Measuring Returns and Risk

Every investment decision comes down to two questions: how much might I earn, and how much could I lose? This chapter gives you the tools to answer both with numbers, not guesses. We will start with the simplest measures of return, then build up the quantitative side of risk so you can compare any two investments fairly.

The Big Picture#

If you put money into a stock, a bond, or a savings account, you need a clear way to judge how well that money performed. A dollar number alone doesn’t help – doubling 200 is very different from doubling 20,000. So we create ratios and percentages that let you compare a small bank deposit with a huge real estate deal side by side. Next, because the future is uncertain, we also need to measure how widely an investment’s returns can swing from year to year. That’s risk. With a handful of simple statistics, you can turn vague feelings of “this is safe” or “this is wild” into precise scores you can act on.

Holding Period Return: How to Measure a Single Investment’s Performance#

The most basic building block is the holding period return, often called HPR. It answers the question: “If I bought this asset, held it for a while, and then sold it, what total return did I get for every dollar I put in?”

You calculate the HPR by adding up everything you receive from the investment during the holding period (including any income like dividends or interest) and comparing it to what you paid at the start:

The result is a factor – a number like 1.12 or 0.95. If you want a percentage, simply subtract one. That percentage is often called the holding period yield, or HPY.

Holding Period Return (HPR): The total return factor for a single investment over a given holding period, computed as (Ending value + Income) ÷ Beginning value.

Holding Period Yield (HPY): The return expressed as a percentage or decimal, equal to HPR – 1.

For example, suppose you bought a share of a company for 2 dividend, and you sold the share for $55 at year-end. Your holding period return is:

The HPY is , or 14%. In plain English, for every dollar you put in, you ended up with $1.14 – so you made 14 cents of gain per dollar invested.

This ratio works for any holding period: a day, a month, a decade. It simply tells you the total growth of your money, including all cash flows you received along the way. Later, when we talk about risk, we’ll often talk in terms of HPY expressed as a decimal (0.14) or percentage (14%).

📝 Section Recap: The holding period return captures the total return on an investment over any period, including both price change and income. Once you can compute HPR and its yield, you can compare the raw performance of completely different investments on equal footing.

From Many Periods to a Single Number: Arithmetic and Geometric Means#

Investments are rarely held for just one period. Over several years, you get a string of annual returns – some up, some down. How do you turn those into a single number that describes the typical return? You have two main choices: the arithmetic mean and the geometric mean. They answer different questions, and using the wrong one can mislead you badly.

The arithmetic mean is the simple average you learned in school: add up all the periodic returns and divide by the number of periods.

The geometric mean, also called the compound annual growth rate, tells you the constant rate that would have given you the same final wealth as the actual, bumpy sequence. It is calculated by multiplying the (1 + return) factors for each period, raising that product to the power of , and then subtracting one:

Let’s see why the geometric mean is the one that tracks your actual money. Imagine an investment that returns +20% in Year 1 and -10% in Year 2. If you started with 120, and after Year 2 you’d have 108. Your two-year total return factor is 1.08, so you earned 8% over two years, not 20%. Now compute the means:

• Arithmetic mean: .
• Geometric mean: , or 3.92%.

That 3.92% is exactly the constant annual rate that would turn 108 after two years: 108. The 5% average would have misled you into thinking you had more growth than you actually did.

A helpful analogy is a bank account that compounds. If you lose 50% one year, you don’t need a 50% gain the next year to break even – you need 100%. The arithmetic mean ignores this compounding, so it always overstates the true long-run growth. As a rule of thumb, use the geometric mean when you want to describe past performance over multiple periods, and use the arithmetic mean when you’re trying to estimate an “average” outcome for next year (more on that in the next section).

📝 Section Recap: The arithmetic mean is the ordinary average of periodic returns; the geometric mean is the compounded growth rate that matches the actual ending wealth. For multi-period performance measurement, always lean on the geometric mean to avoid exaggerating the results.

Looking Forward: Expected Return with Probability-Weighted Outcomes#

History tells you what happened; investing forces you to guess what might happen next. To make that guess systematic, we use the concept of expected return. Instead of one outcome, we imagine several possible scenarios for the economy or the market, each with a probability of occurring, and we compute a weighted average.

Expected Return: The probability-weighted average of all possible returns, giving a forward-looking best guess of the average outcome if we could repeat the investment many times under the same conditions.

Suppose an analyst maps out three economic states for the coming year:

• Strong economy: probability 30%, stock return 15%
• Normal economy: probability 50%, stock return 8%
• Recession: probability 20%, stock return –5%

The expected return is:

That 7.5% doesn’t mean we’ll earn exactly 7.5% next year. It means that if we faced this exact set of probabilities year after year, the long-run average return would settle near 7.5%. Think of it as the centre of gravity around which the actual returns will scatter – and that scattering is what we measure next.

📝 Section Recap: Expected return blends all possible future outcomes by their probabilities, giving you a single number that represents your best estimate of the investment’s typical payoff. It is the foundation for every forward-looking risk measure.

Variance and Standard Deviation: Measuring How Returns Bounce Around#

If expected return is the centre, risk is the size of the bounces around that centre. Investors don’t mind positive bounces, but they hate negative ones – and large bounces in either direction mean big uncertainty. The most common risk measure is variance, which captures the average squared distance of returns from the expected return. Its square root, the standard deviation, puts that measure back into the same units as returns (percent) and is the classic “volatility” number you hear about.

Variance (): The expected value of the squared deviations of returns from the mean, a measure of dispersion.

Standard Deviation (): The square root of variance; it tells you, in the same units as the returns, how widely returns typically vary around the expected value.

Using the same three scenarios and the 7.5% expected return, we first compute the deviation of each possible return from the expected return, square it, and then take the probability-weighted average:

• Strong: , squared = , weight 0.30 →
• Normal: , squared = , weight 0.50 →
• Recession: , squared = , weight 0.20 →

Add them up: . That’s the variance, .

Standard deviation is simply the square root: , or about 6.95%. In plain English, a typical year’s return might be about 7 percentage points above or below the expected 7.5% – a wide range that could put you into positive or negative territory.

Note that you can compute variance and standard deviation from historical data, too. Replace the probabilities with equal weights (1/n) and the expected return with the arithmetic mean, and the same logic applies. No matter the source, a higher standard deviation means a more volatile, less predictable investment.

📝 Section Recap: Variance and standard deviation quantify uncertainty by measuring how far returns tend to stray from the expected value. Standard deviation is the go‑to number for describing an investment’s volatility, purely because it speaks the same language (percent return) as the returns themselves.

Coefficient of Variation: Risk per Unit of Return#

Sometimes you need to compare two investments that offer very different expected returns. A high‑risk investment might also offer a high expected reward – but how much risk are you taking for each unit of return? The coefficient of variation (CV) answers that question by expressing standard deviation relative to the expected return.

Coefficient of Variation (CV): The ratio of standard deviation to the expected return, . It shows how much risk (volatility) you bear per unit of expected reward.

Using the previous example, the CV is . That means for every 1% of expected return, you’re taking on about 0.93% of volatility. The lower the CV, the “cheaper” the expected return is in risk terms.

Let’s compare two different investments:

• Investment A: expected return 10%, standard deviation 15% → CV =
• Investment B: expected return 8%, standard deviation 9% → CV =

Even though A offers a higher raw return, B delivers that return with noticeably less relative volatility. If you had to pick one, and you dislike risk, B might be more attractive despite the lower expected return. The CV is most useful when the expected returns are wildly different; it helps level the playing field.

📝 Section Recap: The coefficient of variation scales risk by reward, showing how many units of volatility you accept for each unit of expected return. It is a simple tool for comparing the efficiency of investments that differ in both their risk and their return.

Downside Risk: Focusing on the Bad Outcomes#

Standard deviation treats every deviation from the expected return – up or down – as equally risky. But most investors don’t worry about unexpectedly high returns; they worry about losses and falling short of a specific goal. Downside risk measures focus only on the negative side of the distribution, giving a picture that matches how people actually feel about risk.

The most common downside measure starts with a target return, often zero (to capture absolute losses) or the return on a safe asset like Treasury bills. Then we compute the average of the squared negative deviations from that target – a number sometimes called semivariance. Its square root is the downside deviation.

Downside Deviation (or semideviation): The square root of the average squared shortfalls below a chosen target return. It ignores all outcomes above the target.

Let’s illustrate with the earlier scenario but use a target of 0% (just not losing money). The returns are 15%, 8%, and –5%. The –5% return in the recession is the only one that falls below the target. Its shortfall is –5% – 0% = –5%, squared = 0.0025. Weighting by probability:

• Strong (0.3): above target → contribution 0
• Normal (0.5): above target → contribution 0
• Recession (0.2): shortfall squared 0.0025 →

So semivariance is 0.0005, and downside deviation is , or 2.24%. Compare that to the overall standard deviation of 6.95% – the downside deviation is much smaller because the big upside of 15% doesn’t count as risk. This measure tells you that the real pain from bad years is about 2.24% below zero, not the wider overall swings.

Analysts often choose a target above zero, like the risk‑free rate, to measure the chance of falling short of what they could have earned safely. The calculation works the same way: only returns below the target enter the calculation.

Downside risk aligns with the intuition that you care about the left tail of the return distribution, not the whole spread. It doesn’t replace standard deviation, but it adds a more realistic lens – especially for investments with uneven or skewed returns.

📝 Section Recap: Downside risk measures isolate the returns that fall short of a target, ignoring upside volatility. This gives a risk picture that matches an investor’s real worry: losing money or missing a minimum acceptable return, rather than simply seeing any movement.

Summary#

We began with the simplest measure – the holding period return – that tells you exactly how much money your investment made, for every dollar you put in. From there, we learned to combine multiple years of returns into a meaningful average using the geometric mean, while keeping the arithmetic mean on hand for forward-looking estimates. Then we shifted our thinking to the future, building expected return from probability-weighted scenarios. With that centre in place, we added tools to measure the scatter around it: variance and standard deviation capture the overall volatility, the coefficient of variation scales that volatility by the expected payoff, and downside risk zeroes in on the part that actually hurts. Together, these numbers let you describe any investment’s track record and prospects with clarity, turning gut feelings into a common language of risk and return.