CFA Level 1 · Module 02 Quantitative Methods · Chapter 1
Every investment decision comes down to one comparison: what you expect to earn against what you have to risk to earn it. Before you can make that comparison you need a clean, common way to say how much an investment earned. That is what a return is, and it sounds simpler than it turns out to be. A property, a government bond, a dividend-paying share, and a fund that charges a fee all deliver value in different shapes, so a single loose phrase like “it went up 10 percent” hides more than it reveals.
This chapter builds the vocabulary of returns from the ground up. You will separate the two ways an asset pays you, combine returns across time, and see why the average return most people quote is usually the wrong one. You will then move from the raw return an asset delivers to the return an investor demands before committing capital, and finish by stripping a headline return down to what actually lands in the investor’s pocket after inflation, fees, taxes, and borrowing.
None of the mathematics here is difficult. What trips candidates up is choosing the right measure for the situation, and keeping the definitions straight under time pressure. Treat this as the foundation reading for all of Quantitative Methods, because every later module, from portfolio mathematics to regression, assumes you can speak this language without hesitating.
Start with what actually pays a return. Financial assets are things you hold for the value or claim they represent: cash, an ownership stake in a company (equity), a loan you have made that the borrower must repay (debt), and hybrids that mix these. Financial instruments are the standardized, tradable versions of those claims, packaged so that investors can buy and sell them in a market. A share and a bond are financial assets that trade as financial instruments. Financial indicators, by contrast, are observable numbers such as an interest rate, an exchange rate, or a market index. You can track them, but you cannot hold one directly, and they do not pay cash flows of their own.
The distinction matters because the source of an investment’s return depends on the kind of cash flows it bundles. A non-dividend share pays you only if its price rises. A bond pays a coupon whether or not its price moves. A rental property pays rent and may also appreciate. Returns are the single number that lets you line up these very different payoff patterns side by side and rank them.
Two more pairs of words will recur throughout the chapter, so fix them now. An expected return, also called an ex ante return, is the return you forecast before the outcome is known. An actual return, also called an ex post return, is what the investment delivered after the fact. The gap between the two is the whole reason risk exists, and later in the chapter it becomes the reason investors demand a risk premium. Returns are expressed as decimals, fractions, or percentages that all say the same thing, and in this chapter we use percentages in prose and figures inside every calculation.
An asset can pay you in two ways, and confusing them is the most common early error. The first is a change in price. Buy an asset for one amount and it is worth more later, and that appreciation is the price return. The second is cash the asset hands you while you hold it: a dividend on a share, a coupon on a bond, rent on a property. That cash is the income return, also called the distribution return. The total return is simply the two added together, measured against what you paid.
where R is the total return for the period, P0 is the price paid at the start, P1 is the price at the end, and Inc is the income (dividend, coupon, or other distribution) received during the period. The first part, ( P1 − P0 ) ÷ P0, is the price return; the second part, Inc ÷ P0, is the income return.
Notice that both components are divided by the same starting price. A common slip is to divide the income by the ending price or by an average price. Keep the denominator anchored to what you actually put in at the start, and the two pieces will always add cleanly to the total.
| Component | What it captures | How it is measured |
|---|---|---|
| Price return | Change in the asset’s market value | (End price minus start price) divided by start price |
| Income return | Cash distributions received while holding | Income divided by start price |
| Total return | Everything the investor earned | Price return plus income return |
Setup. Priya buys one share of Northbank Retail for 200 rupees. One year later the share trades at 224 rupees, and during the year it paid a dividend of 6 rupees. Find the price return, the income return, and the total return.
Answer: price return 12%, income return 3%, total return 15%. The dividend added a fifth of the total return here, which is why ignoring income understates performance for any asset that pays cash.
Price return and income return always share the same denominator, the starting price. If your two pieces do not add up to the total return, you have almost certainly changed the denominator halfway through.
The total return you just calculated over a single year is a holding period return, the return earned over the whole time an asset is held. For one period it is exactly the total return formula. The interesting question is what happens when you hold across several periods and want one number for the entire stretch.
You cannot simply add the yearly returns, because each year’s growth builds on the value left standing at the end of the year before. Money compounds, so returns must be linked by multiplication, not addition. To chain-link, turn each period return into a growth factor by adding 1, multiply the factors together, then subtract 1 at the end.
where R is the holding period return over all n periods, and r1, r2, through rn are the returns in each single period. Each ( 1 + r ) is a growth factor; multiplying them compounds the returns, and subtracting 1 at the end converts the final factor back into a return.
Setup. The Trellis Partners Balanced Fund returns 10% in its first year, loses 5% in its second, and gains 20% in its third. What is the holding period return over the full three years?
Answer: 25.4% over three years. Adding the returns (10 − 5 + 20 = 25%) gets close here only because the numbers are small; with larger swings the additive shortcut drifts badly, which is the whole point of chain-linking.
Adding period returns to get a multi-period return. Addition ignores compounding and is only ever a rough approximation. A return of 100% followed by a loss of 50% adds to plus 50%, but chain-linking gives (2.0 × 0.5) − 1 = 0%, which is the truth: you doubled your money and then halved it, ending exactly where you started.
Once you have a series of period returns, you often want a single average that stands in for the typical year. There are two candidates, and choosing the wrong one is one of the most heavily tested traps in this reading.
The arithmetic mean return is the plain average: add the returns and divide by how many there are. It answers the question, “in a single randomly chosen period, what return should I expect?” It is the right tool for forecasting one period ahead.
where r1 through rn are the returns in each of the n periods.
The geometric mean return is the constant per-period return that would compound to the same ending wealth the actual series produced. It answers a different question, “what steady annual rate did this investment actually deliver over the whole period?” It is the right tool for describing realized, historical performance.
where the bracket is the same chained growth factor from the holding period return, and raising it to the power 1 ÷ n takes the nth root to find the equivalent single-period rate.
The geometric mean is always less than or equal to the arithmetic mean, and the gap widens as returns become more volatile. The two are equal only when every period return is identical. That inequality is not a quirk; it is the mathematical signature of volatility eating into compound growth.
Setup. Meridian Alloys returns positive 50% in one year and negative 40% the next. Compute the arithmetic mean and the geometric mean, and reconcile them with what actually happened to 100 rupees invested.
Answer: arithmetic mean 5%, geometric mean negative 5.13%. The arithmetic mean says the investment gained on average, yet the investor is poorer. Only the geometric mean, negative 5.13% a year, matches the fact that 100 rupees became 90.
| Point in time | Value (rupees) | Note |
|---|---|---|
| Start | 100.00 | Amount invested |
| End of year 1 | 150.00 | Up 50% |
| End of year 2 | 90.00 | Down 40% |
| Geometric mean | −5.13% | Compounds 100 to 90 exactly |
| Arithmetic mean | 5.00% | Overstates realized growth |
Use the arithmetic mean to estimate a single future period. Use the geometric mean to describe what already happened across many periods. The more volatile the returns, the more the arithmetic mean flatters the record.
An investment gains 25% in one year and loses 20% the next. What is its geometric mean annual return, and what does that tell you about the investor’s wealth?
Chain the factors: (1.25)(0.80) = 1.00. The geometric mean is (1.00)1÷2 − 1 = 0%. The investor ended with exactly the starting amount, so the true average annual return is zero, even though the arithmetic mean is (25 − 20) ÷ 2 = 2.5%.
Returns are quoted over all kinds of periods: a monthly fund statement, a quarterly bond payment, a daily money-market rate. To compare them you convert each to a common horizon, usually one year. Annualizing again relies on compounding, not multiplication by the number of periods.
The effective annual rate is the return actually earned over a year once compounding within the year is taken into account. Take the periodic return, add 1, raise it to the number of periods in a year, and subtract 1.
where rperiod is the return for one sub-period and m is the number of such sub-periods in a year (12 for months, 4 for quarters, 365 for days).
The same stated annual rate translates into a higher effective rate the more often it compounds, because you earn return on return sooner. This is why a “12 percent” rate is not one number until you know how often it compounds.
| Compounding | Calculation | Effective annual rate |
|---|---|---|
| Annual | (1 + 0.12)1 − 1 | 12.00% |
| Semiannual | (1 + 0.06)2 − 1 | 12.36% |
| Quarterly | (1 + 0.03)4 − 1 | 12.55% |
| Monthly | (1 + 0.01)12 − 1 | 12.68% |
| Daily | (1 + 0.12÷365)365 − 1 | 12.75% |
| Continuous | e0.12 − 1 | 12.75% |
Reading down the monthly row makes the point concrete: a monthly return of 1% compounds to (1.01)12 − 1 = 12.68%, not 12%. That extra 0.68 percentage point is pure compounding, the interest that each month’s credit earns over the rest of the year. A saver told the account “pays 12 percent” is right about the stated rate but understates what actually lands in the account.
Watch the direction of the conversion. If a question gives a monthly or quarterly figure and asks for the annual equivalent, you compound upward. If it gives an annual figure and asks what is earned in a quarter, you take the root, not a simple division. A stated rate and an effective rate are only equal when compounding is annual.
Push the compounding frequency to its limit, so that return is credited not monthly or daily but at every instant, and you reach continuous compounding. The effective annual rate stops rising and settles at a ceiling given by the exponential function. A continuously compounded return is also called a log return, because you recover it by taking the natural logarithm of the growth factor.
where rcc is the continuously compounded (log) return, R is the ordinary holding period return, ln is the natural logarithm, and e is the base of the natural logarithm (about 2.71828).
Log returns have one property that makes them beloved in quantitative work: they add across time. The continuously compounded return for a year is just the sum of the continuously compounded returns of its parts, with no chain-linking required. That additivity is why later modules on statistics and modeling lean on log returns rather than ordinary ones.
Setup. Over the first half of a year an index has a continuously compounded return of 8%, and over the second half it has a continuously compounded return of 6%. Find the continuously compounded return for the full year, and convert it to an ordinary holding period return.
Answer: a 14% continuously compounded return for the year, equal to a 15.03% ordinary return. The ordinary return is higher because it expresses the same growth after the compounding has been rolled up into a single period.
A stock rises from 100 to 110 over one year. What is its continuously compounded return?
The ordinary holding period return is (110 − 100) ÷ 100 = 10%. The continuously compounded return is ln(1 + 0.10) = ln(1.10) = 0.0953, or 9.53%. The log return is always a little below the ordinary return for a positive gain.
So far every return has been backward looking or mechanical: what an asset paid, and how to average or annualize it. Now turn the question around. Before an investor commits money, what return must an asset promise to be worth buying? That threshold is the required rate of return, and it is built up from the compensation an investor needs for two separate things: giving up the money for a while, and bearing the risk that the outcome disappoints.
The starting block is the risk-free rate, the return on an investment with no default risk, usually taken from short-term government debt. The risk-free rate itself has two parts: the real risk-free rate, which is pure compensation for delaying consumption, and an allowance for expected inflation, which protects the money’s purchasing power. On top of the risk-free rate the investor adds a risk premium for the specific uncertainties of the asset.
where rrequired is the required nominal return, rreal is the real risk-free rate, πexpected is expected inflation, and RP is the risk premium for the asset. The first two terms together make up the nominal risk-free rate. This additive form is an approximation; the exact version multiplies the growth factors (1 + rreal)(1 + π)(1 + RP) and subtracts 1.
| Building block | Compensates the investor for | Rate |
|---|---|---|
| Real risk-free rate | Delaying consumption | 2% |
| Expected inflation | Loss of purchasing power | 4% |
| Nominal risk-free rate | Time value, protected against inflation | 6% |
| Risk premium | Uncertainty of this asset’s payoff | 5% |
| Required return | Everything above | 11% |
Setup. Anand is pricing the shares of Corvus Logistics. The real risk-free rate is 2%, expected inflation is 4%, and he judges the equity risk premium for this company to be 5%. What return should he require before buying?
Answer: about 11% required return, or 11.38% computed exactly. If Corvus shares are priced to return less than this, Anand should pass; the asset does not pay him enough for the time and risk involved.
The risk premium is not a single fixed number. It is the sum of several distinct compensations, each tied to a particular kind of risk an asset carries. Naming them matters, because exam questions often ask which premium a described feature refers to.
An equity risk premium compensates for the extra volatility of owning shares rather than lending. A credit or default premium compensates a lender for the chance the borrower fails to pay. A liquidity premium compensates for an asset that is hard to sell quickly without cutting the price. A maturity premium compensates a bondholder for the added price sensitivity of lending for longer. Stack these onto the risk-free rate and you have the required return for almost any asset.
Closely related is the idea of an excess return, the amount by which an asset’s return exceeds the risk-free rate over the same period. Where the risk premium is what an investor demands in advance, the excess return is what actually showed up after the fact. The two line up in expectation but rarely match in any single realized period, which is exactly the ex ante versus ex post gap from the opening section.
A required return is a sum of parts: a real rate, an inflation allowance, and one or more risk premia. When a question changes a single feature of an asset, such as making it harder to sell, it is usually testing whether you can name the one premium that should rise.
A return figure quoted in ordinary money is a nominal return. It tells you how many more rupees or dollars you have, but not whether you can buy more with them. Once prices in the economy are rising, some of every nominal gain is swallowed just keeping pace with inflation. The return that reflects actual buying power is the real return, and you find it by removing inflation, again through division rather than subtraction.
where rreal is the real return, rnominal is the nominal return, and π is the inflation rate over the same period. Subtracting inflation from the nominal return (rnominal − π) is a quick approximation that drifts once the rates get large.
Setup. Farah’s bond portfolio earned a nominal return of 12% over a year in which inflation ran at 5%. What did she really earn in purchasing power?
Answer: a real return of 6.67%, not the 7% the quick subtraction (12 − 5) suggests. Farah’s spending power grew by two-thirds of the headline number; inflation quietly claimed the rest.
Treating “nominal minus inflation” as the exact real return. The subtraction is a serviceable estimate only when both rates are small. At higher inflation the error grows, and the exam will sometimes offer the subtraction answer as a tempting wrong option sitting right next to the correct division answer.
The return an investment generates before any costs are deducted is the gross return. The return left after the costs of running the investment, such as management fees, administrative charges, and trading commissions, is the net return. The gap between them is money that never reaches the investor, and over long horizons even a modest annual fee compounds into a large slice of final wealth.
Two points are worth holding onto. First, taxes are usually treated separately from the fee deduction, so a “net return” typically still sits before tax unless a question says otherwise. Second, because fees compound just as returns do, a fund that beats a rival on gross return can trail it on net return once a higher fee is applied every year. When you compare funds, compare them net of the same category of costs.
Gross return measures the manager’s raw result; net return measures what the client keeps. Performance claims quoted gross of fees flatter the manager, so always confirm which one you are looking at before ranking funds.
The return before the tax authority takes its share is the pre-tax return. The after-tax return is what remains once tax on the investment’s income and gains is paid. The wrinkle that makes this more than a single subtraction is that different sources of return are often taxed at different rates. Capital gains from price appreciation and income such as dividends or coupons can face separate tax rates, so you must tax each component on its own before adding them back together.
where rprice is the price (capital gain) return, rincome is the income return, tgains is the tax rate on capital gains, and tincome is the tax rate on income distributions.
Setup. Rohit holds shares in Saraswati Textiles. Over the year the price return was 20% and the dividend (income) return was 4%. Capital gains are taxed at 15% and dividend income at 30%. What is his after-tax total return?
Answer: an after-tax return of 19.8%, down from a pre-tax total of 24%. Because the dividend is taxed twice as hard as the gain here, income-heavy investments lose more of their return to tax, which is a real consideration when choosing between them.
Applying one blended tax rate to the whole return. If gains and income are taxed differently, taxing the combined 24% at a single rate gives the wrong answer. Split the return, tax each part at its own rate, then recombine.
An investor earns a 10% price return and a 5% income return. Capital gains are taxed at 20% and income at 40%. What is the after-tax total return?
Price: 10% × (1 − 0.20) = 8.0%. Income: 5% × (1 − 0.40) = 3.0%. After-tax total = 8.0% + 3.0% = 11.0%, compared with a 15% pre-tax total.
Everything so far assumed the investor paid for the asset in full. Borrowing part of the purchase price changes the arithmetic sharply. A leveraged return is the return on the investor’s own money when some of the position is funded with debt. Leverage lifts the return when the asset does well, because the gains accrue on the whole position but the investor only funded part of it. It cuts just as hard the other way, because losses and the cost of the borrowing both fall on the smaller equity stake.
where rleveraged is the return on the investor’s equity, rasset is the return on the underlying asset, rdebt is the borrowing rate, D is the amount borrowed, and E is the investor’s own equity. D ÷ E is the debt-to-equity ratio.
Setup. Blackwater Capital buys a position by putting in 40% of the cost as equity and borrowing the other 60% at an interest rate of 8%. Compare the return on its own money when the asset returns positive 12% against when the asset returns negative 12%.
Answer: the leveraged equity returns positive 18% on the way up and negative 42% on the way down, against the asset’s own swing of plus or minus 12%. Leverage did not just scale the gain; it made the loss far larger than the gain, because the borrowing cost is owed whichever way the asset moves.
| Scenario | Unleveraged return | Leveraged return (D÷E = 1.5) |
|---|---|---|
| Asset rises | +12% | +18% |
| Asset falls | −12% | −42% |
Leverage questions reward a clear head about signs. The borrowing cost is subtracted in every scenario, so on the downside the loss on the asset and the interest owed both work against the shrunken equity base. If your leveraged loss is not larger in magnitude than your leveraged gain for a symmetric asset move, recheck the interest term.
An investor funds half a position with borrowed money at 6% (so debt equals equity). The asset returns 10%. What is the leveraged return on the investor’s equity?
With debt equal to equity, D ÷ E = 1. Leveraged return = 10% + (10% − 6%) × 1 = 10% + 4% = 14%. The 4 percentage point spread between the asset return and the borrowing cost is added once, because the investor borrowed an amount equal to their own stake.
A share is bought at 80, pays a dividend of 4 during the year, and is sold at 88. What is the total return, split into its two components?
Price return = (88 − 80) ÷ 80 = 8 ÷ 80 = 10%. Income return = 4 ÷ 80 = 5%. Total return = 10% + 5% = 15%.
A return measures the total change in value over a period, including price movement and income. A yield usually refers to income alone relative to price, such as a dividend yield or a bond’s coupon yield. Income return in this chapter is close to a yield, but total return adds the price change on top.
Use the arithmetic mean when you want the expected return for a single future period. Use the geometric mean when you want the compound annual rate an investment actually delivered across several periods. For any series with variation the geometric mean is lower, and quoting the arithmetic mean as historical performance overstates the record.
Because a continuously compounded return is the natural logarithm of a growth factor, and logarithms turn multiplication into addition. Chain-linking multiplies ordinary growth factors, so their logs, the continuous returns, add. That additivity is the main reason quantitative models favor log returns.
It is treated as free of default risk, which is why a short-term government instrument is used as the proxy. It is not free of inflation risk or reinvestment risk, and that is precisely why the risk-free rate is split into a real rate plus an allowance for expected inflation.
Because inflation erodes purchasing power. If prices rise 5% while your money grows 12%, part of that 12% only keeps you level with rising prices. Dividing out inflation leaves the real return, which here is about 6.67%, the true growth in what you can buy.
No. Leverage increases the return only when the asset earns more than the borrowing cost. When the asset earns less, or falls, leverage magnifies the shortfall, and the interest is still owed. That is why a symmetric move in the asset produces a larger leveraged loss than leveraged gain.
It depends on the source, which is exactly why the terms matter. Gross return is before costs, net return is after fees, and after-tax return is after tax as well. Always confirm which basis a figure uses before comparing two investments, because the differences compound over time.
Mostly through short calculations that hinge on choosing the correct measure: geometric versus arithmetic mean, compounding upward versus taking a root, dividing out inflation versus subtracting it, and taxing return components separately. The arithmetic is light; the marks are won by picking the right method under time pressure.
Loading comments...
Add your Thoughts: