Risk & Return

Throughout this series we’ve treated return as one number. The DDM told us a stock “implies” 8%. We discounted dividends at a “required return” of 9%. Those numbers went into the formulas and prices came out. Nothing so far says where they come from, or how the risk they are meant to compensate gets measured in the first place.

This post builds the statistical toolkit. We take a set of possible outcomes, compute an expected return and a standard deviation, estimate both from a history of realized returns, model the distribution those numbers imply, put a figure on how bad a bad year gets, and compare investments on a risk-adjusted basis.

Expected Return

Suppose we are analyzing a cyclical stock. Next year’s economy falls into one of three scenarios, each with a probability and a return:

Scenario	Probability	Return
Expansion	25%	18%
Normal growth	50%	7%
Recession	25%	-8%

The expected return is the probability-weighted average of the outcomes:

E [r] = i \sum p_{i} \times r_{i}

E [r] = 0.25 \times 18% + 0.50 \times 7% + 0.25 \times (- 8%) = 4.5 + 3.5 - 2.0 = 6.0%

Our stock’s expected return is 6.0%. That is not a promise of 6.0%, and in this model 6.0% is not even one of the three possible outcomes. It is the center of gravity of the distribution, the number the realized return scatters around. Replay next year a thousand times and the average of those realized returns converges to 6.0%.

Expected return is a forecast. Change the probabilities or the scenario returns and the answer changes with them. Two analysts looking at the same stock can disagree on $E [r]$ because they assign different probabilities to the same events. The formula is precise. The inputs are judgment calls.

Variance and Standard Deviation

The expected return locates the center of the distribution and says nothing about its spread. Whether the realized return falls near 6.0% or swings far from it is what variance measures: the average squared deviation from the mean.

Var (r) = i \sum p_{i} \times (r_{i} - E [r])^{2}

Each term $(r_{i} - E [r])^{2}$ is a squared forecast error: how far outcome $i$ falls from the expected value, squared so that misses above and below both count as positive. The probabilities $p_{i}$ weight those squared errors by how likely each outcome is, and the sum is the average squared miss. Standard deviation $σ = Var (r)$ restates the result in the same units as returns, which makes it easier to interpret.

In practice the true probability distribution is rarely known. What we have instead is a history of realized returns. If those returns are drawn from a reasonably stable distribution, we can estimate the variance from the data, giving each of the $N$ observations equal weight:

Var (r) = \frac{1}{N} t = 1 \sum N (r_{t} - \overset{r}{ˉ})^{2}

where $\overset{r}{ˉ}$ is the sample mean return. The idea is unchanged: measure how far each realized return fell from the average, square it, and take the mean of those squared deviations.

Suppose we have six months of returns for a stock: 3%, -2%, 5%, 1%, -4%, 3%. The sample mean is 1.0%, and the squared deviations from that mean are 4, 9, 16, 0, 25, and 4. Averaging them gives us a variance of 9.67, in units of percent squared, which is awkward to interpret. We take the square root and get $σ = 3.1%$ per month. In a typical month this stock’s return falls about 3 percentage points above or below its average.

Squaring the deviations gives variance a property that matters once we reach portfolios: it is additive across independent variables. A larger $σ$ means more uncertainty about the outcome, and a smaller one means the return is more predictable.

The Normal Model

With $E [r]$ and $σ$ in hand, we can model the entire distribution of returns. The standard assumption is that returns follow a normal distribution, the bell curve, which is completely determined by its mean and standard deviation. Once those two numbers are set, so is everything else: the probability of earning more than 20%, the probability of losing money, the probability of a return inside any range.

The useful rules of thumb:

About 68% of outcomes fall within one $σ$ of the mean ( $E [r] \pm σ$ )
About 95% fall within two standard deviations ( $E [r] \pm 2 σ$ )
About 99.7% fall within three ( $E [r] \pm 3 σ$ )

A stock with $E [r] = 10%$ and $σ = 15%$ puts roughly two-thirds of its annual returns between -5% and 25%, and roughly 95% of them between -20% and 40%.

Drag the two sliders below to set $E [r]$ and $σ$ .

Expected Return E[r]: 10% Std Deviation σ: 15% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50% 60% E[r] 5% VaR E[r] 10.0% σ 15.0% 5% VAR -14.7% 1% VAR -24.9%

Increasing $σ$ flattens and widens the curve, and decreasing it makes the curve tall and narrow. Moving $E [r]$ shifts the whole distribution left or right. The red region is the 5% left tail, and the readouts below the chart report where that tail begins.

Observed return distributions are not perfectly normal. They tend to have fat tails, meaning that extreme events (crashes, spikes) happen more often than the bell curve predicts. The normal model is a useful approximation rather than a law, and for most purposes it works well enough. Tail-risk analysis needs a model that gives those extremes their real weight.

Value at Risk

The normal model assigns a probability to any region of the distribution, and the region risk managers and regulators care about is the left tail. Value at Risk (VaR) marks its boundary: the return that only a stated fraction of outcomes falls below.

The 5% VaR is the return that is worse than 95% of outcomes, breached only 5% of the time. Under the normal model:

VaR_{5%} = E [r] - 1.645 \times σ

The 1.645 comes from the standard normal distribution, where 5% of the area lies more than 1.645 standard deviations below the mean.

Take a stock with $E [r] = 8%$ and $σ = 20%$ :

VaR_{5%} = 8 - 1.645 \times 20 = 8 - 32.9 = - 24.9%

On a 100,000 portfolio, a year among the worst 5% of outcomes costs us roughly 24,900 or more. The 1% VaR is further out: $E [r] - 2.326 \times σ = 8 - 46.5 = - 38.5%$ , a potential loss of 38,500.

VaR is popular because it turns a statistical abstraction into a dollar figure that executives understand. Its weakness is that it says nothing about how bad things get inside that worst 5%. A 5% VaR of -24.9% is consistent with tail losses of -25% (mild) or -60% (catastrophic). The number marks the boundary, not what lies beyond it.

The Sharpe Ratio

VaR describes the downside of one investment. Comparing two takes something else. A stock with an expected return of 13% sounds better than one offering 8%, until the 13% stock turns out to have twice the volatility. The extra return is real and so is the extra risk, and the question is whether one is paying for the other.

The Sharpe ratio answers it by measuring excess return per unit of risk:

S = \frac{E [ r ] - r _{f}}{σ}

where $r_{f}$ is the risk-free rate, the return available at no risk, typically the T-bill yield. The numerator is the extra return earned above the safe option, and the denominator is the price paid for it in volatility.

Take a stock with $E [r] = 11%$ , $σ = 22%$ , and $r_{f} = 3%$ :

S = \frac{11 - 3}{22} = \frac{8}{22} = 0.36

That is 0.36 percentage points of excess return per percentage point of volatility. Whether that is good depends on context, but historically a Sharpe ratio around 0.3 to 0.5 is typical for equity markets. Below 0.2 the compensation for the risk is thin. Above 0.7 is unusually good (and often temporary).

Drag the risk-free rate slider below and watch all three ratios fall together.

Risk-Free Rate: 3% Bond Fund 0.33 Sharpe Ratio E[r] 5% σ 6% EXCESS RETURN 2.0% Balanced Fund 0.42 Sharpe Ratio E[r] 8% σ 12% EXCESS RETURN 5.0% Small-Cap Equity 0.40 Sharpe Ratio E[r] 13% σ 25% EXCESS RETURN 10.0% Higher Sharpe = more return per unit of risk.

As $r_{f}$ rises, all three Sharpe ratios fall, because the excess return in the numerator shrinks while $σ$ stays put. The bond fund’s ratio falls fastest, since it has the smallest standard deviation of the three and each point of lost excess return is divided by it. At a risk-free rate of 5% the bond fund’s excess return reaches zero and its Sharpe ratio goes with it, while the small-cap fund still has a positive Sharpe because it started with more headroom.

The Sharpe ratio is the most commonly used risk-adjusted performance measure. Its limitation is that it treats upside volatility the same as downside volatility. A stock that occasionally shoots up 40% has a high $σ$ , though no investor minds a surprise in that direction. Other measures, the Sortino ratio among them, penalize downside deviations only. For a first pass, the Sharpe ratio is enough.

Annualizing

Each of these measures assumes the data arrives in the units we want, and often it does not: a fund reports monthly returns, and a stock quotes daily closing prices. To compare those figures against annual ones, we annualize them.

Expected returns scale with the number of periods per year:

E [r_{A}] = T \times E [r_{s u b}]

where $T$ is 12 for monthly data and 252 for daily data, the number of trading days in a year.

Standard deviation looks like it should scale the same way, but the correct factor is $T$ :

σ_{A} = T \times σ_{s u b}

Why the square root? Because variance (not standard deviation) is additive across independent periods. We multiply monthly variance by 12 to get annual variance, then take the square root to return to standard deviation: $σ_{A} = 12 \times σ_{M}^{2} = 12 \times σ_{M}$ .

Suppose a fund reports monthly $E [r] = 0.8%$ and monthly $σ = 4.5%$ . We annualize both:

E [r_{A}] = 12 \times 0.8% = 9.6%

σ_{A} = 12 \times 4.5% = 3.46 \times 4.5% = 15.6%

The return scales linearly and the risk scales more slowly, so the risk-return tradeoff improves over longer horizons, assuming returns are independent across periods, which is approximately true.

Arithmetic and Geometric Means

Averaging a run of realized returns raises a separate question, because a sequence of historical returns has two averages and they give different answers.

The arithmetic mean is the simple average. Take five annual returns of +20%, -15%, +10%, +5%, and -10%, and we get:

\overset{r}{ˉ} = \frac{20 + ( - 15 ) + 10 + 5 + ( - 10 )}{5} = \frac{10}{5} = 2.0%

The geometric mean is the constant annual return that would have produced the same cumulative result. We chain the five together and take the fifth root:

r_{g} = (1.20 \times 0.85 \times 1.10 \times 1.05 \times 0.90)^{1/5} - 1

= (1.0603)^{0.2} - 1 \approx 1.2%

Our geometric mean of 1.2% falls below the arithmetic mean of 2.0%, and it always does unless every return in the sequence is identical. The gap widens as volatility increases, because compounding a loss costs more than compounding an equal gain returns.

The two averages answer different questions. For forecasting expected future returns, the arithmetic mean is the unbiased estimator: if next year’s return is drawn from the same distribution, the arithmetic mean is the best guess. The geometric mean says what a buy-and-hold investor earned over the historical period. It is the realized compound growth rate, useful for evaluating past performance but biased downward as a forecast.

Volatility as a Proxy for Risk

Every measure above treats $σ$ as the stand-in for risk, and two long-running objections say it is the wrong stand-in. The first is blunt: “Risk isn’t volatility. Risk is permanent loss of capital.”

The objection is partly right. What investors fear is a loss large enough to disrupt their ability to invest going forward, and a portfolio that drops 50% needs a 100% gain to recover. That is a real and distinct concern from a position that bounces around 2% a month. Nobody disagrees that catastrophic, unrecoverable losses are bad.

The trouble is that permanent loss of capital is not a measurement. It defines what an investor is worried about without supplying a tool for managing it. To put the concept to work, we name a loss threshold, estimate the probability of hitting it, and set a tolerance. That is Value at Risk, and it requires a probability distribution and a standard deviation. The quantitative tools formalize the intuition rather than replacing it.

The second objection runs at the symmetry of $σ$ itself, and it has limits of its own. Standard deviation does treat upside and downside moves alike, which looks wrong for a buy-and-hold investor who welcomes positive surprises. For a professional manager benchmarked against an index, however, risk is symmetric. Missing a stock that doubles damages a track record as much as owning one that halves, and tracking error does not care which direction the miss ran in.

σ

is a measure of risk, not risk itself. True risk is unknowable in any complete sense, so we pick the best available proxy and work with it.

The problem is not in thinking about risk as the potential for loss. The problem is stopping there and never attempting to quantify it. Intuition is a starting point, but it does not say how large a position to hold. For that we need numbers.

The Risk-Return Spectrum

With the caveats attached, these measures describe a pattern that repeats across asset classes. Cash is safe and pays little. Bonds pay more and fluctuate moderately. Stocks are the riskiest of the three and offer the highest expected returns. The scatter plot below places six asset classes in that space, with standard deviation on the horizontal axis and expected return on the vertical. Read it from the bottom left to the top right.

0% 3% 6% 9% 12% 15% 0% 4% 8% 12% 16% 20% 24% 28% T-Bills Govt Bonds Corp Bonds Intl Stock Large-Cap Small-Cap Standard Deviation (σ) Expected Return Approximate long-run averages. Actual returns vary by period and methodology.

The points run from T-bills at the bottom left, near 3% expected return with 3% volatility, up to small-cap stocks at the top right, near 12% with 22%. The upward slope is not a coincidence. Investors demand compensation for bearing risk, and nobody would hold an asset that did not offer a premium for its volatility. The market sets prices so that riskier assets offer higher expected returns, at least on average and over the long run.

The fit is loose rather than exact. International stocks show a higher volatility and a lower expected return than large-cap stocks in the plot, and individual assets can and do disappoint without overturning the general pattern.

This risk-return tradeoff is the central fact of investing. Every decision reduces to which point on that spectrum to choose, and whether the reward there justifies the risk.

What’s Next

We can now measure risk and return for one investment at a time. But nobody holds only one stock. In the next post we work out the arithmetic of a portfolio: how the returns of the parts combine into the return of the whole, and what a margin loan does to that result in both directions. Short selling runs on the same formula, which is what makes borrowing and shorting two sides of the same coin.