Portfolio risk and return

In the next few minutes:

1. How risk aversion and the capital allocation line pick an investor's optimal portfolio
2. How to measure mean, variance, covariance and correlation from past returns
3. Why combining assets that are less than perfectly correlated cuts risk, and what the efficient frontier is

More return comes with more risk

Long-run US data (1926–2017) show the asset classes with the highest average returns also had the highest standard deviations of returns:

Other markets show the same pattern. After inflation, T-bills earned a real return of only about 0.5% a year, against about 7.3% for large-cap stocks.

Mean and variance don't tell the whole story

Judging assets only by expected return and variance assumes returns are normally distributed. They aren't:

• Negative skewness: large downside moves are more likely than large upside moves.
• Excess kurtosis (kurtosis above 3): fat tails, so extreme moves in both directions happen more often.

Liquidity matters too. It affects price and therefore expected return. It's a real concern in emerging markets and for securities that trade rarely, such as low-quality corporate bonds.

Three attitudes to risk

A coin flip pays 100 on heads and nothing on tails. Its expected payoff is 0.5 × 100 + 0.5 × 0 = 50. Would you take the flip or a certain 50?

Financial models assume investors are risk-averse. A risk-averse investor still takes risk, but only when the extra expected return is enough compensation.

Utility puts a number on the trade-off

An investor's utility function scores a portfolio by its expected return, minus a penalty for risk:

• $E(R)$, $\sigma$: expected return and standard deviation, as decimals
• $A$: the risk aversion coefficient

The more risk-averse the investor, the bigger A. A risk-averse investor has A > 0, a risk-neutral investor A = 0, and a risk-seeking investor A < 0.

Indifference curves

An indifference curve joins every combination of risk (σ) and expected return that gives the investor the same expected utility.

• For a risk-averse investor the curves slope upward: more risk is accepted only for more expected return.
• A more risk-averse investor has steeper curves: they need a bigger return increase for each extra unit of risk.
• Curves higher up and to the left give more utility, so they are preferred.

Add a risk-free asset

Combine a risky portfolio P with a risk-free asset (return $R_f$, zero standard deviation, zero correlation with P). Put weight w in P:

Both are linear in w, so every mix lies on a straight line: the capital allocation line (CAL). It runs from $R_f$ through the optimal risky portfolio.

Two-fund separation theorem: every investor's optimal portfolio is some mix of the same optimal risky portfolio and the risk-free asset.

Worked example: points on the CAL

The risk-free rate is 4%. The risky portfolio has E(R) = 10% and σ = 20%.

Each extra 1% of σ adds 0.3% of expected return: (10% − 4%) ÷ 20%. That constant trade-off is why the CAL is a straight line.

The optimal portfolio

The investor's optimal portfolio is the point where the CAL touches the highest reachable indifference curve (a tangency). Scoring our three CAL points with utility shows why investors differ:

The more risk-averse investor (A = 4) prefers 40%. The less risk-averse one (A = 2) prefers 100%. Flatter indifference curves lead to more in the risky portfolio.

Mean and variance from past returns

Three years of returns for an equity fund (S) and a bond fund (B):

We have a sample, so the sample variance divides by T − 1:

• S: (400 + 0 + 400) ÷ 2 = 400, so s = 20%
• B: (100 + 100 + 0) ÷ 2 = 100, so s = 10%

If you had the whole population, you would use the population mean μ and divide by T instead.

Covariance: do returns move together?

Covariance measures how two assets' returns move together. Multiply their deviations year by year, add, and divide by T − 1:

For S and B: (−20 × −10) + (0 × 10) + (20 × 0) = 200, and 200 ÷ 2 = 100 (in %², or 0.0100 in decimals).

• Positive: returns tend to move together.
• Negative: they tend to move in opposite directions.
• Zero: no linear relationship.

Its size depends on the units, so 100 alone says little.

Correlation standardizes covariance

Divide the covariance by the product of the standard deviations:

For S and B: 100 ÷ (20 × 10) = 0.5.

The correlation coefficient has no units and always lies between −1 and +1.

• +1: deviations always move proportionally in the same direction (perfect positive correlation).
• −1: always proportionally opposite (perfect negative correlation).
• 0: no linear relationship. Knowing one return tells you nothing about the other.

Rearranged: $\text{Cov}_{12} = \rho_{12}\sigma_1\sigma_2$.

Portfolio standard deviation

For weights $w_1$ and $w_2 = 1 - w_1$ in two risky assets:

Replace the covariance with $\rho\sigma_1\sigma_2$:

• $\sigma_1$, $\sigma_2$: the assets' standard deviations
• $\rho$: their correlation

Portfolio standard deviation is the square root, $\sigma_P$. Written with ρ, the formula shows how correlation drives the last term.

Worked example: half in S, half in B

S has σ = 20%, B has σ = 10%, and their correlation is 0.5.

σ = √0.0175 = 13.23%. The weighted average of the two standard deviations is 0.5 × 20% + 0.5 × 10% = 15%. Diversification has removed 1.77 percentage points of risk.

Lower correlation, lower risk

Same portfolio, half in S and half in B. Slide the correlation:

At ρ = +1, σ is the weighted average, 15%: no diversification benefit. Any correlation below +1 reduces risk, and the lower the correlation, the bigger the reduction.

Perfect negative correlation

At ρ = −1 the right weights remove risk altogether. Put 1/3 in S and 2/3 in B:

(1/3) × 20% − (2/3) × 10% = 0%. Any other weights leave some risk.

With ρ = 0 the last term of the variance formula drops out, but some risk always remains. Only ρ = −1 allows a zero-variance portfolio. This is why investors spread money across bonds, foreign stocks, real estate and other asset classes.

Minimum-variance and efficient frontiers

With many risky assets, the lowest-risk mix for each level of expected return is a minimum-variance portfolio. Together they form the minimum-variance frontier.

• The global minimum-variance portfolio is the frontier's leftmost point: the least risk of any risky portfolio.
• The efficient frontier is the part of the frontier above it: the highest expected return for each level of risk.

Risk-averse investors choose only efficient portfolios. Any portfolio below the frontier is beaten by one with the same risk and more return.

Remember this