How do the three CFA Program levels build on one another?

Level I establishes the terms, calculations, and concepts used in investment work. Level II asks candidates to analyze and evaluate situations presented in vignettes. Level III emphasizes integrating information and applying judgment, including constructed responses and a specialized pathway. A Level I valuation formula is therefore a starting point: later decisions may require reconciling its assumptions with client needs and competing evidence.

Why does the accounting basis matter when analyzing a company?

Recognition and measurement rules affect reported assets, profit, and ratios even when the underlying business is the same. In Level I Financial Statement Analysis, use IFRS unless a question explicitly specifies U.S. GAAP. First identify the transaction and applicable basis, then trace its effects through the statements before interpreting a ratio.

How should a candidate connect valuation with portfolio risk?

Valuation estimates what an asset is worth under stated cash flow and discount rate assumptions. Portfolio analysis asks how holding that asset changes the investor's total risk and expected return. A stock can appear undervalued on its own yet add concentrated exposure to an existing portfolio. Compare the valuation case with the asset's risk, liquidity, and fit with the investor's objectives.

CFA Program Investment Analysis Study Guide

Start with concept 1 Choose a topic

The CFA Program has three levels that build from investment foundations toward analysis and integrated decisions. This guide develops the 2026 Level I topics through sixty focused lessons. You will trace how cash flows become financial statements, how discount rates turn future payments into present values, and how uncertainty changes an investment decision. The examples also connect subjects that are easy to study in isolation: accounting choices affect valuation, interest rates affect bonds and derivatives, and an attractive return may still be unsuitable for a particular client. Ethics runs through those decisions, from using information responsibly to communicating results fairly. Work through each calculation, then ask what the result means for an investor, issuer, or portfolio.

Level I - Ethical and Professional Standards

1. An ethical decision sequence

An ethical decision sequence begins with the people affected: the client, other clients, the employer, and the analyst. Establish the relevant facts before deciding what to recommend. Next, identify the duties involved, including honest communication, reasonable care, and placing the client’s interests ahead of a benefit to the firm. A payment tied to a recommendation creates a conflict because it gives the firm a reason to favor one product. Clear disclosure lets the client understand that incentive, but it does not make a poor recommendation suitable. Compare the product’s costs, risks, liquidity, and expected role with the client’s objectives and alternatives. Document the basis for the decision and seek guidance when a duty remains unclear. The Level I topic outline places ethical decision making within this subject.

Worked example: An analyst learns that recommending Fund A earns the firm an additional fee. She tells the client the fee’s amount and how it arises, then compares Fund A with suitable alternatives. If the client needs low costs and ready access to cash, and Fund A has higher costs and restricted withdrawals, she recommends a better fitting option. Disclosure alone would not justify Fund A.

Watch the distinction: A common error is to treat disclosure as permission to recommend the product. The analyst must still show that the recommendation serves the client’s needs.

Reference: [1]

2. Material nonpublic information

Information is material when a reasonable investor would consider it important to a decision or when its disclosure could substantially affect a security’s price. Information is nonpublic when it has not been made available to the investing public through a channel that gives investors a fair opportunity to receive it. A private conversation with an officer can therefore differ sharply from analysis of published accounts, industry data, and public interviews. When both conditions hold, the analyst should stop trading or recommending trades based on the information, avoid passing it to someone who might trade, and follow the firm’s compliance process. The analyst can continue independent work using public information, provided the confidential fact does not influence the conclusion. Record what was received, from whom, and when so the response can be assessed.

Worked example: A company officer privately tells an analyst that earnings will miss expectations because a major customer canceled orders. The officer says the announcement is next week. The likely earnings effect makes the information potentially material, and the private disclosure makes it nonpublic. The analyst does not sell the shares or alert clients to sell. She records the conversation and contacts compliance for guidance.

Watch the distinction: An officer’s permission to hear a fact does not make it public. Telling a colleague to trade would transfer the same problem to that colleague.

Reference: [1]

3. Independence under pressure

Independent analysis requires an analyst to judge evidence without giving special weight to an issuer’s hospitality, access, or threats. The practical question is whether a benefit or pressure could reasonably appear to influence the recommendation, even if the analyst believes it will not. Consider its value, timing, source, and connection to a pending decision. A costly benefit offered immediately before a rating review presents a stronger concern than ordinary information supplied equally to analysts. Keep a record of the offer, decline benefits that threaten objectivity, and report the situation through the firm’s process. Then evaluate the issuer using the same financial evidence and valuation assumptions that would apply without the offer. Employer pressure also calls for a documented analytical basis, rather than a rating selected to please a business partner.

Worked example: An issuer offers to pay for an analyst’s luxury trip two weeks before its rating review. The analyst declines and reports the offer. She updates forecasts from filings, public management statements, and market data, recording her assumptions. If the evidence supports a lower rating, she issues it with its rationale. The trip’s rejection helps protect the review from an apparent exchange of favorable coverage for hospitality.

Watch the distinction: Believing you can personally resist influence is insufficient. A costly, well timed issuer benefit can undermine confidence in the analysis even if the rating remains unchanged.

Reference: [1]

4. Matching advice to a client

Suitability begins with a current picture of the client, not the attractiveness of a product in isolation. Identify the investment objective, required return, ability and willingness to take risk, time horizon, liquidity needs, tax circumstances, and any other relevant constraints. Then examine existing holdings: a new position may concentrate a risk the client already owns. Separate money needed soon from capital that can remain invested through losses or a long sale process. For each proposed investment, ask what role it serves, how it could fail, and whether the client could still meet obligations if it does. A recommendation can be unsuitable even when its expected return looks high. Revisit the analysis when a home purchase, retirement date, income source, or other material client circumstance changes.

Worked example: A client has $90,000 in liquid savings and must pay $80,000 for a home in six months. Investing all $90,000 in a volatile private fund leaves no dependable source for the payment. Even reserving $80,000 leaves only $10,000 for other needs. The analyst first protects the near term cash requirement, then evaluates whether any remaining amount fits the client’s broader portfolio and risk capacity.

Watch the distinction: A long term expected return does not solve a six month cash need. Suitability depends on when the client needs money and how reliably it can be accessed.

Reference: [1]

5. Fair treatment in trading

Fair treatment means using a defensible process for clients with comparable eligibility and instructions. Before placing a trade, establish which accounts may participate and how a partial fill will be allocated. Apply that method consistently and keep records of the order, fill, and allocation. Proportional allocation is one reasonable method when accounts request different quantities and the security can be divided into whole shares without a material distortion. Other constraints, such as account mandates or minimum trading sizes, may require a different documented method. An employee account adds a competing personal interest: the employee should not gain an advantage from access to client orders. Fairness does not require identical holdings for every client. It requires that differences arise from relevant client circumstances and a consistently applied process, not favoritism.

Worked example: Client A requests 60 shares and Client B requests 40, for 100 total. Only 50 shares are filled. Under a stated proportional method, A receives 50 × 60/100 = 30 shares and B receives 50 × 40/100 = 20 shares. Both receive half their requests. Giving A all 50 because its account is larger would depart from that method; an employee account should not jump ahead.

Watch the distinction: Equal treatment does not always mean equal share counts. When requests differ, a documented proportional allocation can be fairer than splitting the fill evenly.

Reference: [1]

6. Evidence behind performance claims

Performance evidence should let a reader understand what was earned, by whom, over what period, and with what risk and costs. A single successful account may be real yet unrepresentative of a strategy if similar accounts performed poorly. Check how portfolios were selected, whether returns use comparable periods and methods, whether fees are reflected, and whether a benchmark is appropriate. Explain any material differences rather than implying that one result describes all clients. The Global Investment Performance Standards are included in the Level I ethics scope; citing them in marketing requires support for the specific compliance claim. A return figure by itself provides no such support. A defensible statement describes the selected portfolio accurately and supplies the broader context needed to judge the strategy.

Worked example: A manager advertises an 18% return from one account while three comparable accounts lost money over the same year. The 18% may describe that account, but it cannot stand for the whole strategy. The manager should present the relevant comparable results using consistent periods and return methods. Claiming GIPS compliance would also require evidence that the firm’s applicable presentation and compliance requirements were met.

Watch the distinction: A true number can still mislead when the selection hides other results. One winning account neither establishes typical performance nor proves compliance with a performance standard.

Reference: [1]

Level I - Quantitative Methods

7. The time value of money

The time value of money converts payments at different dates into comparable values. A dollar available now can be invested, so a promised future dollar is worth less today when the required return is positive. For one payment, present value equals future value divided by one plus the discount rate raised to the number of periods: PV = FV/(1 + r)^n. Match the rate’s period to the timing of the cash flow. An annual rate with a payment one year away uses one annual period; a different payment date needs a consistent rate and exponent. The required return reflects the return demanded for the cash flow’s risk and opportunity cost. A higher rate lowers present value because the future payment must compensate for a greater required growth rate. The Level I study guide can help place this calculation among the other foundational quantitative topics.

Worked example: Suppose $110 is received exactly one year from now and the hypothetical annual required return is 10%. PV = $110/1.10 = $100. Investing $100 today at 10% would produce $110 after one year, so the two amounts are equivalent under that assumption. At a higher required return, the same future $110 would have a lower present value.

Watch the distinction: Do not divide by the stated rate alone or mix monthly periods with an annual rate. Timing and rate must use consistent units.

Reference: [1]

8. Arithmetic and geometric returns

The arithmetic mean adds period returns and divides by the number of periods. It describes the average return observed in one period and is often useful when estimating a single future period under suitable assumptions. The geometric mean measures the constant return that would produce the same cumulative growth over the full holding period. To calculate it, multiply each period’s growth factor, take the appropriate root, and subtract one. Losses matter because they shrink the base on which later gains compound. A 10% gain followed by a 10% loss does not restore the starting value: 1.10 × 0.90 = 0.99. When comparing historical multi-period investor outcomes, use the geometric rate and state the period length. Neither average predicts a future result without assumptions about the return process.

Worked example: A $100 investment earns 10% in year one and loses 5% in year two. Its ending value is $100 × 1.10 × 0.95 = $104.50. The arithmetic mean is (10% − 5%)/2 = 2.5% per year. The geometric annual rate is √1.045 − 1 ≈ 2.22%, because $100 × 1.0222² is approximately $104.50.

Watch the distinction: The arithmetic mean of 2.5% is not the annual compounded growth rate. Applying it twice would overstate the investment’s actual ending value.

Reference: [1]

9. Dispersion in return data

Dispersion tells you how widely observations lie around a chosen center. Start with the sample mean, subtract it from each return, square the deviations, and add them. Sample variance divides that sum by n − 1 because the sample mean was estimated from the same observations. The square root is the sample standard deviation. Variance carries squared return units, which are awkward to interpret; standard deviation returns to the original units, such as percentage points. A larger standard deviation signals more variation in the observations, but it does not tell you whether outcomes are skewed or whether losses are unusually severe. Check whether returns refer to the same interval before comparing dispersion across investments. CFA practice questions are useful for checking both the calculation and unit interpretation.

Worked example: Three hypothetical annual returns are 4%, 6%, and 8%. Their mean is 6%, and deviations are −2, 0, and 2 percentage points. The squared deviations sum to 4 + 0 + 4 = 8 percentage points squared. Sample variance is 8/(3 − 1) = 4 percentage points squared; sample standard deviation is √4 = 2 percentage points.

Watch the distinction: Dividing by three produces the population variance for these three observations, not the sample variance. Also, standard deviation is measured in percentage points, not squared units.

Reference: [1]

10. Conditional probability

A signal changes an event probability only after you account for both genuine detections and false alerts. Let D mean default and F mean a flag. Among all borrowers, the proportion who both default and receive a flag is P(D)P(F|D). The proportion flagged without default is P(not D)P(F|not D). Divide the first quantity by the sum to obtain P(D|F). This is Bayes’ rule expressed as a count of two paths into the flagged group. A signal can detect most defaults yet still have a modest positive predictive value when defaults are rare, because the larger nondefault group can produce many false flags. Always identify the base rate, the detection rate, and the false alarm rate before interpreting a flag.

Worked example: In a hypothetical group of 1,000 borrowers, 20% or 200 default. A flag catches 80% of them, so 160 defaulters are flagged. Of the 800 nondefaulters, 10% or 80 are falsely flagged. There are 240 flagged borrowers in total, and 160/240 = 66.7% default. That is the default probability conditional on a flag under these assumptions.

Watch the distinction: An 80% detection rate is P(flag|default), not P(default|flag). The latter also depends on how common default is and how often nondefaulters are flagged.

Reference: [1]

11. Testing a population mean

A hypothesis test asks whether sample evidence is sufficiently inconsistent with a stated population claim under a specified model. First state the null value and whether the alternative is one sided or two sided. Then compute the sample estimate’s distance from the null in standard error units. For a mean with known population standard deviation σ, the z statistic is (sample mean − null mean)/(σ/√n), assuming the sampling model is appropriate. Compare the statistic with the critical value given for the chosen significance level and test direction. Rejection says the observed result would be unusual under the null model; it does not prove the alternative with certainty. Failure to reject also does not prove the null. Consider sample design, independence, measurement quality, and practical effect size alongside the test decision.

Worked example: Test a hypothetical null mean of 50 against a two sided alternative. The sample mean is 52, known population standard deviation is 10, and n = 100. The standard error is 10/√100 = 1, so z = (52 − 50)/1 = 2. With a stated 5% two sided critical value of ±1.96, 2 exceeds 1.96. Reject the null under the stated model.

Watch the distinction: Rejecting the null does not mean the true mean is certainly 52. The test evaluates evidence against 50 using sampling assumptions and a preset decision rule.

Reference: [1]

12. Reading a regression model

A simple estimated regression, ŷ = a + bx, summarizes how the model’s prediction of y changes with x. The intercept a is the predicted y when x equals zero; that interpretation is useful only if zero lies in a meaningful range. The slope b is the predicted change in y for a one unit increase in x. Keep units visible: if x is measured in percentage points and y in dollars, b is dollars per percentage point. A prediction within the observed range is generally less exposed to extrapolation than one far outside it. A fitted relationship may reflect confounding factors, reverse causation, or chance. To judge usefulness, examine residual patterns, uncertainty around estimates, and whether the relationship remains credible outside the sample. The equation alone establishes association, not a causal mechanism.

Worked example: An estimated model is ŷ = 2 + 1.5x, with x and y measured in stated units. At x = 4, ŷ = 2 + 1.5 × 4 = 8 units of y. Raising x from 4 to 5 raises the predicted y from 8 to 9.5, a 1.5 unit change. This does not show that deliberately changing x would cause y to move.

Watch the distinction: A positive fitted slope does not prove causation. Another variable may influence both x and y, or the observed association may be unstable.

Reference: [1]

Level I - Economics

13. Price elasticity and revenue

Price elasticity of demand measures the percentage change in quantity demanded divided by the percentage change in price, holding other relevant influences fixed. Its sign is usually negative: a higher price tends to reduce quantity demanded. When the absolute elasticity exceeds one, quantity changes by a larger percentage than price; a small price increase then tends to reduce revenue. When it is below one, the price effect tends to dominate. Revenue itself is price multiplied by units sold, so calculate both before and after values when actual data are supplied. State the percentage change convention. Using initial values and using midpoint values can give different elasticity estimates for the same finite move. A revenue change by itself does not establish the profit change, because costs may also respond.

Worked example: A hypothetical seller raises price from $10 to $10.50 and sales fall from 100 to 90 units. Using initial values, price rises $0.50/$10 = 5%, while quantity changes by −10/100 = −10%. Elasticity is −10%/5% = −2. Revenue falls from $10 × 100 = $1,000 to $10.50 × 90 = $945, a $55 decline.

Watch the distinction: A higher price does not always mean higher revenue. With elastic demand, the percentage loss of units can outweigh the percentage gain in price.

Reference: [1]

14. The firm's output decision

A firm increases profit by producing an additional unit when that unit’s marginal revenue exceeds its marginal cost, assuming the unit can be sold and other conditions remain unchanged. Marginal revenue is the extra total revenue from the unit; marginal cost is the extra total cost. Their difference is the incremental profit. Continue comparing successive feasible units until the next one would lower profit. In a smooth model, the familiar stopping point is where marginal revenue and marginal cost meet. For discrete units, compare the actual increments instead of forcing an equality. Fixed costs already incurred do not change the incremental decision, though avoidable fixed costs may matter when choosing whether to operate at all. Check that capacity, demand, and price effects from added output are reflected in the supplied marginal figures.

Worked example: The next unit would bring $12 of additional revenue and $9 of additional cost, adding $3 to profit. The following unit would bring $8 but cost $10, reducing profit by $2. Assuming these are consecutive feasible units and no other effects arise, produce the first extra unit and stop before the second. Producing both would add only $1 overall.

Watch the distinction: Comparing the selling price with average cost can miss the decision. The next unit should be assessed using its extra revenue and extra cost.

Reference: [1]

15. Measuring aggregate output

The expenditure approach measures gross domestic product as final consumption, investment, government purchases of goods and services, and net exports: GDP = C + I + G + (X − M). Exports add domestic production sold abroad; imports are subtracted because they may already appear in domestic spending categories but were produced elsewhere. A negative net export figure lowers the expenditure total. Keep the accounting period and currency units consistent across components. Investment here means spending on newly produced capital goods, structures, and inventory changes, rather than a household’s purchase of an existing share. Government transfers are excluded from government purchases because a transfer alone buys no current output. GDP records current domestic production; it is not a direct measure of household welfare or the distribution of income.

Worked example: Suppose hypothetical annual figures, all in billions of currency units, are consumption 500, investment 120, government purchases 180, and net exports −20. GDP = 500 + 120 + 180 − 20 = 780 billion. The negative net exports term reduces the expenditure measure by 20 billion relative to a zero trade balance, assuming the other components stay fixed.

Watch the distinction: Do not add imports as another source of domestic output. Imports are subtracted within net exports to remove foreign production included in spending.

Reference: [1]

16. Aggregate demand and fiscal effects

In a simplified fixed-price expenditure model, an initial increase in government purchases becomes income for someone else. If households spend a fraction of each extra unit of income, that spending creates another round of income and consumption. With marginal propensity to consume c, the successive rounds form 1 + c + c² + …, giving a government spending multiplier of 1/(1 − c) when 0 ≤ c < 1. Multiply the initial spending change by this factor to obtain the modeled output change. The result relies on assumptions such as unused capacity, fixed prices, constant c, and no offset from taxes, imports, interest rates, or displaced private spending. In an economy with supply constraints or monetary responses, actual output may rise by less while prices or rates change.

Worked example: Suppose c = 0.8 in a hypothetical fixed-price model with no crowding out, tax response, or import leakage. The multiplier is 1/(1 − 0.8) = 5. An additional 10 billion currency units of government purchases therefore raises modeled output by 10 × 5 = 50 billion. That is a model result under the stated assumptions, not an assured real world increase.

Watch the distinction: The multiplier is not automatically the observed effect of a policy. Price changes, capacity limits, imports, and private spending responses can weaken the simple chain.

Reference: [1]

17. Reading an exchange-rate quote

Read an exchange quote by naming the units on both sides. A quote of domestic currency units per one foreign currency unit tells you how many domestic units buy one foreign unit. If the number rises, the foreign currency has appreciated against the domestic currency: it now costs more domestic currency to buy the same foreign amount. Equivalently, the domestic currency has depreciated against the foreign currency. For a foreign price held constant, multiply that price by the quote to obtain its domestic cost. Reverse the quote only when you need foreign units per domestic unit. Percentage changes in a quote and its reciprocal do not generally have equal magnitudes, so state which currency and quote you use when reporting appreciation. Real trade costs may also depend on fees and changes in the foreign price.

Worked example: A foreign currency rises from 8.0 to 8.4 domestic units per foreign unit. Its domestic price rises (8.4 − 8.0)/8.0 = 5%. A purchase priced at 100 foreign units costs 100 × 8.0 = 800 domestic units before the move and 100 × 8.4 = 840 afterward. The foreign currency appreciated against the domestic currency.

Watch the distinction: A rising domestic-per-foreign quote signals foreign appreciation, not domestic appreciation. Always read the quote’s denominator before naming the stronger currency.

Reference: [1]

18. Growth and the business cycle

Potential output is an estimate of what an economy can produce sustainably with its available labor, capital, technology, and normal resource use. Actual output can move around that capacity as spending, credit conditions, inventories, and other short-run forces change. An output gap compares actual with potential output, usually as (actual − potential)/potential. A negative gap means production is below the estimated sustainable level; a positive gap means it exceeds that estimate. Potential output itself can grow through more productive labor, capital accumulation, and technological improvement. It is an estimate, so revisions to data or methods can change the measured gap. A negative gap suggests slack but does not, on its own, reveal whether weak demand, a supply disruption, or measurement error produced the observed output.

Worked example: Suppose potential annual output is estimated at 1,000 billion currency units and actual output is 980 billion. The gap is (980 − 1,000)/1,000 = −0.02, or −2%. Actual production is 20 billion below estimated potential. The figure describes the economy’s position relative to estimated capacity; additional evidence is needed to identify the cause.

Watch the distinction: Below-potential output does not mean the economy’s productive capacity has fallen by the same amount. Actual activity and longer-run capacity are distinct concepts.

Reference: [1]

Level I - Financial Statement Analysis

19. How the statements connect

The balance sheet obeys assets = liabilities + equity at a point in time. The income statement measures profit over a period, and profit generally increases retained earnings within equity unless offset by distributions or other equity movements. Cash flow reporting explains cash received and paid during that period, which can differ from profit because some revenues and expenses are noncash or occur at different times from cash settlement. Trace each transaction through all three views. Cash earnings increase an asset and equity; borrowing increases an asset and a liability; a dividend paid reduces cash and retained earnings without reducing profit for the period. For Level I financial statement analysis, IFRS is the stated default unless U.S. GAAP is specified. Use the accounting basis given in a problem before comparing reported figures.

Worked example: A firm begins with assets of 500, liabilities of 300, and equity of 200, all in the same currency units. It earns 20 entirely in cash and pays no dividend. Cash and total assets rise to 520; profit raises retained earnings and total equity to 220. Liabilities remain 300, so the equation checks: 520 = 300 + 220.

Watch the distinction: Profit is not automatically the same as cash flow. This example makes them coincide only because the stated profit is earned entirely in cash.

Reference: [1]

20. Revenue before cash collection

Accrual accounting records revenue when the relevant sale is earned under the applicable recognition conditions, even if the customer pays later. On a credit sale, the seller records a receivable rather than cash. It also removes the sold inventory and records its cost as an expense, so gross profit equals revenue less cost of sales. The balance sheet reflects both changes: the new receivable increases assets, while the inventory reduction decreases them; the net increase matches profit if there are no other effects. When the customer later pays, cash replaces the receivable without creating revenue a second time. For analysis, compare receivables and cash collections with reported sales. Rapid revenue growth accompanied by slow collections can signal credit risk or changing payment terms, although it can also reflect normal growth.

Worked example: A firm sells inventory costing 60 currency units for 100 on credit, with recognition conditions assumed met and no other costs. It records revenue of 100 and cost of sales of 60, giving profit of 40. Receivables rise by 100, inventory falls by 60, and net assets rise by 40. Cash collected at the sale date is zero; later payment converts the receivable into cash.

Watch the distinction: A credit sale is not a cash receipt. Counting the receivable as cash would overstate liquidity, while recording revenue again on collection would double count the sale.

Reference: [1]

21. Inventory cost formulas

Inventory cost formulas assign purchase costs to units sold and units remaining. Under first in, first out (FIFO), the oldest costs enter cost of sales first; ending inventory contains the more recent costs. Under periodic weighted average, divide total cost of goods available by total units available, then apply that cost per unit to sales and closing inventory. Both methods allocate the same total cost, but rising purchase prices usually produce lower cost of sales and higher gross profit under FIFO. The difference also changes the carrying amount of inventory, so compare firms using consistent methods before interpreting margins. Check that cost of sales plus ending inventory equals total available cost. The CFA Institute Level I topic outline places alternative accounting methods within Financial Statement Analysis; it specifies IFRS as the default basis for its questions.

Worked example: A firm buys 10 units at $5 and 10 at $7, then sells 12 at $10. Sales are $120 and available cost is $120. FIFO cost of sales is 10 × $5 + 2 × $7 = $64; ending inventory is 8 × $7 = $56, and gross profit is $56. Periodic average cost is $120 ÷ 20 = $6 per unit. Cost of sales is $72, ending inventory $48, and gross profit $48. Each method allocates the full $120.

Watch the distinction: A higher FIFO profit during rising prices does not mean the firm collected more cash. The formulas change which purchase costs appear in profit and closing inventory, not the sale price or cash receipt.

Reference: [1]

22. Depreciation and carrying amount

Depreciation spreads the depreciable amount of a long lived asset across periods expected to benefit from its use. For straight line depreciation, subtract estimated residual value from cost, then divide by useful life. Each period's charge reduces reported profit and the asset's carrying amount through accumulated depreciation; the charge itself is not a payment made that period. Residual value is the estimated value left at the end of useful life, not an amount depreciated. To read a balance sheet, distinguish original cost, accumulated depreciation, and net carrying amount. Review changes in estimates because a revised useful life or residual value changes future charges. Carrying amount measures unallocated accounting cost under this model; it is not necessarily a current sale price. An impairment assessment may reduce it further if applicable.

Worked example: Equipment costs $100, has an estimated $10 residual value, and a hypothetical three year useful life. Its depreciable amount is $100 − $10 = $90, so straight line depreciation is $30 per year. After year one, accumulated depreciation is $30 and carrying amount is $100 − $30 = $70. After three years, accumulated depreciation totals $90 and carrying amount is $10, assuming no estimate change or impairment.

Watch the distinction: Do not divide the entire $100 cost by three. That would depreciate the $10 residual value as well and leave a carrying amount of zero despite the stated estimate.

Reference: [1]

23. Temporary tax differences

A temporary tax difference arises when financial reporting and tax rules assign different amounts to the same asset or liability, with the difference expected to reverse. For an asset, carrying amount above tax base generally means recovery of the asset will produce more taxable income than accounting income in future periods. The resulting taxable temporary difference gives rise to a deferred tax liability, measured here using the assumed rate expected to apply on reversal. First identify the item, then compare carrying amount with tax base, determine whether the difference is taxable or deductible, and multiply by the relevant rate. A deferred tax liability reflects a future tax consequence rather than a current tax bill. Its measurement depends on the applicable accounting rules and tax assumptions. This simple exercise sets aside exceptions and changes in tax rates.

Worked example: Suppose equipment has an accounting carrying amount of $80 and a tax base of $60. The taxable temporary difference is $80 − $60 = $20. At an assumed 25% rate, the deferred tax liability is $20 × 0.25 = $5. If recovery follows these assumptions, future taxable amounts exceed the related accounting amounts by $20. The $5 is a deferred amount, not tax paid today.

Watch the distinction: A lower tax base does not automatically indicate a deferred tax asset. For an asset whose carrying amount exceeds its tax base, future recovery generally creates taxable amounts and a deferred tax liability.

Reference: [1]

24. Cash flow versus reported profit

Profit follows accrual accounting: revenue can be recorded before collection, while noncash expenses can reduce profit without an immediate payment. Operating cash flow measures cash from operating activities. Under the indirect method, begin with profit, add back noncash charges such as depreciation, and adjust for changes in operating assets and liabilities. An increase in receivables uses cash relative to reported revenue, so subtract it; an increase in payables generally preserves cash, so add it. Track the sign of each change rather than memorizing a single blanket rule. A persistent gap between profit and operating cash flow merits investigation, but its cause can be healthy sales growth, slower collection, or another working capital change. Compare periods, credit terms, and receivable quality before judging earnings. The Level I study guide can help connect this reconciliation to the statements.

Worked example: A firm reports $40 million of profit, $10 million of depreciation, and a $15 million increase in receivables, with no other adjustments. Operating cash flow is $40 million + $10 million − $15 million = $35 million. Cash flow is $35 million ÷ $40 million = 87.5% of profit. The difference signals cash tied up in receivables; it does not by itself show whether customers will pay late.

Watch the distinction: Adding the receivables increase reverses the cash logic. Revenue included in profit but still owed by customers has not entered cash, so the increase is subtracted in this reconciliation.

Reference: [1]

Level I - Corporate Issuers

25. Control and economic ownership

Voting rights govern who can influence directors and major shareholder decisions; economic ownership determines a holder's share of distributions and residual value. The two percentages can diverge when share classes carry different voting power. To assess a dual class company, map votes by class, identify who can control decisions, and separately map claims on cash flows. A controller with a smaller economic stake may receive less of the financial downside from a poor decision than other owners do. That creates a potential agency problem, although concentrated control can also support long term plans. Look beyond the headline ownership percentage to board powers, related party transactions, and protections for minority shareholders. A voting majority does not mean every corporate action is unilateral: specific decisions may require other approvals under the company's governing arrangements.

Worked example: Assume a founder owns shares carrying 60% of votes but only 30% of economic rights. If a hypothetical $10 million distribution is shared strictly by economic ownership, the founder receives $3 million, while other holders receive $7 million. The founder may nevertheless determine ordinary shareholder votes requiring a simple majority. The gap between 60% control and 30% cash flow exposure is the governance issue to investigate.

Watch the distinction: Do not treat 60% voting power as a claim to 60% of dividends. Voting and distribution rights are separate attributes, and a dual class structure can assign them in very different proportions.

Reference: [1]

26. Stakeholder risks in company decisions

Operational decisions can affect customers, workers, suppliers, and owners at the same time. Expected cost gives one starting point for a measurable risk: multiply the probability of an event by the loss if it occurs. Then compare the reduction in expected loss with the recurring or upfront cost of mitigation on a consistent time basis. This calculation requires a clear baseline, the measure's effect on event probability or severity, and an estimate of any side effects. A low probability, severe disruption may deserve attention even when its expected value looks modest, particularly if the firm cannot survive the loss. Also ask who bears each cost: a supplier failure can create delayed customer deliveries and reputational damage beyond the firm's direct repair expense. Expected value supports a decision; it does not replace scenario and stakeholder analysis.

Worked example: Suppose a supplier disruption has a hypothetical 10% chance each year and would cost $50 million. Expected annual loss is 0.10 × $50 million = $5 million. A measure costing $3 million annually is assumed to eliminate the event, reducing expected loss by $5 million. Its expected net benefit is $5 million − $3 million = $2 million per year, before estimation error or other effects.

Watch the distinction: A $5 million expected loss does not mean the firm will lose $5 million every year. The outcome may be no disruption or a much larger loss; expected value averages those possibilities.

Reference: [1]

27. Capital budgeting with NPV

Net present value (NPV) compares cash paid today with the present value of future project cash flows. Discount each expected cash flow by the required return for its date, then subtract the initial investment. A positive NPV indicates value added under the stated cash flow and discount rate assumptions. Use incremental cash flows caused by the project, including relevant operating, investment, and terminal effects; accounting profit is not a direct substitute. The discount rate should reflect the risk of the project rather than merely the firm's preferred hurdle. When projects compete for limited resources, a positive standalone NPV is only the first step: compare alternatives on a consistent basis. Before accepting the result, test sensitivity to volume, costs, timing, and terminal value, because small forecast changes can overturn a narrow positive estimate.

Worked example: A project costs $100 million now and pays $60 million at each of the next two year ends. At a hypothetical 10% annual required return, the payment values are $60 million ÷ 1.10 = $54.55 million and $60 million ÷ 1.10² = $49.59 million. NPV is −$100 million + $54.55 million + $49.59 million = about $4.13 million. Accept it if these cash flows, risk, and financing assumptions are appropriate.

Watch the distinction: Adding the two $60 million payments and subtracting $100 million gives $20 million, but ignores timing. Discounting reduces their combined value to about $104.13 million under the stated rate.

Reference: [1]

28. Weighted average cost of capital

Weighted average cost of capital (WACC) combines the required returns of financing sources in proportion to their market value weights. Multiply each weight by its cost, then sum the components. In a simplified setting, interest is tax deductible, so the debt component uses its cost times one minus the assumed tax rate. Equity has no corresponding interest deduction. Confirm that weights sum to one and that rates and tax assumptions use the same period. WACC is a discount rate for projects with risk similar to the firm's existing operations and a comparable financing mix. Applying it mechanically to a much riskier project can overstate that project's NPV. Market weights reflect current investor claims more directly than historical book amounts. The Level II study guide offers a related route for further analysis.

Worked example: Assume market value financing is 60% equity and 40% debt. Equity costs 12%, debt costs 6%, and the hypothetical tax rate is 25%, with interest fully deductible. Equity contributes 0.60 × 12% = 7.2 percentage points. After tax debt contributes 0.40 × 6% × (1 − 0.25) = 1.8 points. WACC is 7.2% + 1.8% = 9% annually under these assumptions.

Watch the distinction: Multiplying the entire WACC by one minus the tax rate gives a false deduction on equity. Apply the assumed interest tax effect only to the debt cost component.

Reference: [1]

29. Financial leverage and earnings

Financial leverage introduces fixed financing charges, such as interest, between operating profit and earnings available after those charges. If interest remains constant, a change in earnings before interest and tax (EBIT) produces a larger percentage change in pretax income whenever pretax income is positive and below EBIT. Calculate the new EBIT, subtract the same interest amount, then compare pretax income with its starting level. The degree of financial leverage at a particular operating level is EBIT divided by EBIT less interest in this simple model. That ratio changes as EBIT changes; it is not a permanent property of the company. Leverage raises the sensitivity of earnings in both directions and can turn a modest operating decline into a much larger earnings loss. Examine debt obligations and cash generation alongside the percentage calculation.

Worked example: A firm has EBIT of $100 million and fixed annual interest of $20 million, leaving $80 million pretax income. EBIT then falls 20% to $80 million. With interest unchanged, pretax income becomes $60 million. Its decline is ($60 million − $80 million) ÷ $80 million = −25%. Initial degree of financial leverage is $100 million ÷ $80 million = 1.25, consistent with 1.25 × −20% = −25%.

Watch the distinction: A 20% EBIT decline does not imply a 20% pretax income decline when interest is fixed. The same $20 million charge consumes a larger share of reduced operating profit.

Reference: [1]

30. The cash conversion cycle

The cash conversion cycle estimates the time between cash paid to suppliers and cash collected from customers. Add days inventory is held to days receivables remain outstanding, then subtract days payables remain unpaid. Each measure should use a consistent period and a suitable average balance rather than a single unusual closing balance. A shorter cycle often reduces working capital financing needs, but the reason matters. Faster collections can improve cash flow; cutting inventory too far can cause stockouts; stretching supplier payments can damage terms or relationships. Read the components separately before concluding that a change is favorable. The cycle is an operating timing measure, not a direct profit margin. It can even be negative when firms collect quickly and pay suppliers later. For a seasonal firm, compare like periods to avoid mistaking timing for structural improvement.

Worked example: Suppose receivable days are 45, inventory days are 60, and payable days are 30. The cash conversion cycle is 45 + 60 − 30 = 75 days. If collections improve and receivable days fall to 35 while the other measures stay constant, the cycle becomes 35 + 60 − 30 = 65 days. Cash is tied up for 10 fewer days under this simplified timing measure.

Watch the distinction: Subtracting inventory days reverses the operating sequence. Inventory ties cash up before sale, so its holding period is added; supplier credit delays payment, so payable days are subtracted.

Reference: [1]

Level I - Equity Investments

31. Sources of equity return

A share's holding period return combines the change in its price with cash distributions received during the period. Subtract purchase price from ending price to find the capital gain or loss, add dividends, then divide by purchase price. Splitting the result into price return and dividend yield shows what drove performance. Use the same share and period for every input, and adjust for stock splits or other changes in share count before calculating. The simple formula assumes dividends are received as cash and does not add returns from reinvesting them. If a dividend is paid during a long holding period, its timing matters for annualized or money weighted measures. Compare total returns when assessing investments because a falling share price can outweigh a dividend, while a firm paying no dividend can still deliver a positive price return.

Worked example: An investor buys one share for $50, receives a $2 cash dividend, and can sell it at period end for $55. The price gain is $55 − $50 = $5, or 10% of cost. Dividend yield is $2 ÷ $50 = 4%. Total holding period return is ($55 + $2 − $50) ÷ $50 = 14%, before taxes and trading costs.

Watch the distinction: Using only the $5 price gain understates this investor's return. The $2 dividend is part of the period's payoff even though it is absent from the ending share price.

Reference: [1]

32. Bid, ask, and trading cost

A bid is the price at which a buyer is willing to purchase; an ask is the price at which a seller is willing to sell. A trader seeking immediate execution typically buys at the ask and sells at the bid. Their difference is the quoted spread, a component of trading cost and compensation for supplying liquidity. Multiply the per share spread by the number of shares to estimate the cost of an immediate round trip when quotes do not move. The midpoint is useful as a reference price, but it is not necessarily executable. A displayed quote may cover only a limited quantity, so a large order can face additional price impact. Compare spreads across securities with attention to price level, trading conditions, and order size. CFA practice questions can reinforce which side of a quote each trade uses.

Worked example: A stock is quoted at a $20.00 bid and $20.10 ask, with enough shares available at both prices. Buying 100 shares immediately costs 100 × $20.10 = $2,010. Selling them immediately at the unchanged bid brings 100 × $20.00 = $2,000. The round trip loses $10, equal to the $0.10 spread times 100 shares, before commissions, taxes, or quote changes.

Watch the distinction: The midpoint of $20.05 is not the quoted price for both immediate trades. Assuming a buy and sale there removes the very spread cost the quote reveals.

Reference: [1]

33. Market-value-weighted indexes

A market value weighted equity index gives larger companies greater influence on index return. At the start of a measurement period, calculate each constituent's index weight as its eligible market value divided by the total, then multiply that weight by its period return. Add the weighted returns to obtain the index return when constituents and shares remain unchanged. This is a weighted average, so a small firm's large move may matter less than a large firm's modest move. Distinguish full market capitalization from float adjusted value: some indexes exclude shares unavailable to public investors. Index providers also adjust for splits, issuance, and constituent changes to preserve continuity. For a simple two stock problem, stated starting weights are sufficient. For a real index, check its weighting and rebalancing rules before drawing conclusions from a headline index move.

Worked example: At the start of a period, stock A has an 80% index weight and stock B has 20%. A rises 10%; B falls 10%. A contributes 0.80 × 10% = 8 percentage points, while B contributes 0.20 × −10% = −2 points. The index return is 8 − 2 = 6%. Equal weighting would have produced 0%, showing why size matters here.

Watch the distinction: Averaging 10% and −10% to get zero assumes equal weights. With an 80% weight, stock A's gain has four times the index influence of stock B's equal sized percentage loss.

Reference: [1]

34. Industry concentration

Industry concentration describes how market sales are distributed among firms. One common measure sums squared market shares expressed as whole percentages. Squaring gives larger firms disproportionate influence: moving share from a small firm to a dominant one usually raises the measure. Before calculating, define the product and geographic market; a narrow definition can make an industry appear more concentrated than a broader one. Concentration can affect bargaining power and competitive behavior, but it cannot by itself establish profitability. Entry barriers, substitutes, regulation, buyer power, and the ability of firms to coordinate or differentiate also matter. The measure is a snapshot; rapid entry or changing technology can make it less predictive. Use it as a diagnostic cue for deeper industry analysis, not as a valuation conclusion. Report the share convention so readers can interpret the number.

Worked example: Four hypothetical firms hold 40%, 30%, 20%, and 10% of a defined market. Their shares sum to 100%. Squaring percentage shares gives 40² + 30² + 20² + 10² = 1,600 + 900 + 400 + 100 = 3,000. This describes a market led by two firms with 70% combined share. It does not establish their margins; investigate entry costs, substitutes, and pricing conduct next.

Watch the distinction: A concentration score of 3,000 is not a 3,000% market share or a profit forecast. It is a sum of squared percentage shares whose meaning depends on the market definition.

Reference: [1]

35. Comparing company margins

A common size income statement expresses each line as a percentage of revenue, making firms of different sizes easier to compare. Gross margin equals revenue less cost of sales, divided by revenue. A higher gross margin can reflect stronger pricing, cheaper inputs, a different product mix, or different accounting choices. It does not establish higher operating profit because selling, research, and administrative costs sit below gross profit. Compare firms serving similar products and use consistent periods and accounting methods. A margin change should prompt a cause check: did selling prices rise, did input costs change, or did the sales mix shift? When the difference is small, review whether it persists rather than treating one period as decisive. Gross margin is useful because it isolates the relationship between sales and directly assigned product costs, but it is only one layer of profitability.

Worked example: Firm A and Firm B each report $100 million of revenue. A has $60 million cost of sales, leaving $40 million gross profit and a 40% gross margin. B has $65 million cost of sales, leaving $35 million and a 35% margin. A's advantage is five percentage points. If A also spends $10 million more on operating overhead than B, that gross advantage would not carry through to operating profit.

Watch the distinction: The firm with the higher gross margin is not necessarily more profitable overall. Gross margin excludes overhead, financing costs, and taxes, any of which can reverse the comparison.

Reference: [1]

36. Constant-growth dividend value

The constant growth dividend discount model values a share as the present value of dividends expected to grow at a fixed rate indefinitely. If next year's dividend is D1, required return is r, and perpetual growth is g, value equals D1 divided by r minus g. The condition r greater than g keeps the infinite series finite and positive. Identify whether a problem supplies the most recent dividend or next year's: a current dividend must first be grown to D1. The model is most useful when dividends reflect a sustainable distribution policy and growth can reasonably settle at a stable rate. It is highly sensitive to a small gap between r and g, so test alternative assumptions. A calculated value is an estimate, not an observable market price or a guarantee of return. Examine payout capacity before accepting the dividend path.

Worked example: Assume a stock is expected to pay a $3 dividend next year, the required annual return is 9%, and dividends can grow 3% yearly indefinitely. The estimated value is $3 ÷ (0.09 − 0.03) = $50 per share. The six percentage point spread keeps the formula valid. If the $3 instead referred to the dividend just paid, next year's dividend would be $3 × 1.03 = $3.09.

Watch the distinction: Using the dividend just paid as D1 understates value when growth is positive. First identify the dividend's date, then grow a current dividend once before applying the formula.

Reference: [1]

Level I - Fixed Income

37. Reading a bond's promised cash flows

A plain annual coupon bond promises periodic interest and repayment of principal at maturity. Multiply face value by the stated annual coupon rate to find each full year's coupon. At the final payment date, add principal to the last coupon; earlier payments contain coupons alone. Lay out the cash flow timeline before calculating price or yield, because a missing final coupon changes both. Bond terms can differ: payments may be semiannual, principal may amortize, and an issuer may hold options to call the bond. Always use the stated contract rather than assuming every bond is plain. Promised payments are distinct from expected receipts when default is possible. The CFA Program exam information identifies Level I as the foundation level; the Level I topic outline specifically includes fixed income securities, yields, valuation, and risk.

Worked example: A three year bond has $1,000 face value and a 4% annual coupon, paid at each year end. Each coupon is $1,000 × 0.04 = $40. The year one and year two cash flows are $40 each. At year three, the bond promises the final $40 coupon plus $1,000 principal, totaling $1,040. Total promised payments are $1,120 across three years, assuming no default or other contract feature.

Watch the distinction: Listing $1,000 alone at maturity omits the last coupon. Unless the terms say otherwise, a coupon bond's final scheduled payment contains both the period's coupon and principal.

Reference: [1]; [2]

38. Bond price and required yield

A bond's price is the present value of its remaining promised cash flows discounted at the yield investors require for that bond. For a one year bond with a single payment, divide the final coupon plus principal by one plus the annual yield. The coupon rate determines the promised coupon; the required yield determines the price paid today. When required yield exceeds the coupon rate on a bond near a simple par repayment, price generally falls below par so the buyer can earn the higher yield through both coupon and price gain. Conversely, a lower required yield raises price. The price yield relationship is inverse because a higher discount rate reduces present value. This simple valuation assumes the cash flow is received as promised and excludes accrued interest conventions, transaction costs, and any special bond features.

Worked example: A one year bond promises a $50 coupon and $1,000 principal at year end. With a hypothetical 6% annual required yield, its sole payment is $1,050 and its value today is $1,050 ÷ 1.06 = $990.57, rounded to cents. The $50 coupon is 5% of face value, below the 6% required yield, so the bond trades $9.43 below its $1,000 par amount under these assumptions.

Watch the distinction: Discounting only the $1,000 principal ignores the coupon and understates price. Combine all cash flows due at the same date before applying the one year discount factor.

Reference: [1]

39. Spot and forward rates

A spot rate prices a cash flow at one specified maturity. A forward rate is the rate implied today for a future borrowing or investment interval by the current spot rate curve. With annual compounding, investing for two years at the two year spot rate must produce the same terminal amount as investing for one year at the one year spot rate and then earning the implied forward rate for year two, under the simplified no arbitrage setup. Thus, divide the two year accumulation factor by the one year factor and subtract one. Keep maturity dates, compounding conventions, and rate units aligned. An implied forward is a break even rate embedded in current prices; the eventual one year spot rate can differ. If the rates include different credit or liquidity risks, a direct comparison needs care.

Worked example: Assume one year and two year annual spot rates of 4% and 5%. One unit invested for two years becomes 1.05² = 1.1025 units. One year at 4% gives 1.04 units, so the implied year two factor is 1.1025 ÷ 1.04 = 1.060096. The implied forward rate is about 6.01% annually. That rate equates the two accumulation paths today; it does not predict the future spot rate with certainty.

Watch the distinction: Subtracting 4% from 5% gives one percentage point, not the year two forward rate. The spot rates cover different total horizons, so equate their compounded accumulation factors.

Reference: [1]

40. Duration and convexity

Modified duration estimates the first order percentage change in a bond's price for a small yield change: negative modified duration times the yield change in decimal form. Convexity adds a curvature adjustment of one half times convexity times the squared yield change. Together they approximate the price yield relationship more accurately than duration alone for a moderate move. Start with the sign: a yield rise normally lowers a conventional bond's price. Then calculate the duration term, add the convexity term, and multiply the resulting percentage change by the starting price. The estimate depends on how duration and convexity were defined and on a parallel yield change. Large moves, changing credit spreads, or embedded options can make the approximation less reliable. For bonds with positive convexity, the curvature term cushions a yield rise and enhances the gain from an equal yield fall.

Worked example: A bond costs $1,000, has modified duration 5 and convexity 40, and its yield rises by 0.5 percentage point, or 0.005. The duration estimate is −5 × 0.005 = −0.025. Convexity adds 0.5 × 40 × 0.005² = 0.0005. Estimated price change is −0.0245, or −2.45%. Estimated new price is $1,000 × (1 − 0.0245) = $975.50, subject to the approximation's assumptions.

Watch the distinction: Entering 0.5 rather than 0.005 for a half percentage point yield change makes the estimate wildly wrong. Convert percentage points to a decimal yield change before using duration or convexity.

Reference: [1]

41. Expected credit loss

Credit loss starts with a promised cash flow and asks how much of it may go unpaid. If default occurs with probability p and the investor recovers fraction R of the promised amount, expected loss is promised amount × p × (1 − R). This is an average across possible outcomes, not the amount lost in every default. Check what the recovery percentage applies to: principal, a particular payment, or the whole claim. Also keep the time horizon consistent with the default probability. The Level I Fixed Income outline includes basic credit analysis. When moving from expected loss to a bond price, you must also consider when losses occur, discount rates, uncertainty around the estimate, and compensation investors require for bearing risk.

Worked example: Suppose a payment of $1,000 is due in one year. Default probability is a hypothetical 2%, and recovery upon default is 40% of that payment, or $400. Expected payment is 0.98 × $1,000 + 0.02 × $400 = $988. Expected shortfall is $1,000 − $988 = $12. This one-year expected shortfall alone cannot establish a required credit spread.

Watch the distinction: A 40% recovery does not mean the investor expects to lose 60% of the payment. That loss occurs only in the default outcome; its probability must also enter the calculation.

Reference: [1]

42. Loss allocation in securitization

Securitization divides claims on an asset pool into tranches with different priorities. A junior tranche absorbs losses first under a simple first-loss structure, protecting the senior tranche until the junior claim is exhausted. To trace a loss, start with the pool's value, subtract the loss, then allocate the remaining value according to the stated payment priority. The senior claim is safer against modest pool losses, but its protection is limited by the junior tranche's size and the transaction's actual terms. Interest, fees, timing, collateral tests, and changes in expected future losses can affect market values even before a realized loss reaches the senior claim. Keep the example's book allocation separate from a prediction of what either tranche could be sold for today.

Worked example: A loan pool initially worth 100 units backs senior and junior claims of 80 and 20 units. Assume losses reduce the junior claim first and there are no other changes. A 6-unit pool loss leaves assets worth 94. Allocate 80 to the senior claim and 14 to the junior claim, since 20 − 6 = 14. The senior claim absorbs none of this loss.

Watch the distinction: Dividing the 6-unit loss between tranches in proportion to their initial sizes ignores the stated priority. Proportional ownership and loss priority are different features of a securitization.

Reference: [1]

Level I - Derivatives

43. A forward's no-arbitrage price

A forward locks in a price for delivery at a future date. For an asset with no income, storage cost, or other carrying benefit, its forward price follows the cost of buying and financing it until delivery: F = S(1 + r)^T with annual compounding. The comparison comes from two ways to obtain the asset later: buy it today with borrowed funds, or agree to take delivery under a forward. If their future costs differ, an investor able to transact and finance on the assumed terms could construct an arbitrage. Match the financing rate, compounding convention, and time period before calculating. Real assets may pay income or incur storage costs, which alter the relation. The Level I study guide provides a useful place to connect this pricing idea with spot and forward markets.

Worked example: An asset costs $100 now. Assume it pays no income, has no storage cost, and can be financed for one year at a hypothetical 5% annual rate. Buying it with $100 borrowed today requires $100 × 1.05 = $105 repayment in one year. Under these assumptions, the one-year no-arbitrage forward price is $105 per asset.

Watch the distinction: Adding 5% to a forward price because the asset is expected to rise confuses financing with a price forecast. The simplified no-arbitrage calculation requires no prediction of next year's spot price.

Reference: [1]

44. Futures marking to market

A futures contract is settled through a margin account as its settlement price changes. For a long position, a rise in price creates a gain; for a short position, the same move creates a loss. Multiply the change in the quoted futures price by the contract multiplier and the number of contracts, keeping the quote's unit consistent with the multiplier. Daily settlement credits or debits the account rather than waiting until expiry to recognize all gains and losses. Initial margin is collateral, not the contract's purchase price or the maximum possible loss. If the price moves against the position, additional funds may be required under the contract's margin arrangements. A correct calculation therefore begins by identifying position direction, price change, contract size, and settlement period.

Worked example: A trader is long one futures contract entered at 100, with a multiplier of 50 currency units for each one-point price move. Today's settlement price is 103. The gain is (103 − 100) × 50 × 1 = 150 currency units, credited through daily settlement. A short position on the same contract would instead have a 150-unit loss.

Watch the distinction: A three-point futures rise is not automatically a three-unit cash gain. The multiplier converts the quoted price change into the contract's settlement amount, and position direction determines its sign.

Reference: [1]

45. Interest-rate swap cash flows

An interest-rate swap exchanges interest cash flows calculated on a stated notional principal. In a plain fixed-for-floating swap, one party pays a fixed rate and receives a floating rate, while the other takes the opposite side. For one settlement period, calculate each leg as notional × applicable annual rate × year fraction, using the rates and day-count assumptions specified in the question. Net the two payments when they share a currency and payment date. The notional is generally a reference amount for interest calculations, not cash exchanged at inception or settlement. A floating-rate payer benefits from a lower floating fixing for that period, while a floating-rate receiver benefits from a higher one. Swap value over its full life also depends on the remaining expected payments and discounting, not just the latest net settlement.

Worked example: Assume a one-year settlement on a $1 million notional, with a party paying 4% fixed and receiving a 5% floating rate. Its fixed payment is $1,000,000 × 0.04 = $40,000; its floating receipt is $1,000,000 × 0.05 = $50,000. Netting produces a $10,000 receipt. This simplified swap does not exchange the $1 million notional.

Watch the distinction: Treating the $1 million notional as an amount owed makes the swap cash flow vastly too large. Apply each rate to the notional for the relevant period, then net the interest legs.

Reference: [1]

46. Option payoff and profit

A call option gives its holder the right to buy an underlying asset at the strike price. At expiry, its payoff per share is max(ending price − strike, 0). The maximum with zero matters: the holder can let an out-of-the-money call expire rather than buy above market price. Profit for the buyer subtracts the premium paid from that payoff, before financing and trading costs. A call can therefore finish in the money yet still produce a loss if the exercise value does not cover its premium. The break-even ending price is strike plus premium for this simple expiry calculation. Identify whether a question asks for payoff or profit before inserting numbers. The CFA practice questions can help reinforce that distinction across different ending prices.

Worked example: A call costs $3 per share and has a $50 strike. At expiry, the share price is $58. Exercise payoff is max($58 − $50, 0) = $8 per share. After subtracting the $3 premium, buyer profit is $5 per share before other costs. The break-even expiry price is $53, so $58 is $5 above it.

Watch the distinction: Calling the $8 exercise value an $8 profit omits the premium. The holder paid $3 for the option, so the expiry profit in this scenario is $5 per share.

Reference: [1]

47. Put-call parity

Put-call parity links a European call and put with the same underlying, strike, and expiry. For a stock paying no dividends before expiry, call price + present value of strike = put price + stock price. At expiry, both sides deliver the same value in every stock-price outcome: either the stock or the strike amount, whichever is larger. Their current costs must therefore match under the stated financing and trading assumptions. Rearrange the equation for the missing option price, and discount the strike using a rate consistent with its maturity and compounding convention. Parity does not claim that a put and call have the same price. Dividends, early exercise rights, and transaction costs require adjustments or change the equality used in this simple form.

Worked example: Suppose a non-dividend-paying stock costs $100, and matching one-year European options have a $100 strike. At an assumed 5% annual risk-free rate, the strike's present value is $100 ÷ 1.05 = $95.24. If the call costs $10, parity implies put price = $10 + $95.24 − $100 = $5.24, rounded to cents.

Watch the distinction: Using the undiscounted $100 strike would give a $10 put and break the matching-cash-flow argument. The strike is paid at expiry, so its current value must be discounted.

Reference: [1]

48. Hedging equity market exposure

An index-futures hedge can reduce a portfolio's exposure to broad market moves. Estimate the portfolio's market-linked dollar exposure as portfolio value × beta, then divide by the dollar exposure represented by one futures contract. Sell contracts to offset a long equity portfolio's positive market beta; buy them to add market exposure. This sizing is approximate because beta is estimated from past relationships and can change. The futures index may also differ from the portfolio, creating basis and tracking risk. Other sources of return, such as security selection and industry-specific events, remain after the market hedge. Check the contract multiplier and current futures price when deriving exposure per contract, and treat the resulting contract count as a hedge estimate rather than a guaranteed neutral position.

Worked example: A $10 million equity portfolio has estimated beta of 1.2. Assume each index futures contract represents $200,000 of market exposure at the current price. Estimated beta-adjusted exposure is $10 million × 1.2 = $12 million. Selling $12 million ÷ $200,000 = 60 contracts approximately offsets that measured exposure, while company-specific and hedge-basis risks remain.

Watch the distinction: Hedging only the portfolio's $10 million market value overlooks its beta of 1.2. The estimated market sensitivity is $12 million, so 50 contracts would leave measured exposure.

Reference: [1]

Level I - Alternative Investments

49. Liquidity and reported value

A private asset's reported value is an estimate based on available evidence and a valuation method. Liquidity asks a different question: how much cash could be obtained, and how quickly, through an actual sale. A thin market may have few buyers, lengthy negotiation, and sizable price concessions for immediate execution. The distinction matters when a portfolio has near-term spending needs or must meet redemptions. Compare the valuation date and assumptions with the terms of a real offer, including transaction costs and restrictions. An immediate offer below an appraisal may reflect urgency, buyer bargaining power, changed information, or an appraisal that has not caught up with market conditions. Neither figure automatically proves the other wrong; each answers a different question under different conditions.

Worked example: A private holding has a hypothetical appraisal of $100 million. The only buyer willing to close immediately offers $90 million, with no additional sale costs assumed. Selling now would produce $90 million, which is $10 million or 10% below the appraisal. An investor needing immediate cash must plan around the executable offer, while a patient seller may obtain another price.

Watch the distinction: Recording an appraisal of 100 does not make 100 available for tomorrow's withdrawal. Estimated value and immediately realizable cash can differ materially when buyers are scarce.

Reference: [1]

50. Private investment multiples and IRR

A money multiple compares total cash received with cash invested, without accounting for how long the money was committed. For a single investment and a single final payment, the multiple is exit proceeds divided by entry cost. Annualized internal rate of return solves entry cost × (1 + r)^T = exit proceeds, so r = multiple^(1/T) − 1 for T years. This shortcut works only when there are no interim cash flows. With contributions and distributions on several dates, IRR must instead use the dated cash-flow schedule. A higher multiple does not necessarily mean a higher annualized return if it took longer to achieve. Fees, taxes, and unrealized values also affect what the investor actually earns, so check which cash flows the stated measure includes.

Worked example: A private investment costs $100 million today and pays $150 million exactly three years later, with no interim flows or fees. The money multiple is $150 million ÷ $100 million = 1.5. Its annualized IRR is 1.5^(1/3) − 1 ≈ 14.47%. Compounding $100 million at about 14.47% for three years returns approximately $150 million.

Watch the distinction: Dividing the 50% total gain by three gives 16.67%, which ignores compounding. The annualized rate must compound for three years to reproduce the ending value.

Reference: [1]

51. Real estate income value

Direct capitalization values a property by relating a representative annual net operating income, or NOI, to a capitalization rate: value = NOI ÷ cap rate. NOI reflects property revenue less operating expenses under the stated assumptions; it is generally measured before financing costs and income taxes. The rate captures the market's required relationship between stabilized income and price, so a higher rate produces a lower indicated value for unchanged NOI. Use an income figure and rate that describe the same property interest and operating condition. A one-year income spike or temporary vacancy may need adjustment before calling NOI stabilized. The method is sensitive to both inputs and does not separately model future rent growth, large capital expenditures, or eventual sale proceeds. Those omitted factors may be embedded imperfectly in the chosen rate.

Worked example: A property is expected to produce a stabilized annual NOI of $8 million. At a hypothetical 8% capitalization rate, indicated value is $8 million ÷ 0.08 = $100 million. If the same NOI is capitalized at 10%, value becomes $80 million. The $20 million difference shows the estimate's sensitivity to the assumed rate.

Watch the distinction: An 8% cap rate does not mean the owner receives an 8% guaranteed total return. It relates a specified year's NOI to indicated value, excluding several future cash-flow effects.

Reference: [1]

52. Commodity futures and contango

A commodity market is in contango when a futures price exceeds the current spot price for the same commodity and relevant delivery terms. This price pattern can reflect financing, storage, and other carrying effects; it is not by itself a forecast that spot will rise. A long futures position entered at price F₀ gains or loses the difference between the spot price at expiry and F₀, assuming convergence and ignoring frictions. To evaluate its outcome, compare the entry futures price with the eventual spot price, not merely with today's spot price. A collateralized futures investment can also earn or lose value on the collateral, while a rolling strategy faces prices on successive contracts. Thus a single contract's settlement gain is only one part of a broader investment return.

Worked example: Assume a one-year commodity future is entered at $105 per unit while today's spot price is $100. At expiry, spot is $100 and the futures contract settles at that price. The long's contract result is $100 − $105 = −$5 per unit. This is a $5 loss before collateral return and trading costs, despite unchanged spot.

Watch the distinction: Contango does not guarantee a long futures loss. The loss in this example arises because expiry spot is below the $105 entry price; a sufficiently higher expiry spot would produce a gain.

Reference: [1]

53. Infrastructure operating sensitivity

Infrastructure assets may have substantial fixed operating and maintenance commitments. When revenue changes but those costs remain fixed, cash available after them changes by a larger percentage than revenue. Begin with revenue, subtract operating costs and required maintenance spending, then compare the resulting cash under each scenario. This simple operating sensitivity shows why a seemingly small change in usage, tariffs, or availability can matter greatly to owners. It does not measure distributable cash if interest, taxes, capital spending beyond the stated maintenance amount, or contractual reserve requirements still have to be paid. The stability of revenue also depends on contract terms and counterparty performance. State which costs truly stay fixed over the scenario's time horizon; over longer periods, management may adjust expenses or defer some spending.

Worked example: An infrastructure asset has annual revenue of $10 million, operating costs of $4 million, and required maintenance spending of $2 million. Cash after these items is $10 million − $4 million − $2 million = $4 million. If revenue falls to $9 million and both costs stay fixed, cash becomes $3 million, a decline of $1 million ÷ $4 million = 25%.

Watch the distinction: A 10% revenue fall does not imply a 10% cash-flow fall when costs stay fixed. The $1 million revenue loss comes entirely out of the original $4 million residual.

Reference: [1]

54. Hedge fund fees and investor return

Hedge fund fee arrangements can charge both a management fee and a performance fee. The investor's net result depends on the agreement's fee bases and the order of calculation. A useful sequence is to find ending value before fees, deduct the management fee as defined, determine the gain eligible for a performance fee, then deduct that fee. Some agreements use high-water marks, hurdle rates, different valuation dates, or different bases, so the sequence must follow the given terms. Fees can reduce investor return sharply when gross gains are modest. The performance fee is usually calculated from a specified gain, not automatically from the entire ending account value. Also distinguish a return percentage from a currency amount: divide the final net gain by the investor's initial capital to report net return.

Worked example: Under a simplified hypothetical agreement, $100 million grows to $112 million before fees. A $2 million management fee leaves $110 million, or a $10 million gain over initial capital. A performance fee of 20% of that remaining gain is $2 million. Final value is $108 million, so the investor's net gain is $8 million and net return is $8 million ÷ $100 million = 8%.

Watch the distinction: Charging 20% on the full $112 million would produce a $22.4 million fee, contrary to the stated agreement. Apply the performance rate to the defined eligible gain.

Reference: [1]

Level I - Portfolio Management

55. Turning needs into portfolio constraints

Portfolio planning starts by turning an investor's spending plans into amounts and dates. The initial withdrawal rate is expected withdrawals over a period divided by starting portfolio value. That rate describes planned cash use; it does not tell you what markets will return or whether the plan can last indefinitely. Next, identify the cash needed before investments can be sold without unacceptable timing or price risk. Other constraints may include investment horizon, tolerance for losses, tax circumstances, and obligations that arrive later. Match near-term payments with suitably liquid resources, then decide how much remaining capital can bear longer-term risk. A spending need can also change the appropriate risk level even when the investor says they are comfortable with volatility. The portfolio must meet cash obligations in adverse market scenarios as well as expected ones.

Worked example: An investor begins with a $1 million portfolio and expects to withdraw $50,000 during the next year. Initial withdrawal rate is $50,000 ÷ $1,000,000 = 5%. If the payment is due in three months, the plan needs accessible cash or assets that can reliably fund it then. The 5% rate does not assume or guarantee a 5% investment return.

Watch the distinction: Calling the 5% withdrawal rate a required portfolio return overlooks timing and market losses. Even a portfolio expected to earn 5% can lack cash when the withdrawal is due.

Reference: [1]

56. Weighted portfolio return

A portfolio's one-period return is the weighted sum of its holdings' returns when weights are measured at the period's start and there are no external cash flows during the period. For each holding, multiply its starting weight by its return; add the contributions. Positive and negative contributions can offset each other, and the starting weights must sum to one for a fully invested portfolio. If the holdings move differently, their ending weights drift, so using ending weights to reconstruct the same period's return gives the wrong answer. Rebalancing during the period or investor deposits and withdrawals require more careful treatment, often by splitting the period at each event. The weighted calculation explains which holding contributed what to the realized return, but it does not by itself measure portfolio risk.

Worked example: A portfolio starts with 60% in stocks and 40% in bonds. Over one period, stocks return 10% and bonds return −2%, with no cash flow or interim rebalancing. Stock contribution is 0.60 × 10% = 6.0 percentage points; bond contribution is 0.40 × −2% = −0.8 percentage points. Total portfolio return is 5.2%.

Watch the distinction: Taking the simple average of 10% and −2% gives 4%, but the holdings did not start equally sized. Their 60% and 40% starting weights determine the return.

Reference: [1]

57. Correlation and portfolio volatility

Portfolio volatility depends on each asset's volatility and how their returns move together. For two assets, variance is w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ, where ρ is their correlation. Compute variance first using returns expressed as decimals, then take its square root to obtain volatility. When correlation is below +1, combining assets can reduce volatility relative to a perfectly synchronized mix. Zero correlation removes the covariance term, but it does not remove either asset's own risk. Negative correlation can provide greater diversification, while correlations observed in one period may change in stressed markets. The formula measures variability under the assumed weights, volatilities, and correlation; it does not establish a worst-case loss. Check that the resulting volatility has return units, while variance has squared return units.

Worked example: Two assets each have a 50% starting weight. Their annual volatilities are 20% and 10%, with assumed correlation zero. Portfolio variance is 0.5² × 0.20² + 0.5² × 0.10² + 0 = 0.0125 in squared annual-return units. Volatility is √0.0125 ≈ 0.1118, or 11.18% annually, below the 15% weighted average of their individual volatilities.

Watch the distinction: A zero correlation does not make portfolio volatility zero. Both assets still fluctuate; zero correlation only eliminates the covariance term in this two-asset variance calculation.

Reference: [1]

58. Beta and required return

The capital asset pricing model relates required return to market risk: required return = risk-free rate + beta × expected market risk premium. Beta estimates how strongly an asset's return moves with the market portfolio; it is not a measure of all risks that asset bears. Use rates for the same horizon and currency, multiply beta by the premium, and compare the result with a forecast return. A forecast below the CAPM required return suggests the asset offers too little expected return for its estimated market risk under this model and those inputs. That comparison is sensitive to beta, the market premium, and the forecast itself. CAPM is a benchmark for systematic risk, so company-specific uncertainty may still be important to a real investment decision even though diversified investors can reduce that risk.

Worked example: Assume a 3% annual risk-free rate, a 5% annual expected market risk premium, and beta of 1.2. CAPM required return is 3% + 1.2 × 5% = 9% annually. If an analyst forecasts an 8% annual return, the forecast is one percentage point below the model's required return. Under these assumptions, its expected compensation for market risk is insufficient.

Watch the distinction: Adding beta itself to the risk-free rate gives a meaningless mix of units. Beta has no percent unit; multiply it by the market risk premium first.

Reference: [1]

59. Choosing a risky-asset allocation

Combining a risky portfolio with a risk-free asset lets an investor adjust market exposure without changing the risky portfolio's internal mix. If weight w goes into the risky portfolio and the rest into the risk-free asset, expected return is wE(Risky) + (1 − w)Rf. Assuming the risk-free return is certain over the chosen horizon, combination volatility is |w| times the risky portfolio's volatility. For a long-only allocation between zero and one, this scales risk directly with w. Choose the risk target or acceptable loss exposure first, then solve for the risky weight and check the resulting expected return. A risk-free asset must match the investor's currency and investment horizon closely enough for the assumption to be meaningful. Forecast returns and volatilities are estimates, not promises about realized results.

Worked example: A risky portfolio has expected annual return of 8% and annual volatility of 12%. A matching risk-free asset yields 2% annually. Investing 75% in the risky portfolio and 25% in the risk-free asset gives expected return 0.75 × 8% + 0.25 × 2% = 6.5%. Assuming a certain risk-free return, volatility is 0.75 × 12% = 9% annually.

Watch the distinction: A 25% allocation to the risk-free asset does not eliminate all risk. The remaining 75% risky allocation still produces an estimated 9% annual portfolio volatility.

Reference: [1]

60. Risk-adjusted performance

The Sharpe ratio compares return above a risk-free benchmark with total return volatility: (portfolio return − risk-free rate) ÷ portfolio standard deviation. Use rates and volatility measured over the same period. A higher ratio means more measured excess return per unit of total volatility for the observations or forecasts supplied. It does not mean higher absolute return, smaller worst-case loss, or superior performance in every market. Total volatility includes both market and asset-specific variation, making the ratio useful when judging an entire portfolio. Before comparing portfolios, check whether returns are gross or net of fees, whether measurement periods match, and whether unusually smooth reported values understate risk. A negative excess return also changes interpretation: dividing by volatility cannot turn underperformance of the benchmark into a favorable result.

Worked example: Assume an annual risk-free rate of 2%. Portfolio A returns 10% annually with 16% annual volatility, so its Sharpe ratio is (10% − 2%) ÷ 16% = 0.50. Portfolio B returns 8% with 10% volatility, giving (8% − 2%) ÷ 10% = 0.60. B delivers more excess return per unit of measured total volatility, although A's return is higher.

Watch the distinction: Selecting A solely because its 10% return exceeds B's 8% ignores the question's risk-adjusted measure. Their different volatilities reverse the ranking by Sharpe ratio.

Reference: [1]

Official references and further reading

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CFA Level I

CFA Level II

CFA Level III: Portfolio Management

CFA Level III: Private Markets

CFA Level III: Private Wealth

Level I - Ethical and Professional Standards

1. An ethical decision sequence

2. Material nonpublic information

3. Independence under pressure

4. Matching advice to a client

5. Fair treatment in trading

6. Evidence behind performance claims

Level I - Quantitative Methods

7. The time value of money

8. Arithmetic and geometric returns

9. Dispersion in return data

10. Conditional probability

11. Testing a population mean

12. Reading a regression model

Level I - Economics

13. Price elasticity and revenue

14. The firm's output decision

15. Measuring aggregate output

16. Aggregate demand and fiscal effects

17. Reading an exchange-rate quote

18. Growth and the business cycle

Level I - Financial Statement Analysis

19. How the statements connect

20. Revenue before cash collection

21. Inventory cost formulas

22. Depreciation and carrying amount

23. Temporary tax differences

24. Cash flow versus reported profit

Level I - Corporate Issuers

25. Control and economic ownership

26. Stakeholder risks in company decisions

27. Capital budgeting with NPV

28. Weighted average cost of capital

29. Financial leverage and earnings

30. The cash conversion cycle

Level I - Equity Investments

31. Sources of equity return

32. Bid, ask, and trading cost

33. Market-value-weighted indexes

34. Industry concentration

35. Comparing company margins

36. Constant-growth dividend value

Level I - Fixed Income

37. Reading a bond's promised cash flows

38. Bond price and required yield

39. Spot and forward rates

40. Duration and convexity

41. Expected credit loss

42. Loss allocation in securitization

Level I - Derivatives

43. A forward's no-arbitrage price

44. Futures marking to market

45. Interest-rate swap cash flows

46. Option payoff and profit

47. Put-call parity

48. Hedging equity market exposure

Level I - Alternative Investments

49. Liquidity and reported value

50. Private investment multiples and IRR

51. Real estate income value

52. Commodity futures and contango

53. Infrastructure operating sensitivity

54. Hedge fund fees and investor return

Level I - Portfolio Management

55. Turning needs into portfolio constraints

56. Weighted portfolio return

57. Correlation and portfolio volatility

58. Beta and required return

59. Choosing a risky-asset allocation

60. Risk-adjusted performance

Official references and further reading

Frequently Asked Questions

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