Risk and return: the trade-off explained

In everyday speech, risk is the chance that something goes wrong. In finance it is the spread of everything that could happen, good and bad. An investment described as returning 8 percent a year with a standard deviation of 15 points is not promising 8 percent. It is describing a distribution. If that distribution were a bell curve, roughly two years in three would land between -7 and +23 percent, and about one year in twenty would land outside -22 to +38 percent.

That framing puts the upside inside the definition. A wider spread means worse bad years and better good years at the same time. Nobody would pay for a wider spread of losses on its own, and the market does not price one.

Two warnings come with the measure. Actual market returns have fatter tails than a bell curve, so the extreme years arrive more often than those bands suggest, which makes the ranges above a sketch rather than a forecast. And volatility is not the same thing as permanent loss. A price moving about while the asset behind it is intact is a problem you can wait out. A borrower defaulting or a business failing is a problem with nothing left to recover. Only the second is fatal to a plan, which is why the useful question is not how much a holding wobbles but how much of it could be gone for good.

Picture two investments that each pay off in a year. One hands over $10,000 for certain. The other hands over $10,000 on average, sometimes a good deal more, sometimes a good deal less. Ask which one people bid more for and everything else follows.

Almost everyone prefers the certain payment, so buyers compete for it and its price rises. A higher price for the same payoff is a lower return. The uncertain one has to be cheaper before anyone will hold it, and a lower price for the same expected payoff is a higher expected return. The gap between the two prices is the market's standing offer: accept the spread of outcomes and you are paid extra for doing it.

Two things follow that the slogan about risk and reward tends to hide.

The payment is expected, not scheduled. If the extra return arrived reliably it would not be compensation for anything, and the price gap would close. It shows up as an average across many outcomes and many investors, while any one investor over one lifetime gets a single draw. A decade of results below the average is inside the deal, not evidence the deal was broken.

Only risk you cannot shed for free is paid for. Holding one company instead of several hundred widens your range of outcomes a great deal and raises your expected return by nothing, because anyone can remove that risk by owning more names, which is why it is called unsystematic risk. Risk everyone is stuck with, such as the whole economy turning, is systematic risk, and that is the kind that carries a premium. This is the entire argument for spreading holdings: the free part of the risk is worth removing precisely because nobody pays you to hold it.

Every expected return is quoted from a baseline: what you could earn while taking no default risk at all. In practice that baseline comes from short-dated government debt in the currency you spend, which for a dollar investor in the United States means Treasury bills. That substitution holds only for a government borrowing in a currency it issues and trusted to repay it. Debt from a weaker sovereign, or from one borrowing in a currency it does not control, carries genuine default risk and is not a risk-free baseline anywhere. Wherever you are, the rate moves with policy and with expectations, so it is a level to look up rather than a number to memorise.

Everything else is built on top of it:

$$E(r) = r_f + \text{risk premium}$$

Read that risk premium term as a shopping list rather than one number, and read every item on the list as an expectation rather than a promise. Lend for ten years instead of three months and you expect a term premium for tying money up while rates could move against you, though that one can price below zero when short rates are widely expected to fall. Lend to a company instead of a government and you expect a credit spread for the chance of not being repaid. Own the company instead of lending to it, standing last in the queue if it fails, and you expect an equity risk premium. Hold something you cannot sell quickly and you expect an illiquidity premium. Each one is a specific uncertainty with a price attached, and each price is set by what buyers will pay for it rather than by a rule.

Risk-free is also a narrow claim. It means free of default risk over the term of the loan, and nothing more. It is not free of inflation risk: 4 percent while prices rise 3 percent leaves under 1 percent of real gain, which the fourth example below works through in money. It is not free of reinvestment risk, because a bill matures and the next one may pay less. And it is not free of currency risk if you spend a different currency from the one you lent in. The calculator at the top of this page turns any quoted rate into the real one behind it.

Averages are worked out by adding and dividing. Money is worked out by multiplying. That difference is the whole of volatility drag.

Take $10,000 for two years. Gain 30 percent, then lose 20 percent, and the two annual returns average a respectable 5 percent, but the balance ends at $10,400, which is compound growth of 1.98 percent a year. Earn a flat 5 percent in both years instead and the same $10,000 ends at $11,025. Identical average, more money, because nothing had to be recovered.

The bottom row is the same point in its sharpest form: an average of 25 percent a year and not a cent of gain. A loss hurts more than the matching gain helps, because the gain works on a smaller base. Down 50 percent needs up 100 percent just to break even.

The approximation worth carrying is:

$$r_{\text{compound}} \approx r_{\text{average}} - \frac{\sigma^2}{2}$$

That formula is a small-swing approximation, and it is worth knowing where it stops working. On ordinary portfolio volatility it is close. On the rows in the table above it is not: the plus 30 then minus 20 path gives 1.875 percent against a true 1.98, and the plus 100 then minus 50 path gives minus 3.125 percent against a true zero. Inside its range it carries the useful part, which is that the penalty rises with the square of the spread, so roughly doubling the volatility quadruples the drag. It also means cutting volatility without cutting the average return is a genuine gain rather than a matter of temperament. When you compare two investments, compare compound growth rates, which is what the CAGR calculator works out. An advertised average is not the rate a balance grew at.

Differences in expected return turn into much larger differences in money, because they compound. The two rates below are illustrative, picked to show the arithmetic rather than to describe what any asset pays as you read this. Put $10,000 into a safe asset paying 4 percent and leave it for 25 years and it reaches $26,658.36. Put the same amount into a portfolio with an expected 8 percent and the same 25 years produce $68,484.75, roughly two and a half times as much. The mechanism is ordinary compounding applied to a higher rate for a long time.

Inflation takes a bite out of both. At 3 percent a year, the safe pot is worth $12,732.18 in today's money and the risky pot $32,708.70. Note what that says about the safe asset: a quarter of a century of an outcome you could predict added a bit over a quarter to buying power. Certainty is not free, and its price is most of the growth.

The warning attached to the second figure is the same one from the top of the page. An expected 8 percent is the middle of a distribution, not a rate anyone has agreed to pay. Realised returns over any particular 25 years can land well below it, and the balance will not travel there in a straight line.

What the arithmetic does show is why the horizon changes the answer. Over a few years the spread of outcomes easily swamps the premium, so the risky asset can comfortably finish behind the safe one. Over decades the premium has far more room to show up, though it never hardens into a guarantee.

The right amount of risk is not a personality quiz. It starts with when the money is needed and what breaks if it is not there.

Money with a job in the next few years has no time to recover from a bad draw, so it belongs where the range of outcomes is narrow, even though narrow outcomes pay less. An emergency fund is the clearest case: its entire purpose is being the right size on the worst possible day, which rules out anything that might be down 30 percent on that day. Money that will not be touched for decades can carry dispersion, because there is room for a bad stretch to be followed by a good one.

Two further things change the answer.

• Capacity against tolerance. Capacity is how much loss a plan can absorb before it fails. Tolerance is how much you can watch without selling. The smaller of the two governs, because a portfolio abandoned at the bottom turns a temporary fall into a permanent one.
• Paying in and drawing out. An untouched balance only cares about the compound rate. A balance you are drawing from also cares about the order of returns, since selling into a fall removes units that are not there to recover later. The same average return can support or exhaust a withdrawal plan depending on when the bad years arrive.

Borrowed money deserves its own line. It widens both tails of the distribution and adds a repayment schedule that does not care what markets are doing. It raises expected return only while the holding out-earns the cost of the borrowing, and it raises the chance of being forced to sell at the worst possible moment either way, which is the one risk that converts a wide spread into a permanent loss.