Expected Return and Standard Deviation

Why this matters

Part 5 built portfolios with judgment: spread your money, match the mix to your time horizon, don't let one holding dominate. This track covers the quantitative tools professionals use for the same job, and it's equally interested in where they fail. Everything in it starts from two numbers. Harry Markowitz, who shared the 1990 Nobel Memorial Prize for the idea, showed that under certain conditions the choice of a portfolio can be reduced to balancing two things: its expected return and its variance, which is the square of its standard deviation.¹

Expected return

An investment's expected return is the average of its possible returns, each weighted by its probability. It's the expected value from Probability and Expected Value, applied to returns.

Scenario	Probability	Return	Probability × return
Strong economy	30%	+25%	+7.5%
Ordinary year	50%	+8%	+4.0%
Recession	20%	−20%	−4.0%
Expected return	100%		+7.5%

The expected return is 7.5%, and it isn't one of the three outcomes. Nobody will earn 7.5% next year. It's the centre of the possibilities, weighted by how likely each one is.

Standard deviation

Standard deviation measures how far the outcomes sit from the expected return. With scenarios, take each outcome's distance from the expected return, square it, weight it by its probability, add them up, and take the square root.

Scenario	Distance from 7.5%	Squared	× probability
Strong economy	+17.5	306.25	91.88
Ordinary year	+0.5	0.25	0.13
Recession	−27.5	756.25	151.25
Variance			243.25
Standard deviation			15.6%

The square root of 243.25 is 15.6. Most of the variance comes from the recession: it's the least likely scenario and the furthest from the centre, and squaring makes distance count heavily.

The sum before the square root is the variance. Standard deviation is used for reporting because it's in the same units as the return. Variance is used in the formulas because variances can be added and standard deviations can't, as the next lesson shows.

Estimating both from history

In practice nobody knows the scenarios or their probabilities. The usual substitute is the past: take the average of historical returns as the estimate of expected return, and their standard deviation as the estimate of risk.

These are arithmetic averages, the right kind for a single year's expected return, as Arithmetic and Geometric Averages explained. Fees, taxes, and inflation are left out.²

How good is the estimate?

An average of noisy numbers is itself uncertain. Its uncertainty, called the standard error, is the standard deviation divided by the square root of the number of observations. For shares, that's 19.4% ÷ √98 = 2.0%. A common rule is that the true figure is probably within two standard errors of the estimate.

Years of data	Standard error	Likely range for the true expected return
98	2.0%	8.0% to 15.7%
50	2.7%	6.5% to 17.2%
20	4.3%	3.4% to 20.4%
10	6.1%	−0.2% to 23.9%

Each row centres on the 11.9% average and uses the 19.4% standard deviation. Almost a century of data leaves a range nearly eight points wide. Ten years of data can't tell a market that should return nothing from one that should return 20%. That's assuming the future resembles the past at all, which is a second and larger uncertainty.

This is a practical fact that's easy to lose sight of once the formulas start: the inputs are estimates with wide error bars. A fund's five-year record says almost nothing about its expected return.

From one investment to a portfolio

A portfolio's expected return is simple. It's the weighted average of the expected returns of what's in it.

Portfolio expected return = Sum of (each holding's weight × its expected return)

A portfolio of 60% shares and 40% bonds, using the historical averages: 0.60 × 11.9% + 0.40 × 4.8% = 9.0%.

Risk doesn't work that way. The weighted average of the two standard deviations is 0.60 × 19.4% + 0.40 × 7.9% = 14.8%. The standard deviation of the 60/40 mix over the same 98 years, rebalanced each year, was 12.1%.² The portfolio was less risky than the average of its parts, because shares and bonds didn't have their bad years together. That gap is diversification, measured. Covariance, Correlation, and Beta shows where it comes from.

What longer holding periods change

A single year of shares has ranged from −43.8% to +52.6%. Hold for longer and the yearly average narrows.

Calculated for this lesson from the same data, before fees, taxes, and inflation.² The periods overlap, so there are far fewer independent 30-year stretches than the count suggests, and one country's past century isn't a guarantee.

It's tempting to read the table as saying that risk disappears with time. Look at what the 30-year row means in money. At 8.0% a year, $1 becomes about $10. At 13.6%, it becomes about $46. The yearly averages are close together, and the ending wealth differs by more than four times. Time narrows the range of average returns and widens the range of outcomes in dollars. Which of those matters depends on what the money is for.

Using the two numbers

• Treat expected returns as rough. Use a range, and prefer a cautious figure for planning.
• Trust differences in risk more than differences in return. That shares swing more than bonds is plain in the table above. That one fund's expected return is a point higher than another's can't be read from a few years of results.
• Remember what standard deviation leaves out. It treats gains and losses alike and understates extremes, as Distributions, Standard Deviation, and Fat Tails showed.
• Don't expect the expected return. It's a centre of gravity for many possible futures, and you'll live through one of them.

Key takeaways

• Expected return is the probability-weighted average of possible returns. Standard deviation is the spread around it.
• A portfolio's expected return is the weighted average of its holdings' expected returns.
• A portfolio's standard deviation is lower than the weighted average of its holdings' whenever they don't move in step.
• Historical estimates of expected return are uncertain by several percentage points, even with a century of data.
• Longer holding periods narrow the range of yearly averages and widen the range of ending wealth.

Check your understanding

A portfolio holds 70% in a fund with an expected return of 10% and 30% in a fund with an expected return of 4%. What's the portfolio's expected return?

1. 7.0%, the simple average of the two funds' expected returns.
2. 8.2%, the average of the two weighted by the share in each.
3. 10%, since the larger holding sets the return of the whole.

A fund has returned an average of 14% a year for ten years, with a standard deviation of 20%. How much does that record tell you about its expected return?

1. A great deal, since ten years is long enough for luck to even out.
2. Everything, since the average return is the expected return.
3. Little. The likely range runs from about 1% to about 27% a year.

What's next

The 60/40 portfolio was less risky than the average of its parts. Covariance, Correlation, and Beta explains why, with the formula that the rest of portfolio theory is built on.