CFA Level 1 Quantitative Methods: notes and free practice
CFA Level 1 Quantitative Methods connects returns, time value of money, probability and statistical inference. Match dates and units before calculating, and distinguish what a statistical result shows from what it cannot establish.
The key ideas, at a glance
Move cash to one date
Read the visual, then follow the worked example below. Use the stated case assumptions.
Illustrative worked case
$
PV today
1,000
$
Year 3 cash flow
1,331
$
One payment of $1,331 in year 3, discounted at 10% annually, has a present value of $1,000.
| Concept | Rule or decision | Common mistake |
|---|---|---|
| Move cash to one date | PV = FV / (1 + r)^n | A nominal annual rate with monthly compounding is not the monthly rate. |
| Choose the return measure | Holding return = (end - start + income) / start | Subtracting the two percentage changes does not compound wealth. |
| Weight the possible outcomes | Expected value = Σ probability × outcome | A probability-weighted average need not be one of the possible outcomes. |
| Read what the estimate says | Reject H0 when p ≤ chosen significance level | Failing to reject a null is not proof that it is true. |
01Move cash to one date
PV = FV / (1 + r)^n
Watch for: A nominal annual rate with monthly compounding is not the monthly rate.
02Choose the return measure
Holding return = (end - start + income) / start
Watch for: Subtracting the two percentage changes does not compound wealth.
03Weight the possible outcomes
Expected value = Σ probability × outcome
Watch for: A probability-weighted average need not be one of the possible outcomes.
04Read what the estimate says
Reject H0 when p ≤ chosen significance level
Watch for: Failing to reject a null is not proof that it is true.
Selected 2026 study concepts. Match your preparation to your registered exam-year curriculum. These notes are not complete curriculum coverage.
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CFA Level I: 20 practice MCQs
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Put the method to work
One worked example, from facts to answer
Given
$1,331 arrives in three years; the annual discount rate is 10%.
Method
1331 ÷ 1.10^3. Calculator: N=3, I/Y=10, PMT=0, FV=1331; compute PV.
Answer and interpretation
PV is $1,000; the calculator reports -1,000 under that sign convention.
Original independent teaching example. Assumptions are stated above; this is not an official exam question or scoring key.
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10 free Quantitative Methods questions
Choose an answer, check the reasoning, then use ‘Review the method’ to revisit the exact idea. Selected practice, not a complete syllabus or pass prediction.
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- Question 1 of 10Level I topic sampleQuantitative MethodsMultiple choice
A deposit of $2,000 compounds annually at 6% for two years with no withdrawals. What is its future value?
Answer and explanations
$2,000 × 1.06² = $2,247.20 because the first year’s interest also earns interest.
Review the method- A: $2,240 uses simple interest and omits interest on the first year’s interest.
- B - correct: $2,000 × 1.06² = $2,247.20 because the first year’s interest also earns interest.
- C: $2,120 is the value after only one year.
- Question 2 of 10Level I topic sampleQuantitative MethodsMultiple choice
A certain $1,210 payment arrives in two years. Using a 10% annual discount rate, what is its present value?
Answer and explanations
$1,210 / 1.10² = $1,000; discount twice to move the payment to today.
Review the method- A - correct: $1,210 / 1.10² = $1,000; discount twice to move the payment to today.
- B: $1,100 discounts only one of the two years.
- C: $1,464.10 compounds the future payment instead of discounting it.
- Question 3 of 10Level I topic sampleQuantitative MethodsMultiple choice
An account quotes a 12% nominal annual rate compounded monthly. Its effective annual rate is closest to:
Answer and explanations
(1 + 0.12 / 12)^12 - 1 = 12.6825%, approximately 12.68%.
Review the method- A: 12% ignores the effect of earning interest on monthly interest.
- B: 1% is the monthly periodic rate, not the annual effective rate.
- C - correct: (1 + 0.12 / 12)^12 - 1 = 12.6825%, approximately 12.68%.
- Question 4 of 10Level I topic sampleQuantitative MethodsMultiple choice
A share bought for $80 is sold for $84 after paying a $2 dividend. Ignore costs and taxes. What is the holding-period return?
Answer and explanations
Total gain is $84 - $80 + $2 = $6; $6 / $80 = 7.5%.
Review the method- A: 5% counts the $4 capital gain but omits the dividend.
- B - correct: Total gain is $84 - $80 + $2 = $6; $6 / $80 = 7.5%.
- C: 2.5% counts dividend income only.
- Question 5 of 10Level I topic sampleQuantitative MethodsMultiple choice
An investment earns +25% in year one and -20% in year two, with no external cash flows. What is the cumulative return?
Answer and explanations
1.25 × 0.80 - 1 = 0%; the second loss applies to the larger first-year value.
Review the method- A - correct: 1.25 × 0.80 - 1 = 0%; the second loss applies to the larger first-year value.
- B: Adding the returns gives 5% but does not track compounded wealth.
- C: 2.5% is the arithmetic annual average, not cumulative growth.
- Question 6 of 10Level I topic sampleQuantitative MethodsMultiple choice
An asset returns 14% in a favorable scenario with probability 0.6 and -4% otherwise. What is its expected return?
Answer and explanations
0.6 × 14% + 0.4 × (-4%) = 6.8%; both outcomes contribute.
Review the method- A: 10% adds the two scenario returns without probability weights.
- B: 8.4% includes only the favorable scenario contribution.
- C - correct: 0.6 × 14% + 0.4 × (-4%) = 6.8%; both outcomes contribute.
- Question 7 of 10Level I topic sampleQuantitative MethodsMultiple choice
Of 200 borrowers, 40 are in a high-risk group and 10 of those 40 default. What is the observed default proportion conditional on being in that group?
Answer and explanations
Conditional on the group, the denominator is 40: 10 / 40 = 25%.
Review the method- A: 5% uses all 200 borrowers rather than conditioning on the 40 high-risk borrowers.
- B - correct: Conditional on the group, the denominator is 40: 10 / 40 = 25%.
- C: 20% is the high-risk group’s share of the total borrower population.
- Question 8 of 10Level I topic sampleQuantitative MethodsMultiple choice
Return variance is 0.0225 when returns are measured as decimals. What is return standard deviation?
Answer and explanations
The square root of 0.0225 is 0.15, or 15%, in return units.
Review the method- A: 2.25% restates the variance without taking its square root.
- B: 22.5% incorrectly rescales the variance instead of converting units through a square root.
- C - correct: The square root of 0.0225 is 0.15, or 15%, in return units.
- Question 9 of 10Level I topic sampleQuantitative MethodsMultiple choice
A test has p = 0.03 and a prespecified significance level of 0.05. Which decision follows the stated rule?
Answer and explanations
The p-value is below 0.05, so reject the null under the chosen decision rule.
Review the method- A - correct: The p-value is below 0.05, so reject the null under the chosen decision rule.
- B: A statistical decision does not prove the alternative with certainty.
- C: The comparison is with the significance threshold, not with zero.
- Question 10 of 10Level I topic sampleQuantitative MethodsMultiple choice
A simple regression predicts annual sales growth y, in percentage points, as y = 1.5 + 0.8x. For x = 4, predicted y is:
Answer and explanations
Substitute x: 1.5 + 0.8 × 4 = 4.7 percentage points.
Review the method- A: 3.2 is the slope contribution and omits the intercept.
- B - correct: Substitute x: 1.5 + 0.8 × 4 = 4.7 percentage points.
- C: 6.3 does not follow the model’s stated intercept and slope.
Four ideas to put to work
Find the method you need
Start with: Percentages, basic algebra and distinguishing a cash flow from a rate.
METHOD 01
Move cash to one date
PV = FV / (1 + r)^n
- FV = PV × (1 + r)^n; match the rate period to n.
- Effective annual rate = (1 + nominal annual rate / m)^m - 1, with m compounding periods.
- Use opposite cash-flow signs for payments and receipts in a financial calculator.
Worked example
- Given
- $1,331 arrives in three years; the annual discount rate is 10%.
- Calculate / decide
- 1331 ÷ 1.10^3. Calculator: N=3, I/Y=10, PMT=0, FV=1331; compute PV.
- Result
- PV is $1,000; the calculator reports -1,000 under that sign convention.
Watch for: A nominal annual rate with monthly compounding is not the monthly rate.
Practise this ideaMETHOD 02
Choose the return measure
Holding return = (end - start + income) / start
- Include income, not just the price change.
- Multiply wealth relatives when linking successive returns.
- Use time-weighted returns to reduce the effect of external cash-flow timing.
Worked example
- Given
- A $100 investment returns +20% and then -10%, with no cash flows.
- Calculate / decide
- 100 × 1.20 × 0.90 = 108.
- Result
- Cumulative return is 8%; the arithmetic average is not the two-year growth rate.
Watch for: Subtracting the two percentage changes does not compound wealth.
Practise this ideaMETHOD 03
Weight the possible outcomes
Expected value = Σ probability × outcome
- Check that mutually exclusive scenario probabilities total one.
- Condition on the relevant group before choosing a denominator.
- Keep variance in squared units and standard deviation in original units.
Worked example
- Given
- A project pays $200 with probability 0.3 and $50 with probability 0.7.
- Calculate / decide
- 0.3 × 200 + 0.7 × 50.
- Result
- Expected payoff is $95, not a guaranteed cash flow.
Watch for: A probability-weighted average need not be one of the possible outcomes.
Practise this ideaMETHOD 04
Read what the estimate says
Reject H0 when p ≤ chosen significance level
- Set the testing threshold before examining the result.
- A Type I error rejects a true null hypothesis.
- In y = a + bx, b measures the predicted change in y per unit of x.
Worked example
- Given
- A fitted model is y = 2 + 0.5x and x = 6.
- Calculate / decide
- 2 + 0.5 × 6.
- Result
- Predicted y is 5; prediction is not certainty or proof of causation.
Watch for: Failing to reject a null is not proof that it is true.
Practise this ideaQuestions about this topic
- When should I use an arithmetic or geometric mean return?
- An arithmetic mean averages individual period returns. A geometric mean summarizes compound growth across periods. Use the measure the question asks for; arithmetic averaging does not reproduce a multi-period compounded wealth path.
- Why do rate and period units matter?
- A rate per year cannot be used unchanged with a period count in months. Convert under the stated compounding convention and use matching rate, payment and period units before valuing the cash flows.
- Does statistical significance establish an investment opportunity?
- No. A test evaluates evidence under a model and assumptions. Economic magnitude, costs, model validity and out-of-sample performance are separate questions.
Sources and curriculum scope
Official outlines establish curriculum scope. The explanations and worked examples are independent teaching material; the sources do not verify or endorse them.
- 2026 CFA Level I outline: Quantitative Methods
- 2026 CFA Level 1 curriculum outline
- 2027 CFA Level 1 curriculum outline
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