0.1, a 50/50 mix has an expected return of 5.75% and a volatility of 9.6%, well below the 11.5% you would get if risk simply averaged. Even more surprising, there is a mix with lower risk than Birch Bonds alone: about 5% in Alder Equities gives the minimum-variance portfolio. Add a little of the riskier asset and the total gets safer, because it moves differently. Drag the correlation to +1 and the curve straightens into a line; the benefit of mixing vanishes. Correlations are not fixed: they tend to rise when markets fall together, so diversification is weakest when most wanted.

One risk-free asset and one best mix: the Sharpe ratio

Add a third ingredient: a risk-free asset paying r_f (in practice, short government bills in a stable currency; in some countries no asset is truly risk-free). James Tobin showed in 1958 that something neat follows. Among all mixes of risky assets there is one best mix, the tangency portfolio, the one with the highest reward per unit of risk. Everyone who agrees about the numbers holds that same mix of risky things; they differ only in how much they put in it versus the risk-free asset. Choosing the mix and choosing how much risk to carry are separate decisions.

The yardstick is the Sharpe ratio, which William Sharpe introduced in 1966 as "reward to variability": excess return over the risk-free rate, divided by volatility.

With three invented funds (Alder Equities, Birch Bonds, Cedar Property Trust) and a risk-free rate of 2%, the best long-only mix on a 1% grid is 20% Alder, 72% Birch, 8% Cedar, with a Sharpe ratio of 0.43, against 0.38 for an equal split. Raise the risk-free rate to 4% and the best mix swings to mostly Alder, because Birch's 3.5% no longer clears the bar. That swing is a warning: the "optimal" weights are very sensitive to the inputs. The next section is about what that does.

Alpha, beta and factors: how long to tell skill from luck

Regress a fund's excess returns on the market's and you get a line: the slope is beta (how much of the market's movement the fund carries) and the intercept is alpha, the average return left over (Michael Jensen's 1968 study of mutual funds used exactly this). Alpha is the claim of skill. Beta is exposure you could get cheaply. CAPM (see 34/51) says only beta is paid.

The evidence since has added factors: Fama and French (1993) added size (small minus big, SMB) and value (high minus low book-to-market, HML) to the market; their 2015 model adds profitability (RMW) and investment (CMA); Mark Carhart (1997) added momentum. Much of what looked like manager alpha turned out to be exposure to these factors. Why the factors pay is argued: Fama and French read them as compensation for risk; behavioural economists read them as investor mistakes; and Harvey, Liu and Zhu (2016) counted hundreds of published factors and argued that many are statistical accidents of too many tests.

A statistician's question matters more than any model: how long would it take to see alpha if it were real? The t-statistic of an average is its size divided by its noise, and the noise shrinks only with the square root of the years:

A manager who truly adds 2% a year with a tracking error of 6% needs about 36 years of results before the record clears that bar. After 10 years, even this genuinely skilled manager has only about a one-in-six chance of showing t above 2. And with thousands of managers, as in the coin-flip tool of 34/51, some will clear it by luck. Track records are weak evidence, which is why serious investors ask for a process and a reason, not just a number.

The optimiser's curse: estimation error and 1/N

The tools above need expected returns, volatilities and correlations. The future does not hand them over; they have to be estimated from history, and expected returns are the hard part. If a fund has a 6% premium and 18% volatility, ten years of data pin the average down only to within about ±5.7 points (18% ÷ √10), a quantity as big as the premium itself. An optimiser treats those noisy estimates as truth, and then it loads up on whichever asset happened to look best. It has been called an error maximiser.

The test is out of sample. DeMiguel, Garlappi and Uppal (2009), in the Review of Financial Studies, compared many refined optimisation rules with the naive "1/N" rule, which puts the same weight on each of N assets. They found no refined rule that reliably beat 1/N outside the sample, because the estimation error ate the gain, and they estimated that a many-asset optimiser would need an impractically long history before it pays. Remedies exist: shrinking estimates toward a common value, limiting weights, and Black and Litterman's 1990 method of starting from market-implied returns. Optimisation is fragile, not useless.

Try it. The tool draws random histories from the true numbers of the three invented funds, lets an optimiser pick weights from each history, then scores those weights with the true numbers.

At ten years of history the optimiser's weights score a true Sharpe ratio of about 0.36, worse than the equal split's 0.38 and well short of the best attainable 0.43, and it beats the equal split in under a third of the histories. It takes roughly 40 years of history to pass the equal split on average. The lesson graduates keep is not "never optimise" but: a simple rule that ignores the data can beat a sophisticated rule that trusts it, until the data is plentiful.

Costs: the arithmetic of active management

William Sharpe's short 1991 paper "The Arithmetic of Active Management" makes a point that needs no data. The market is the sum of all investors' holdings. So before costs, the average dollar managed actively earns exactly what the average dollar held passively earns: the market return. Active managers can beat it only by taking from other active managers. After costs, which are higher for active managers (fees, trading, spreads), the average active dollar must fall behind. Some individuals still win, as the coin-flip logic says some will, but the group cannot. S&P Dow Jones Indices' SPIVA scorecards, which compare active funds with their benchmarks in many markets, have repeatedly found most active funds trailing over long periods, with the share varying by year and market.

This is the intellectual root of index funds. Early index funds for institutions started in 1973; John Bogle started the First Index Investment Trust for individuals on 31 December 1975, which became the Vanguard 500 Index Fund. Fees compound against you, since a cost is paid every year on a base that would have grown:

At a 7% gross return and $10,000, thirty years at a 0.1% cost leaves about $74,000. A 1.1% cost leaves about $55,800. The 1-point difference in fee takes about 28% of everything the money earned over the thirty years. Slide it yourself.

The counterpoint: nothing here says no manager adds value, and critics of heavy indexing argue it could weaken price discovery, a debate that is open. The arithmetic says the burden of proof sits with the fee.

Tails and drawdowns: the bell curve flatters

Most of this theory describes risk as one number, volatility, and implicitly pictures a bell curve. Real returns have fat tails: extreme days happen much more often than a normal curve says. Benoit Mandelbrot argued this for cotton prices in 1963, and Eugene Fama found it in stocks in 1965. A common stand-in is the Student-t distribution, whose single parameter, the degrees of freedom, sets how fat the tails are; daily returns on many assets are often fitted with values around 3 to 5.

Take a fall of at least three standard deviations in one day. On a normal curve that is about 1 day in 741, or once every three years of trading. With 4 degrees of freedom it is about 1 day in 151, roughly once every seven months. Five standard deviations: once every 14,000 years on the bell curve, against roughly once every 4 years at 4 degrees of freedom. The numbers are model outputs, and a model fitted to one market and decade will not transfer exactly, but the direction is robust: volatility alone understates the chance of a very bad day.

A second thing volatility hides is the drawdown: the fall from a peak. It matters because investors live along a path and because of the repair arithmetic from the start of this piece. A 20% fall needs +25%, a 50% fall +100%, a 70% fall +233%.

What a graduate just sees

• An average return is not what you keep. Compounding keeps roughly the average minus half the variance.
• Risk does not add up. Two assets that move differently make a portfolio safer than either's average, and sometimes safer than the safer one.
• Choosing the mix of risky assets and choosing how much risk to carry are separate decisions.
• Reward per unit of risk (Sharpe) is the fair comparison; a higher return alone is not.
• A track record is weak evidence. It takes decades to separate modest skill from luck.
• Optimised weights inherit the noise in their inputs; a naive split is a serious rival.
• Costs are certain and returns are not. The average active dollar cannot beat the average passive dollar after fees.
• Extreme days are more common than the bell curve says, and a deep fall is much harder to climb out of than it was to fall.

Where it shows up next

• Derivatives & Risk Management (36/50): the tools that reshape a portfolio's risk, and the same no-arbitrage logic that prices them.
• Financial Statement Analysis & Valuation (36/52): where the expected cash flows behind the numbers come from.
• Financial Markets & Institutions (36/51): pension funds, insurers and sovereign funds are the large holders of portfolios, and their rules shape markets.
• Behavioral Economics (34/46): why real investors chase last year's winners, flee in a panic and hold home-country shares far above a diversified share; French and Poterba documented this home bias in 1991.
• In the world: a pension plan's asset mix, the cost line on a fund fact sheet, a robo-adviser's questionnaire, and any headline saying a fund "beat the market".

If you want to go deeper

• 📘 Textbook. Zvi Bodie, Alex Kane and Alan Marcus, Investments (McGraw Hill, 13th edition, 2024). The standard undergraduate text; broad and clear on portfolio theory and CAPM.
• 📘 Textbook. Edwin Elton, Martin Gruber, Stephen Brown and William Goetzmann, Modern Portfolio Theory and Investment Analysis (Wiley). More mathematical, strong on optimisation.
• 📘 Gentler. John Bogle, The Little Book of Common Sense Investing (Wiley), the case for low costs, and Peter Bernstein, Capital Ideas (Free Press, 1992), a history of how the theory was invented.
• 🌐 Free course. MIT OpenCourseWare 15.401 Finance Theory I (Fall 2008, Andrew Lo), which covers diversification, portfolio selection, equilibrium pricing and efficient markets; and 15.433 Investments (Spring 2003) with lecture notes.