Lesson 1 of 8 · All lessons in this module
  1. Asset classes: return, risk and other characteristics
  2. Risk aversion and utilityVideo lesson
  3. Indifference curves and the capital allocation lineVideo lesson
  4. Measuring asset and portfolio riskVideo lesson
  5. Correlation and the power of diversificationVideo lesson
  6. Integrated cases: a foreign bond with currency, and pairs from a scenario table
  7. Minimum-variance and efficient frontiersVideo lesson
  8. Optimal risky portfolio and the optimal investor portfolioVideo lesson

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Lesson 1 of 8 · 13 min

Asset classes: return, risk and other characteristics

Over long periods, asset classes with more risk have earned more return, but mean and variance alone miss skewness, fat tails and trading costs.

In short

  • Historical return is what was earned; expected return is what investors anticipate. Historical means are often used as estimates of expected returns, with caution.
  • Expected nominal return compounds three parts: 1+E(R)=(1+rrF)[1+E(π)][1+E(RP)]1 + E(R) = (1 + r_{rF})[1 + E(\pi)][1 + E(RP)].
  • Long-run ranking by both risk and return: small-company stocks > large-company stocks > bonds > T-bills. This positive risk–return trade-off reflects risk-averse investors.
  • Real returns remove inflation and make periods comparable; the risk premium is the return above the risk-free rate.
  • Returns are not normal: equity returns show negative skewness and fat tails (kurtosis), risks a mean–variance view misses.
  • Liquidity drives trading costs through the bid–ask spread and price impact.

Historical vs expected return

A historical return is what an investor actually earned over a past period. An expected return is what investors anticipate earning in the future. The two are easy to confuse but they are different things: for any single period, a risky asset's actual return will almost never equal its expected return.

Expected return is built from three pieces. Investors want a real risk-free rate rrFr_{rF} for postponing consumption, compensation for expected inflation E(π)E(\pi), and a risk premium E(RP)E(RP) for bearing the asset's risk. The pieces compound: 1+E(R)=(1+rrF)[1+E(π)][1+E(RP)]1 + E(R) = (1 + r_{rF})[1 + E(\pi)][1 + E(RP)]. Because each piece is normally positive, expected returns are normally positive.

Over a long enough horizon, average realised returns should approach expected returns, but nobody knows how long is long enough. In practice analysts often use the historical mean as a stand-in for expected return, accepting that it may be a poor estimate for a particular decade.

0%2.5%5%7.5%10%1.5%Real risk-free+2.5%Inflation+5%Risk premium+0.24%Compounding9.24%E(R)
The first worked example: 1.5% + 2.5% + 5.0% gets you to 9.0%, and compounding the three pieces adds the last 0.24 points, for 9.24%. The risk premium is the biggest piece.

What the long-run record shows

Long-run data for major markets point the same way. Equities have earned the highest returns and shown the highest standard deviations; bonds come next; short-term government bills earn the least and are the least volatile. Within equities, small-company stocks have been riskier and more rewarding than large-company stocks. Individual decades can break the pattern: stocks have had decades with negative returns.

Nominal returns are hard to compare across periods because inflation varies a great deal. Real returns strip out inflation, so they are the better basis for comparison. Compounded over a century, even a few percentage points of real return per year turn into enormous differences in ending wealth.

Long-run performance is usually summarised with the geometric mean, which is the right measure for compounding over many periods and is never larger than the arithmetic mean. Risk is measured by the standard deviation of returns.

Two further observations matter later. First, risk for an asset class tends to be fairly stable from one decade to the next, even when returns swing a lot, so historical risk is a reasonable proxy for future risk. Second, a global equity portfolio has historically shown lower risk than either the domestic or the foreign component on its own: diversification at work.

The long-run pattern (stylised ranking, highest first)
Asset classReturnRisk (standard deviation)
Small-company stocksHighestHighest
Large-company stocksHighHigh
Long-term bondsModerateModerate
Treasury billsLowest (rarely negative nominal)Lowest

Risk–return trade-off and risk premiums

The risk–return trade-off is the positive relation between expected risk and expected return: in efficient markets and over long periods, you cannot expect a higher return without accepting more risk. The historical data are consistent with it, and they show that market prices are set by investors who are, on balance, risk averse.

A risk premium is the extra return investors expect for taking risk, measured above the risk-free rate. The nominal risk premium is the nominal risky return minus the nominal risk-free rate (which itself contains expected inflation and the real risk-free rate). The real risk premium is the real risky return minus the real risk-free rate. Equities have historically earned a premium over bonds because they are riskier.

Beyond mean and variance

Judging investments only by mean and variance quietly assumes two things: that returns are normally distributed (so mean and variance describe them completely) and that markets are operationally efficient (trading is cheap and easy). Both assumptions fail to some degree.

Skewness is asymmetry. A negatively (left) skewed distribution has most of its mass on the right and a long tail of large losses on the left; a positively (right) skewed one has a long right tail. Equity returns tend to be negatively skewed.

Kurtosis (fat tails) means extreme returns happen more often than a normal distribution predicts. Fat tails add risk that standard deviation does not capture, so investors use tools such as value at risk (VaR) and conditional tail expectations to gauge it.

Liquidity is the main operational issue. Trading costs have three parts: brokerage commission, bid–ask spread and price impact. Liquidity affects the last two: illiquid securities have wider spreads, and large orders move their price more. Liquidity concerns are greater in emerging markets and in corporate bonds, especially low-quality issues that may not trade for days. Other market features, such as analyst coverage, information availability and firm size, also influence decisions.

The spread is a percentage cost, so the same EUR 0.05 spread is cheap on a EUR 50 share (0.1%) and expensive on a EUR 5 share (1%).

Formulas

Components of expected return
1+E(R)=(1+rrF) [1+E(π)] [1+E(RP)]1 + E(R) = (1 + r_{rF})\,[1 + E(\pi)]\,[1 + E(RP)]

rrFr_{rF} = real risk-free rate, E(π)E(\pi) = expected inflation, E(RP)E(RP) = expected risk premium. The pieces multiply; adding them is only an approximation.

Real return
1+rreal=1+rnominal1+π1 + r_{\text{real}} = \dfrac{1 + r_{\text{nominal}}}{1 + \pi}
Risk premium
Risk premium=risky return−risk-free rate\text{Risk premium} = \text{risky return} - \text{risk-free rate}

Use nominal with nominal, real with real.

Bid–ask spread as a cost
Spread cost=ask−bidprice\text{Spread cost} = \dfrac{\text{ask} - \text{bid}}{\text{price}}

Worked examples

Example 1 Building up an expected return

An analyst estimates a real risk-free rate of 1.5%, expected inflation of 2.5% and a risk premium of 5.0% for a Swedish equity fund. What expected nominal return do these imply?

Example 2 Real return and risk premiums

A bond fund returned 7.5% in nominal terms in a year when inflation was 2.5%. What was its real return? If the nominal risk-free rate was 3.0%, what was the nominal risk premium?

Example 3 Liquidity and trading cost

Share P trades at EUR 40 and Share Q at EUR 8. Both have a bid–ask spread of EUR 0.08. Which is cheaper to trade, and by how much in percentage terms?

On the exam

Mostly conceptual: the ranking of asset classes by risk and return, what the risk–return trade-off says about investors, skewness and kurtosis, and which trading-cost components liquidity affects.

  • Treating a historical mean as if it were the expected return with certainty; it is only an estimate.
  • Adding the real rate, inflation and risk premium when the question asks for the exact compounded figure.
  • Saying liquidity drives brokerage commissions; it affects the bid–ask spread and price impact.
  • Mixing up skew directions: a long left tail of losses is negative skewness.

Check yourself

  1. Q1A real risk-free rate of 2%, expected inflation of 3% and a risk premium of 4% imply an expected nominal return closest to:

  2. Q2Over long historical periods, which US asset class has shown both the highest return and the highest risk?

  3. Q3A share priced at EUR 5.00 has a bid–ask spread of EUR 0.05. The spread as a percentage of price is closest to:

  4. Q4If a return distribution has fat tails, an analysis based only on mean and variance most likely:

  5. Q5A risky asset returned 8.5% (nominal), the nominal risk-free rate was 3.0% and inflation was 2.0%. The nominal risk premium is closest to:

0/5Check yourself

Answer the 5 questions: +10 XP for each one you get right first time.

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