PM 1 Portfolio Risk and Return: Part I

Before looking at any data, separate two ideas that are easy to blur. Historical return is what an asset actually delivered in the past. Expected return is what an investor anticipates earning going forward. They are different concepts, and the reading flags every time historical figures are used as a stand-in for expectations.

Expected return is the nominal figure that would just persuade the marginal investor to buy, built from the real risk-free interest rate, expected inflation, and the expected risk premium for the asset’s risk. All three components are usually positive, so expected return is generally positive. Their relationship is multiplicative, not additive.

A historical mean, by contrast, comes from returns actually earned. Because the investment is risky, the realized return rarely equals the expected return in any single period. Over a long enough horizon the average realized return should converge on the expected return, but no one knows whether that horizon is ten years, fifty, or a hundred, so treating the historical mean as the expected return is a convenient assumption rather than a proven fact.

Nominal and real returns across asset classes

Over the 1926 to 2017 period in the United States, the ranking of nominal returns tracked risk closely. Small company stocks earned the most and were the most volatile, large company stocks came next, then long-term bonds, and Treasury bills sat at the bottom on both counts. Treasury bills posted a negative annual return only once (1938, at −0.02 percent), because nominal short-term rates almost never turn negative and bills carry little interest rate or inflation risk.

Returns are annualized geometric means; risk annualizes monthly standard deviations. Source: 2018 SBBI Yearbook.

Because annual inflation swung from −10.30 percent to +13.31 percent over these years, comparing nominal returns across eras is misleading, so real returns are the cleaner lens. On a real basis from 1900 to 2017, one dollar invested in US equities grew to 1,654 dollars, against 10.20 dollars for bonds and 2.60 dollars for bills: a 6.5 percent real annual return for equities compounding into a vast lead over the 2.0 percent earned by bonds.

GM: geometric mean. SD: standard deviation. All returns measured in US dollars. Source: Credit Suisse Global Investment Returns Sourcebook, 2018.

The geometric mean is never above the arithmetic mean, and the analysis leans on the geometric mean because it represents multi-period returns more accurately. Notice diversification even at the index level: US equity risk is 20.0 percent and world excluding US equity risk is 18.9 percent, yet blending them into the world index pulls risk down to just 17.4 percent.

The risk-return trade-off

The phrase describes the positive link between expected risk and expected return: in efficient markets over long horizons, a higher return cannot be had without accepting more risk. A useful way to view it is through the risk premium, the extra return earned for bearing risk above the risk-free rate. The nominal risk premium takes the nominal risky return and subtracts the nominal risk-free rate; the real version uses real figures throughout. Over long spans, higher risk did reward investors with higher mean returns, which is exactly the pattern a market populated by risk-averse investors produces.

Judging investments on mean and variance alone rests on two assumptions: that returns are normally distributed, so two parameters describe them fully, and that markets are efficient both informationally and operationally. Where those assumptions break down, other characteristics matter.

Distribution shape: skewness and kurtosis

A normal distribution has its mean equal to its median, is fully defined by mean and variance, and is symmetric, with roughly 68 percent of observations within one standard deviation of the mean, 95 percent within two, and 99 percent within three.

Real returns depart from normality in two ways. They are skewed, so the two halves on either side of the mean are not mirror images. A distribution is negatively (left) skewed when most of the mass sits to the right with a long left tail, and positively (right) skewed in the opposite case. US large company stock returns from 1926 to 2017 are negatively skewed. Returns also show kurtosis, or fat tails, meaning extreme outcomes occur far more often than a normal distribution would predict. Both features argue against relying only on mean and variance.

Market characteristics: liquidity

Operational frictions also shape choices, and liquidity is the main one. Trading costs have three parts: brokerage commission, the bid-ask spread, and price impact, and liquidity drives the last two. The bid-ask spread is the gap between the buying and selling price. A 10 cent spread is only 0.1 percent on a 100 dollar stock but a full 1 percent on a 10 dollar stock, so the cheaper stock is dearer to trade and an investor needs about 0.9 percent of extra return to break even against the pricier one.

Price impact is how much an order moves the price. A small order in a liquid name may not move it at all, but an order to buy 100,000 shares can push the price up as the buyer coaxes more holders to sell, and the thinner the stock, the larger that impact. Liquidity is a sharper concern in emerging markets and in lower-quality corporate bonds, where a single issue may not trade for days or weeks, a problem that surfaced starkly during the global financial crisis. Analyst coverage, information availability, and firm size round out the market traits that feed investment decisions.

Offer an investor a choice between 50 pounds for certain and a gamble paying 100 pounds or nothing on a coin flip. Both have an expected value of 50 pounds, but the responses sort investors into three types. Someone who takes the gamble is risk seeking and will even accept a lower expected value, say 45 pounds, for the thrill of uncertainty, the instinct behind lottery tickets and casino bets. Someone indifferent between the two is risk neutral and cares only about return, ignoring risk entirely. Someone who prefers the certain 50 pounds is risk averse and may accept a guaranteed 45 pounds instead of the risky 50.

Historical data show a positive risk-return relationship, which means market prices were set by risk-averse investors. For that reason the reading assumes a representative risk-averse investor throughout, the standard assumption across the investment industry. Risk tolerance is simply the flip side: the more risk an investor can bear, the higher the tolerance, and tolerance moves opposite to risk aversion.

The utility function

Utility measures the satisfaction an investor takes from a portfolio. A simple and widely used form ranks investments by combining expected return and variance.

A is the marginal reward an investor demands to accept more risk, so it is larger for the more risk averse. Several points follow. Utility is unbounded above and below. Higher return raises utility. Higher variance lowers it, and the penalty is amplified by A. Utility only ranks investments; it does not measure satisfaction in absolute terms, so a utility of 4 is not twice as good as a utility of 2, and utility cannot be compared or summed across people. For a risk-averse investor A is positive, so risk cuts utility; for a risk-neutral investor A is zero, so risk is irrelevant; for a risk lover A is negative, so risk adds to utility. A risk-free asset has zero variance and therefore yields the same utility, equal to its return, for everyone.

Indifference curves

An indifference curve traces every risk-return pair that leaves an investor equally satisfied, so the investor does not care which point on a single curve is chosen. For a risk-averse investor the curves slope upward from southwest to northeast, because added risk must be repaid with added return to hold utility constant, and they are convex, because the marginal utility of return diminishes, so each extra unit of risk demands ever more return and the curve steepens. Utility rises as you move northwest toward higher return with lower risk, and curves for the same investor never touch or cross.

The slope of the curve encodes risk aversion. The most risk-averse investor has the steepest curve, demanding sharply higher returns as risk grows. The least risk-averse investor has the flattest upward curve. A risk-neutral investor has a horizontal curve, since risk does not affect utility, and a risk lover has a downward-sloping curve, happy to swap return for risk.

The simplest use of utility theory combines two assets: a risk-free asset with return R_f and zero risk, and one risky asset with standard deviation sigma sub i and a higher expected return, since the risky asset must pay more to be held. Write w sub 1 for the fraction placed in the risk-free asset and (1 minus w sub 1) for the fraction in the risky asset. Because the risk-free asset has zero variance, two of the three variance terms drop out, leaving a clean result.

Sweeping w sub 1 from 100 percent down through zero and into negative values traces a straight line in risk-return space. Substituting the risk expression back into the return expression gives the equation of that line, the capital allocation line.

The line starts at the risk-free asset on the vertical axis, where all wealth sits in the safe asset. Putting 100 percent in the risky asset lands at its return and risk. To push further up the line for more return, an investor borrows at the risk-free rate and overinvests in the risky asset: borrowing 50 percent makes w sub 1 equal to −0.50 and places 150 percent in the risky asset, giving a return of 1.50 times E(R sub i) minus 0.50 times R sub f, which exceeds E(R sub i).

Which point on this line an investor picks comes from overlaying the personal indifference curves on the capital allocation line. The line is the feasible set; the indifference curves rank the feasible points. Points below the line are attainable but never chosen, because a higher return is available at identical risk simply by rising to the line, and points above the line are desirable but unreachable with the available assets.

The optimal portfolio is where the highest achievable indifference curve just touches the line, that is, the point of tangency. A curve lying entirely above the line has unreachable points; a curve cutting the line at two points is beaten by a tangent curve that offers a higher return at the same risk. The tangency point maximizes return per unit of risk and simultaneously delivers the most utility the investor can reach. Because indifference curves differ, that tangency lands in a different place for each investor: a less risk-averse investor such as Kelly (A = 2) has a curve sitting to the right of a more risk-averse investor such as Jane (A = 4), so Kelly’s optimal portfolio carries more risk than Jane’s.

A portfolio return is always the weighted average of the component returns. If asset i returns R sub i and carries weight w sub i, with weights summing to one, the portfolio return is their weighted sum. For two assets weighted 25 percent and 75 percent with returns of 20 percent and 5 percent, the portfolio earns (0.25 × 20%) + (0.75 × 5%) = 8.75 percent.

Risk does not average the same way. Unless the assets are perfectly correlated, a portfolio standard deviation is not the weighted average of the component standard deviations. Portfolio variance is the sum of each asset’s variance contribution plus the covariance terms between pairs. Because an asset’s covariance with itself is its variance, the two-asset case reduces to a compact expression.

Covariance and correlation carry the same information about co-movement, linked by a simple identity. Covariance is hard to read because it is unbounded, while the correlation coefficient is bounded between −1 and +1, so it is easier to interpret.

A correlation of +1 means the two assets move together all the time, −1 means they move in exact opposition, and 0 means the movement of one says nothing about the other. When correlation equals +1, the portfolio variance collapses to a perfect square and the standard deviation becomes the plain weighted average, so no risk is diversified away.

Whenever correlation is below +1, the portfolio standard deviation falls short of that weighted average, which is diversification at work. Drive correlation all the way to −1 and, at the right weights, the portfolio can be made risk free.

How weights and correlation reshape the opportunity set

Now let the two assets differ. Asset 1 returns 7 percent with 12 percent risk; Asset 2 returns 15 percent with 25 percent risk. The table below varies the weight in Asset 1 and shows the portfolio risk under four correlations. Return depends only on the weights, never on correlation, but risk shrinks at every weight as correlation falls, reaching its lowest values at correlation −1.

Plotted, the pair traces a straight line when correlation is +1 and bows further to the left, toward lower risk, as correlation falls. This leftward bulge is the signature of every investment opportunity set except at the extremes of correlation equal to +1 or −1.

Using correlations of −0.32 (equities to Treasuries), −0.06 (equities to currency), and 0.33 (Treasuries to currency), the three-asset calculation gives a portfolio risk of about 8.4 percent. So swapping gilts for US Treasuries raises the return to 3.9 percent but nudges risk up to 8.4 percent, because owning a foreign bond drags in exchange-rate risk even though the Treasury is less volatile than gilts in local-currency terms.

Extend the two-asset logic to N assets. Portfolio variance is the sum of every asset’s own variance contribution plus every pairwise covariance. A clean insight appears if all N assets are held at equal weight, using the average variance and average covariance.

As N grows, the first term shrinks toward zero, so any single asset’s own variance becomes a negligible part of portfolio risk. The second term approaches the average covariance. For a portfolio of many assets, then, covariance among the holdings accounts for almost all the risk. If instead every asset shares the same variance and the same mutual correlation, the standard deviation takes an even sharper form.

Again the first term fades as N rises, leaving correlation as the dominant driver of portfolio risk. If the assets were truly unrelated, the portfolio could approach zero risk. Real assets, unfortunately, mostly show high positive correlations, so the diversification challenge is finding holdings whose correlation is well below +1.

Historical correlations and stability

Risk for an asset class tends to be stable from one decade to the next, so historical risk is a reasonable proxy for future risk. Correlations are likewise fairly stable within a country, though correlations between countries have crept up with globalization. A correlation above 0.90 offers little diversification, while correlations below roughly 0.50 are the ones worth seeking. Among major US asset classes from 1970 to 2017, the strongest link is between US large and small company stocks, and the weakest links, some of them negative, are between stocks and bonds.

Source: 2018 SBBI Yearbook. Intl stk: international stocks. LT: long-term.

Ways to diversify

Several practical routes spread risk. Diversify across asset classes, since correlations among major classes such as domestic and foreign stocks and bonds, real estate, and commodities are generally low. Use index funds or ETFs when building broad exposure directly would need hundreds of securities and become costly to trade and track. Diversify across countries, whose returns differ by industry mix, policy, and currency, and whose currency returns, being uncorrelated with stock returns, can trim the risk of even a volatile emerging market. Avoid overloading on your employer’s stock, since your human capital and earnings are already tied to that firm. Weigh the trading and tracking cost of each new holding, and add it only when it genuinely improves the portfolio. A clean rule for that last test compares risk-adjusted returns.