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Set out how the main asset classes have performed historically in terms of risk and return

CFA® Level I · Portfolio Construction · Portfolio Risk and Return: Part I · PM.8.c

The idea

Long runs of historical data show a consistent ordering. Small-company stocks have earned the highest average returns with the highest volatility, followed by large-company stocks, then long-term government bonds, with short-term Treasury bills lowest on both counts; bills have barely kept pace with inflation, so their real return has been close to zero. The trade-off holds on average over long horizons, not in every period, and a decade in which bonds beat equities does not overturn it. Several cautions apply when reading these figures. Nominal returns should be converted to real by (1 + nominal) / (1 + inflation) - 1. The geometric mean, which is what an investor compounds, falls below the arithmetic mean by more the more volatile the asset. Equity returns show fat tails and negative skewness rather than a normal distribution. And long records drawn from one successful market carry survivorship bias.

Formulas

Real return from a nominal return

real return = (1 + nominal return) ÷ (1 + inflation) − 1

nominal return
the return before allowing for inflation
inflation
the rise in prices over the same period
real return
the gain in purchasing power

The exact form divides rather than subtracts. Nominal minus inflation is only a close approximation, and it gets worse when either number is large.

Geometric mean against arithmetic mean

geometric mean ≈ arithmetic mean − σ² ÷ 2

geometric mean
the compound growth rate per period
arithmetic mean
the simple average return per period
σ²
the variance of the periodic returns

Volatility drags down the compounded return. The two are equal only when every period earns the same, and the gap widens as variance rises.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which ranking of average returns from highest to lowest is consistent with long-run historical data?

  1. Small stocks, large stocks, bonds, bills - correct
  2. Large stocks, small stocks, bills, bondsWhy not: Reverses both pairs: small stocks have out-earned large, and bonds have out-earned bills.
  3. Bonds, large stocks, small stocks, billsWhy not: Puts long-term government bonds above equities, the reverse of the long-run record.

Higher average return has historically come with higher volatility: small-cap equity at the top, then large-cap equity, long-term government bonds and Treasury bills.

A harder one

Over one ten-year period, long-term government bonds outperformed equities. What is the best inference?

  1. Equities are no longer compensated for their higher riskWhy not: One decade of data does not erase a premium that has been observed over long periods.
  2. Bonds have become riskier than equitiesWhy not: A single decade of returns does not change the underlying risk of an asset class.
  3. Short periods can deviate from the long-run trade-off - correct

The historical risk-return relationship is a long-run average. Because equities are risky, there will be periods, even long ones, in which they underperform safer assets.

The hardest tier

An equity index returned +40% in one year and -20% in the next. What was its geometric mean annual return?

  1. 10.00%Why not: This is the arithmetic mean, which overstates what an investor actually compounded.
  2. 5.83% - correct
  3. 20.00%Why not: Adds the two returns without averaging or compounding them.

Geometric mean = (1.40 x 0.80)^0.5 - 1 = 1.12^0.5 - 1 = 5.83%. The gap from the 10% arithmetic mean widens with volatility, which is why equities show the largest gap.

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