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Concepts 'n' Clarity®
CFA Level I
Quantitative Methods
Core Concepts + 50 Practice Questions Per Chapter
CFA Level I  •  FRM Part I  •  Quantitative Methods

What's Inside

  • Ch 1 — Rates & Returns (75 Questions)
  • Ch 2 — Time Value of Money (75 Questions)
  • Ch 3 — Statistics & Quantitative Methods
  • Ch 4 — Probability Concepts
  • Ch 5 — Portfolio Mathematics
  • Ch 6 & 7 — Simulation & Estimation
  • Ch 8 — Hypothesis Testing
© Concepts 'n' Clarity®  |  For Educational Use Only
Concepts 'n' Clarity®
Ch 1 — Rates & Returns
CFA Level I  •  FRM Part I
PART 1 — CORE CONCEPTS

1. Holding Period Return (HPR)

HPR = (Ending Value − Beginning Value + Income) / Beginning Value
  • Measures total return over any holding period (not annualized).
  • For multi-period HPR: chain-multiply sub-period gross returns — HPR = (1+r₁)(1+r₂)···(1+rₙ) − 1.

2. Annualizing Returns

Annualized = (1 + HPR)^(c/t) − 1   where c = periods per year, t = holding period length
  • Days: raise to (365/days). Months: raise to (12/months). Weeks: raise to (52/weeks).

3. Arithmetic vs. Geometric Mean

  • Arithmetic mean — simple average; has known statistical properties; overstates compound growth.
  • Geometric mean — GM = [(1+r₁)(1+r₂)···(1+rₙ)]^(1/n) − 1; best for multi-period compound growth; always ≤ AM.
  • Harmonic mean — HM = n / Σ(1/rᵢ); used for dollar-cost averaging (average price paid); always ≤ GM.
  • Key identity: AM × HM = GM²; and AM ≥ GM ≥ HM.

4. Time-Weighted vs. Money-Weighted Return

  • Time-Weighted Return (TWR) — chains sub-period returns; eliminates effect of external cash flow timing; standard for evaluating manager skill.
  • Money-Weighted Return (MWR) — equals the IRR of all portfolio cash flows; reflects impact of contribution/withdrawal timing; preferred when investor controls cash flows.

5. Continuously Compounded Returns

r_cc = ln(1 + HPR)  |  HPR = e^(r_cc) − 1
  • For a price move from P₀ to P₁: r_cc = ln(P₁/P₀).
  • CC EAR = e^(stated rate) − 1 (always > discrete compounding EAR).
  • For positive returns: HPR > CC return.

6. Return Components

Nominal Rate = Real Risk-Free Rate + Inflation + Default + Liquidity + Maturity Premiums
  • Real return = (1 + nominal) / (1 + inflation) − 1 → measures purchasing power gain.
  • T-bill yield = nominal risk-free rate (includes inflation premium; no default/liquidity risk).
  • Real risk-free rate = pure time preference; no inflation, no risk premiums.
  • Risk premium = equity return − risk-free rate.
  • Gross return (before fees) is used for manager-to-manager comparison.
  • Net return = gross return − management fees.
PART 2 — PRACTICE QUESTIONS (75 Questions)

Click Show Answer to reveal the answer and explanation.

LO-A: Holding Period Return & Time-Weighted Return — Q1 to Q20
Q1
An investor purchased 150 shares of a stock for $20 per share. After three months, all shares were sold at $21 per share. The investor received a $300 dividend before selling. The holding period return is closest to:

A. 3.7%

B. 5.0%

C. 15.0%

C. 15.0% — HPR = (Price gain + Dividends) / Cost = (150×1 + 300) / (150×20) = 450/3000 = 15.0%.
Q2
A stock is currently worth $75. It was purchased one year ago for $60, and paid a $1.50 dividend during the year. The holding period return is closest to:

A. 22.0%

B. 24.0%

C. 27.5%

C. 27.5% — HPR = (75 − 60 + 1.50) / 60 = 16.50 / 60 = 27.5%.
Q3
A fund had annual returns of −12%, 8%, and 15%. The fund's three-year holding period return is closest to:

A. 3.0%

B. 3.7%

C. 9.3%

C. 9.3% — HPR = (0.88)(1.08)(1.15) − 1 = 1.09296 − 1 = 9.3%. (Holding period return chains all sub-period returns.)
Q4
An investor buys a share of stock for $100. After year one she buys three more shares at $89 each. At end of year two she sells all four shares for $98. The stock paid a $1.00 per share dividend at the end of both years. The investor's time-weighted return is closest to:

A. 0.06%

B. 6.35%

C. 11.24%

A. 0.06% — Sub-period 1: (89 + 1 − 100)/100 = −10%. Sub-period 2: (4×98 + 4×1 − 4×89)/(4×89) = 40/356 = 11.24%. TWR = √(0.90 × 1.1124) − 1 = √1.0012 − 1 ≈ 0.06%.
Q5
Using the same data as Q4, the investor's money-weighted return is closest to:

A. 0.06%

B. 5.29%

C. 6.35%

C. 6.35% — CF0 = −100; CF1 = +1 − 267 = −266; CF2 = 4×98 + 4×1 = +396. Solve IRR: 100x² + 266x − 396 = 0 → x ≈ 1.0635 → r ≈ 6.35%.
Q6
An investor buys a share at $200 at t=0. At t=1 she buys another at $225. At t=2 she sells both at $235. Each year the stock pays a $5/share dividend. The time-weighted return is closest to:

A. 9.4%

B. 10.8%

C. 15.0%

B. 10.8% — HPR1 = (225+5−200)/200 = 15%. HPR2 = (2×235+2×5−2×225)/(2×225) = 30/450 = 6.67%. TWR = √(1.15×1.0667)−1 = √1.2267−1 ≈ 10.8%.
Q7
Using the data from Q6, the money-weighted return is closest to:

A. 9.4%

B. 10.8%

C. 7.7%

A. 9.4% — CF0 = −200; CF1 = +5−225 = −220; CF2 = 2×235+2×5 = +480. Solve: 200x² + 220x − 480 = 0 → x ≈ 1.094 → MWR ≈ 9.4%.
Q8
On January 1, an investor deposits $50,000. By March 31 it grows to $51,000. On April 1 she adds $10,000 (account = $61,000). By June 30 account is $60,000. On July 1 she withdraws $30,000 (balance $30,000). By Dec 31 it grows to $33,000. The annual time-weighted return is closest to:

A. 5.5%

B. 7.0%

C. 10.4%

C. 10.4% — HPR1 = 51/50−1 = 2%. HPR2 = 60/61−1 = −1.64%. HPR3 = 33/30−1 = 10%. TWR = (1.02)(0.9836)(1.10)−1 = 10.4%.
Q9
An investor purchases a share of stock for $50. After year one she purchases an additional share for $75. After year two she sells both shares at $100. Year 1 dividend was $5/share; Year 2 was $7.50/share. The time-weighted return is closest to:

A. 23.2%

B. 51.4%

C. 51.7%

B. 51.4% — HPR1 = (75+5−50)/50 = 60%. HPR2 = (2×100+2×7.5−2×75)/(2×75) = 65/150 = 43.3%. TWR = √(1.60×1.433)−1 = √2.293−1 ≈ 51.4%.
Q10
Using the same data as Q9, the money-weighted return is closest to:

A. 16.1%

B. 48.9%

C. 51.4%

B. 48.9% — CF0 = −50; CF1 = +5−75 = −70; CF2 = 2×100+2×7.5 = +215. Solve: 50x² + 70x − 215 = 0 → x ≈ 1.489 → MWR ≈ 48.9%.
Q11
A portfolio manager is responsible for a $1 million account. Quarterly dividend yield and capital gains are: Q1: 2.5%, −10.4%; Q2: 1.8%, 5.7%; Q3: 2.0%, 8.9%; Q4: 3.0%, 12.0%. The annualized geometric mean return is closest to:

A. 6.00%

B. 6.08%

C. 26.27%

B. 6.08% — Quarterly returns: Q1=(1.025)(0.896)−1=−7.9%; Q2=(1.018)(1.057)−1=7.6%; Q3=(1.02)(1.089)−1=11.1%; Q4=(1.03)(1.12)−1=15.4%. Product = (0.921)(1.076)(1.111)(1.154) = 1.2627. Geometric mean = 1.2627^(1/4)−1 = 6.08%.
Q12
A fund had annual returns of 10%, 18%, −2%, and −8% over four years. The annualized holding period return (geometric) is closest to:

A. 4.01%

B. 4.50%

C. 17.03%

A. 4.01% — Product = (1.10)(1.18)(0.98)(0.92) = 1.17027. Annualized = 1.17027^(1/4)−1 ≈ 4.01%.
Q13
An investor earns annual returns of 15.5%, 6.4%, −5.8%, 8.9%, −7.7% over five years. The geometric mean annual return is closest to:

A. 0.23%

B. 1.16%

C. 3.08%

C. 3.08% — Product = (1.155)(1.064)(0.942)(1.089)(0.923) = 1.1636. GM = 1.1636^(1/5)−1 ≈ 3.08%.
Q14
An asset manager reports a return of −5.34% for a 3-year period with annual returns of +6%, −37%, and +27%. The manager has reported the:

A. Arithmetic mean return

B. Geometric mean return

C. Holding period return

B. Geometric mean return — GM = [(1.06)(0.63)(1.27)]^(1/3) − 1 = [0.8483]^(1/3) − 1 ≈ −5.34%. The geometric mean correctly compounds the sub-period returns.
Q15
A portfolio manager records AUM-weighted returns for 4 years: Year 1 ($35M) = 10%, Year 2 ($40M) = 18%, Year 3 ($28M) = −2%, Year 4 ($45M) = −8%. For performance evaluation independent of cash flow timing, the appropriate return is the time-weighted return, which equals:

A. 4.01%

B. 4.50%

C. 17.03%

A. 4.01% — TWR uses geometric mean of sub-period returns ignoring AUM. Product = (1.10)(1.18)(0.98)(0.92) = 1.17027. Annualized = 1.17027^(0.25)−1 ≈ 4.01%.
Q16
An analyst has the following annual returns: 5.26%, −2.10%, 3.86%, 8.18%. The arithmetic mean return is closest to:

A. 3.73%

B. 3.80%

C. 3.76%

B. 3.80% — AM = (5.26 − 2.10 + 3.86 + 8.18) / 4 = 15.20 / 4 = 3.80%.
Q17
An investor buys a stock on March 24 for $63.25. The stock pays quarterly dividends of $0.54 on May 1 and August 1. On September 27 the stock is sold for $62.80. The holding period return is closest to:

A. 1.0%

B. 2.0%

C. 2.5%

A. 1.0% — HPR = (62.80 − 63.25 + 0.54 + 0.54) / 63.25 = 0.63 / 63.25 ≈ 1.0%.
Q18
Stock XYZ is purchased on January 2 at $12/share. The investor receives a quarterly dividend of $0.60 on April 1. The stock closes on June 30 at $13. The holding period return is closest to:

A. 8.33%

B. 13.33%

C. 18.33%

B. 13.33% — HPR = (13 − 12 + 0.60) / 12 = 1.60 / 12 = 13.33%.
Q19
A 10% coupon bond was purchased for $1,000. One year later it was sold for $915. The investor's holding period yield is closest to:

A. 1.5%

B. 9.0%

C. 18.5%

A. 1.5% — HPR = (915 − 1000 + 100) / 1000 = 15 / 1000 = 1.5%. The bond's coupon income partially offsets the capital loss.
Q20
A bond was purchased for $910 and sold one year later for $1,020. Two semi-annual coupon payments of $30 were received during the year. The holding period return is closest to:

A. 6.0%

B. 12.1%

C. 18.7%

C. 18.7% — HPR = (1020 − 910 + 60) / 910 = 170 / 910 = 18.7%.
LO-B: Annualized Returns & Continuously Compounded Returns — Q21 to Q40
Q21
Over a period of 18 months an investor has earned a return of 28.4%. The investor's annualized return is closest to:

A. 18.1%

B. 20.4%

C. 25.5%

A. 18.1% — Annualized = (1.284)^(12/18) − 1 = (1.284)^(2/3) − 1 ≈ 18.1%. Raise to the power of periods per year.
Q22
An investor has earned a return of 3.4% for 45 days. The annualized return is closest to:

A. 31.1%

B. 40.8%

C. 49.4%

A. 31.1% — Annualized = (1.034)^(365/45) − 1 = (1.034)^8.111 − 1 ≈ 31.1%.
Q23
If an investor's weekly return is 0.3%, then his compound annual return is closest to:

A. 3.6%

B. 15.6%

C. 16.8%

C. 16.8% — Annual = (1.003)^52 − 1 ≈ e^(0.003×52) − 1 = e^0.156 − 1 ≈ 16.8%.
Q24
Jennet's portfolio return for the last 120 days is 6.8% and Adamson's return for the last 18 months is 16%. The annualized return for Adamson's portfolio is closest to:

A. 9.2% higher than Jennet's

B. 11.7% lower than Jennet's

C. 22.7% higher than Jennet's

B. 11.7% lower than Jennet's — Jennet annualized = (1.068)^(365/120) − 1 ≈ 22.1%. Adamson annualized = (1.16)^(12/18) − 1 ≈ 10.4%. Difference = 22.1 − 10.4 = 11.7%, so Adamson is 11.7% lower.
Q25
If a stock decreases from $90 to $80, the continuously compounded rate of return for the period is closest to:

A. −0.1250

B. −0.1000

C. −0.1178

C. −0.1178 — CC = ln(80/90) = ln(0.8889) = −0.1178. The CC return is always ln(ending/beginning).
Q26
A stock was priced at €44.23 one year ago and is now €42.00 (no dividend). The continuously compounded rate of return is closest to:

A. −5.17%

B. −5.04%

C. +5.17%

A. −5.17% — CC = ln(42/44.23) = ln(0.9496) ≈ −5.17%. For a price decline the CC rate is negative and more negative than the simple HPR.
Q27
A portfolio declines from $127,350 to $108,427 over one year. The continuously compounded rate of return is closest to:

A. −14.86%

B. −13.84%

C. −16.09%

C. −16.09% — CC = ln(108427/127350) = ln(0.8514) ≈ −16.09%. The simple HPR (−14.86%) is the holding period return; the CC rate is more negative.
Q28
The continuously compounded rate of return that will generate a one-year holding period return of −6.5% is closest to:

A. −5.7%

B. −6.3%

C. −6.7%

C. −6.7% — CC = ln(1 + HPR) = ln(1 − 0.065) = ln(0.935) ≈ −6.7%.
Q29
A stated annual interest rate of 9% compounded continuously results in an effective annual rate closest to:

A. 9.42%

B. 9.20%

C. 9.67%

A. 9.42% — EAR = e^0.09 − 1 = 1.09417 − 1 = 9.42%.
Q30
For a given stated annual rate of return, compared to the effective rate with discrete compounding, the effective rate with continuous compounding will be:

A. The same

B. Higher

C. Lower

B. Higher — Continuous compounding yields the highest EAR for a given stated rate because interest compounds infinitely. EAR_continuous = e^r − 1 > (1 + r/m)^m − 1 for any finite m.
Q31
Given a holding period return of R, the continuously compounded rate of return is:

A. e^R − 1

B. ln(1 + R)

C. ln(1 + R) − 1

B. ln(1 + R) — By definition: HPR = e^(r_cc) − 1 → r_cc = ln(1 + HPR). This is the natural log of the gross return.
Q32
A stock increased in value last year. Which return measure will be greater: its continuously compounded return or its holding period return?

A. The continuously compounded return

B. The holding period return

C. Neither; they are equal

B. The holding period return — For any positive return R, HPR > CC return because ln(1+R) < R when R > 0. The HPR always exceeds the CC return for positive performance.
Q33
An investor buys a non-dividend paying stock for $100 with 50% initial margin. At year-end the stock is $95. Deflation of 2% occurred. Which return measure is greatest?

A. Leveraged return

B. Real return

C. Nominal return

B. Real return — Nominal HPR = −5%. Real = (0.95/0.98) − 1 = −3.06%. Leveraged = 2 × (−5%) = −10%. Real > Nominal > Leveraged.
Q34
An investor sold a 30-year bond at $850 after purchasing it at $800 one year ago and received $50 interest at the time of sale. The annualized holding period return is closest to:

A. 12.5%

B. 15.0%

C. 6.25%

A. 12.5% — HPR = (850 − 800 + 50) / 800 = 100 / 800 = 12.5%. This is already a one-year return, so no annualization adjustment is needed.
Q35
An investor buys a stock at $32 and sells nine months later at $37.50 after receiving $2 in dividends. The holding period return is closest to:

A. 17.19%

B. 23.44%

C. 32.42%

B. 23.44% — HPR = (37.50 − 32 + 2) / 32 = 7.50 / 32 = 23.44%. Note: this is the HPR for the 9-month period, not annualized.
Q36
An investor expects a $20 stock to reach $25 by year-end. Last year's dividend was $1 but this year's is expected to be $1.25. The expected holding period return is closest to:

A. 24.00%

B. 28.50%

C. 31.25%

C. 31.25% — HPR = (25 − 20 + 1.25) / 20 = 6.25 / 20 = 31.25%.
Q37
An investor begins with a $100,000 portfolio. At end of period 1, it generates $5,000 income (not reinvested). At end of period 2, $25,000 is contributed. At end of period 3, the portfolio is $123,000. The money-weighted return per period is closest to:

A. 1.20%

B. 0.94%

C. −0.50%

B. 0.94% — CF0 = −100,000; CF1 = +5,000; CF2 = −25,000; CF3 = +123,000. Solving for IRR per period ≈ 0.94%.
Q38
An investor with a buy-and-hold strategy makes quarterly deposits into an account. To evaluate portfolio performance, the investor should most appropriately use the portfolio's:

A. Arithmetic mean return

B. Geometric mean return

C. Money-weighted return

C. Money-weighted return — When the investor controls the timing of contributions, the MWR captures the impact of those decisions and is the most appropriate performance measure.
Q39
Which return measure is most appropriate for evaluating and comparing the investment skills of asset managers?

A. Net return

B. Gross return

C. Pre-tax return

B. Gross return — Gross return measures the manager's investment skill before fees. It allows fair comparison across managers with different fee structures.
Q40
A security portfolio earns a gross return of 7.0% and a net return of 6.5%. The 0.5% difference most likely results from:

A. Inflation

B. Fees

C. Taxes

B. Fees — Gross return minus net return equals management fees and operating expenses. Inflation affects real returns; taxes apply after-investment, not gross-to-net.
LO-C: Return Components, Real Returns & Risk Premiums — Q41 to Q57
Q41
The real risk-free rate is best described as:

A. Approximately the nominal risk-free rate plus expected inflation

B. Approximately the nominal risk-free rate reduced by expected inflation

C. Exactly the nominal risk-free rate reduced by expected inflation

B. Approximately the nominal risk-free rate reduced by expected inflation — Real risk-free rate ≈ Nominal risk-free rate − Inflation premium. The exact relationship uses (1+nominal)/(1+inflation)−1, so 'approximately' is the correct qualifier.
Q42
Which return measure is best described as purely representing time preference?

A. Real risk-free interest rate

B. Nominal risk-free interest rate

C. Total rate of return

A. Real risk-free interest rate — The real risk-free rate reflects only the pure time value of money — compensation for delaying consumption — stripped of inflation, default, liquidity, and maturity premiums.
Q43
T-bill yields can best be thought of as:

A. Nominal risk-free rates because they contain an inflation premium

B. Nominal risk-free rates because they do not contain an inflation premium

C. Real risk-free rates because they contain an inflation premium

A. Nominal risk-free rates because they contain an inflation premium — T-bills are default-free and highly liquid, making them 'risk-free.' However, they incorporate an inflation premium, so their yield is a nominal (not real) risk-free rate.
Q44
Which of the following best describes the components of the required interest rate on a security?

A. Real risk-free rate + inflation + default risk premium + liquidity premium + maturity premium

B. Nominal risk-free rate + inflation + default risk premium + liquidity premium + maturity premium

C. Real risk-free rate + default risk premium + liquidity premium + maturity premium

A. Real risk-free rate + inflation + default risk premium + liquidity premium + maturity premium — The nominal risk-free rate = real risk-free rate + inflation premium. So the full decomposition starts with the real risk-free rate, then adds all four risk premiums.
Q45
The following data applies: Treasury Bills = 2.5%, Equities = 7.6%, Inflation = 0.8%. The real rate of return for equities is closest to:

A. 4.30%

B. 6.75%

C. 7.6%

B. 6.75% — Real return = (1.076 / 1.008) − 1 = 1.0675 − 1 = 6.75%. Use the exact Fisher formula for precision.
Q46
Using the same data (T-Bills = 2.5%, Equities = 7.6%, Inflation = 0.8%), the risk premium for equities is closest to:

A. 4.30%

B. 4.98%

C. 6.80%

B. 4.98% — Risk premium = (1.076 / 1.025) − 1 = 1.0498 − 1 = 4.98%. This is the excess return over the risk-free rate using the exact formula.
Q47
The following data applies: Risk Premium of Stock A = 4.5%, Risk-Free Rate = 2.3%, Inflation = 0.8%. The real return for Stock A is closest to:

A. 5.34%

B. 6.90%

C. 7.76%

B. 6.90% — Real return = (1 + real risk-free)(1 + risk premium) − 1 = (1.023)(1.045) − 1 ≈ 6.90%. The 2.3% risk-free rate here is the real risk-free rate.
Q48
An investor projects a net return of 15.0% on his $1 million portfolio. Tax rate is 25% on dividends and capital gains. Inflation is 4%. The expected after-tax real return is closest to:

A. 6.97%

B. 7.93%

C. 15.70%

A. 6.97% — After-tax nominal = 15% × (1 − 0.25) = 11.25%. Real = (1.1125 / 1.04) − 1 = 6.97%.
Q49
The most appropriate measure of the increase in purchasing power of a portfolio's value over time is:

A. After-tax return

B. Real return

C. Holding period return

B. Real return — The real return strips out inflation and therefore directly measures the gain in purchasing power. The nominal or HPR includes the effects of inflation.
Q50
An asset management firm generated the following returns: 2017: 24%, 2018: 13%, 2019: −7.6%. The 2020 return needed to achieve a trailing four-year geometric mean of 10% is closest to:

A. 9.82%

B. 13.08%

C. 15.67%

B. 13.08% — (1.24)(1.13)(0.924)(1+x) = (1.10)^4 = 1.4641. (1.24)(1.13)(0.924) = 1.2947. 1+x = 1.4641/1.2947 = 1.1308 → x = 13.08%.
Q51
An asset management firm generated returns: 2016: 8.84%, 2017: −4.57%, 2018: 11.69%, 2019: 15.50%. The 2020 return needed to achieve a trailing five-year geometric mean of 10% is closest to:

A. 17.86%

B. 20.20%

C. 43.51%

B. 20.20% — Product of known years = (1.0884)(0.9543)(1.1169)(1.155) = 1.3397. Need: 1.3397(1+x) = (1.10)^5 = 1.6105. 1+x = 1.2022 → x = 20.20%.
Q52
Wei Zhang has funds earning 6% interest. If he withdraws $15,000 to buy a car, the 6% interest rate is best thought of as a(n):

A. Discount rate

B. Financing cost

C. Opportunity cost

C. Opportunity cost — The 6% represents the return he is giving up by withdrawing funds. When a rate reflects forgone returns, it is an opportunity cost.
Q53
A bank wants to raise funds with 2-year CDs. Investors require 1.0% more than savings for a 1-year CD and 1.5% more for a 2-year CD. If the savings rate is 3%, the 2-year CD yield must be at least:

A. 4.0% — required rate of return

B. 4.5% — discount rate

C. 4.5% — required rate of return

C. 4.5% — required rate of return — Investors require 3% + 1.5% = 4.5% to invest for 2 years. This is a required rate of return — the minimum return they demand to make the investment.
Q54
Selmer Jones wants to set aside money for a vacation in Hawaii one year from now. His bank pays 5%. To determine how much to set aside today, he should use the 5% as a:

A. Discount rate

B. Opportunity cost

C. Required rate of return

A. Discount rate — He is computing a present value — finding how much today grows to the needed amount at 5%. Used to find PV, the interest rate is the discount rate.
Q55
An analyst reviews two periods for U.S. equities — 2011–2015 (real return 3%, risk premium 12%, inflation 5%) and 2015–2020 (real return 5%, risk premium 8%, inflation 7%). For a useful return comparison, the period with a higher nominal equity return is most likely:

A. 2011–2015

B. 2015–2020

C. Neither; both periods earned equivalent returns

C. Neither; both periods earned equivalent returns — Nominal equity return ≈ real return + risk premium + inflation. 2011–2015: 3+12+5 = 20%. 2015–2020: 5+8+7 = 20%. Both yield ≈20%, so neither period dominates.
Q56
Mike Roland gathered 10 years of historical U.S. equity returns to estimate expected future returns. This approach of using historical data to estimate forward-looking expected returns is best described as:

A. Applying a discount rate approach

B. Using historical returns as proxies for expected returns

C. Applying a required return framework

B. Using historical returns as proxies for expected returns — When historical returns are used to estimate future expected returns, the analyst is using past data as a proxy. This is common practice but assumes the past is representative of the future.
Q57
An investor's portfolio has declined from $127,350 to $108,427. The continuously compounded rate of return is closest to:

A. −14.86%

B. −16.09%

C. −13.84%

B. −16.09% — CC = ln(108427/127350) = ln(0.8514) ≈ −16.09%. The simple HPR = −14.86%, but CC is more negative for a decline.
LO-D: Mean Returns — Arithmetic, Geometric & Harmonic — Q58 to Q75
Q58
An analyst evaluates returns of 5.26%, −2.10%, 3.86%, 8.18%. The arithmetic mean is closest to:

A. 3.73%

B. 3.80%

C. 3.76%

B. 3.80% — AM = (5.26 − 2.10 + 3.86 + 8.18) / 4 = 15.20 / 4 = 3.80%.
Q59
A dataset has no equal values. The arithmetic mean is 13.25 and the geometric mean is 12.75. The harmonic mean will be:

A. Less than 12.75

B. Between 12.75 and 13.25

C. Greater than 13.25

A. Less than 12.75 — For any dataset with unequal values: AM ≥ GM ≥ HM. Since GM = 12.75, the HM must be less than 12.75.
Q60
The relationship between arithmetic, geometric, and harmonic means for a dataset with at least some variation is:

A. AM > GM > HM

B. GM > AM > HM

C. HM > GM > AM

A. AM > GM > HM — AM ≥ GM ≥ HM always holds (with equality only when all values are identical). Equivalently: Harmonic ≤ Geometric ≤ Arithmetic.
Q61
The product of the arithmetic mean and the harmonic mean equals:

A. The square root of the geometric mean

B. The square of the geometric mean

C. The geometric mean itself

B. The square of the geometric mean — AM × HM = GM². This identity allows computation of any one mean given the other two.
Q62
An analyst calculates the geometric mean of 8 values as 8.50 and the arithmetic mean as 8.90. The harmonic mean is closest to:

A. 8.12

B. 8.63

C. 9.30

A. 8.12 — HM = GM² / AM = 8.50² / 8.90 = 72.25 / 8.90 = 8.12.
Q63
Based on the advice of his advisor regarding dollar-cost averaging, a client invests $2,000 monthly. Stock prices over four months were $12, $14, $11, $9. Using the harmonic mean, the average cost per share is closest to:

A. $11.50

B. $11.75

C. $11.20

C. $11.20 — HM = 4 / (1/12 + 1/14 + 1/11 + 1/9) = 4 / (0.0833 + 0.0714 + 0.0909 + 0.1111) = 4 / 0.3568 ≈ $11.21 ≈ $11.20.
Q64
An investor employs dollar-cost averaging. The most appropriate mean to use when computing the average purchase price per unit is:

A. Arithmetic mean

B. Geometric mean

C. Harmonic mean

C. Harmonic mean — Dollar-cost averaging buys a fixed dollar amount each period. The average cost per unit equals the harmonic mean of the purchase prices — not the arithmetic mean.
Q65
Time-weighted returns are used by the investment management industry because they:

A. Take all cash inflows and outflows into account using IRR

B. Result in higher returns than money-weighted returns

C. Are not affected by the timing of cash flows

C. Are not affected by the timing of cash flows — TWR divides the overall period into sub-periods around each cash flow and compounds sub-period returns, eliminating the distortion caused by the timing and magnitude of external cash flows.
Q66
If funds are contributed to a portfolio just prior to a period of favorable performance, the:

A. Money-weighted return will tend to be depressed

B. Money-weighted return will tend to be elevated

C. Time-weighted return will tend to be elevated

B. Money-weighted return will tend to be elevated — MWR (IRR) is sensitive to the timing of cash flows. A contribution immediately before strong performance means more dollars earned good returns, pushing the MWR higher versus TWR.
Q67
Computing the internal rate of return of the inflows and outflows of a portfolio gives the:

A. Money-weighted return

B. Net present value

C. Time-weighted return

A. Money-weighted return — The money-weighted return (MWR) is defined as the IRR of the portfolio's cash flows — setting the NPV of all inflows and outflows to zero.
Q68
An analyst is using the money-weighted rate of return to assess portfolio performance. This measure will best assess which objective?

A. The return earned on the money invested

B. A comparison of returns between similar portfolios

C. A return comparison between different investment opportunities

A. The return earned on the money invested — MWR measures the return on the actual dollars invested, accounting for the timing of contributions and withdrawals. It is NOT suitable for manager-to-manager comparison because it's affected by cash flow timing.
Q69
Which of the following is NOT an advantage of the arithmetic mean return?

A. It accurately estimates compound growth over multiple periods

B. It has well-known statistical properties

C. It can test whether the mean return is statistically different from zero

A. It accurately estimates compound growth over multiple periods — The arithmetic mean overestimates compound (multi-period) growth when returns are volatile. The geometric mean is the correct measure for compound growth. Options B and C are genuine advantages of AM.
Q70
A portfolio manager generated annual returns of 24%, −15%, 32%, and −8%. Which return measure is the most appropriate single estimate of the portfolio's compound annual growth?

A. Arithmetic mean

B. Geometric mean

C. Harmonic mean

B. Geometric mean — The geometric mean correctly captures the effect of compounding. AM = (24−15+32−8)/4 = 8.25%, but this overstates compound growth. GM = [(1.24)(0.85)(1.32)(0.92)]^(1/4)−1 ≈ 6.1%.
Q71
Jennet and Adamson are portfolio managers. Jennet earned 6.8% over 120 days; Adamson earned 16% over 18 months. Which statement about their annualized returns is most accurate?

A. Adamson's annualized return exceeds Jennet's by about 9.2%

B. Adamson's annualized return is about 11.7% lower than Jennet's

C. Their annualized returns are approximately equal

B. Adamson's annualized return is about 11.7% lower than Jennet's — Jennet annualized = (1.068)^(365/120)−1 ≈ 22.1%. Adamson annualized = (1.16)^(12/18)−1 ≈ 10.4%. Difference ≈ 11.7%, with Adamson lower.
Q72
An investor buys one share at $100. At end of year 1 she buys three more at $89. At end of year 2 she sells all four at $98. The stock pays $1/share dividend at end of each year. The money-weighted return is closest to:

A. 0.06%

B. 5.29%

C. 6.35%

C. 6.35% — CF0 = −100; CF1 = 1−267 = −266; CF2 = 4(98+1) = 396. Solve IRR: 100x² + 266x − 396 = 0 → x ≈ 1.0635 → 6.35%.
Q73
Using the data from Q72, the time-weighted return is closest to:

A. 0.06%

B. 6.35%

C. 11.24%

A. 0.06% — HPR1 = (89+1−100)/100 = −10%. HPR2 = (4×99−4×89)/(4×89) = 40/356 = 11.24%. TWR = √(0.90×1.1124)−1 = √1.0012−1 ≈ 0.06%. The MWR (6.35%) is higher because more dollars were invested during the favorable second period.
Q74
Which of the following statements about money-weighted vs. time-weighted returns is most accurate?

A. TWR is preferred when the manager controls cash flow timing

B. MWR is preferred when evaluating manager skill across different portfolios

C. TWR reflects the impact of investor contributions and withdrawals

A. TWR is preferred when the manager controls cash flow timing — TWR removes the impact of external cash flows and is the standard for evaluating investment manager skill. MWR is appropriate when the investor (not the manager) controls cash flow timing.
Q75
An investor sold a bond for $850 one year after buying it for $800. She received $50 interest at the time of sale. Simultaneously, inflation was 3% during the year. The real holding period return is closest to:

A. 9.2%

B. 12.5%

C. 15.0%

A. 9.2% — Nominal HPR = (850−800+50)/800 = 12.5%. Real HPR = (1.125/1.03)−1 = 1.092−1 = 9.2%.
Concepts 'n' Clarity®
For Educational Use Only
Ch 1 — Rates & Returns
Concepts 'n' Clarity®
Ch 2 — Time Value of Money
CFA Level I  •  FRM Part I
PART 1 — CORE CONCEPTS

1. Interest Rates — Interpretation

  • Required rate of return — minimum return an investor must receive to accept an investment.
  • Discount rate — rate used to find the present value of future cash flows.
  • Opportunity cost — return forgone by choosing a particular investment.

2. Interest Rate Components

Nominal Rate = Real Risk-Free Rate + Inflation Premium + Default Risk Premium + Liquidity Premium + Maturity Premium

3. Effective Annual Rate (EAR)

EAR = (1 + Stated Rate / m)^m − 1   [discrete]
EAR = e^(r_cc) − 1   [continuous]

4. Future Value & Present Value

FV = PV × (1 + r)^N  |  PV = FV / (1 + r)^N
Continuous: FV = PV × e^(rN)

5. Annuities & Perpetuities

  • Ordinary annuity — cash flows at END of each period.
  • Annuity due — cash flows at START of each period (BGN mode on BA II Plus).
  • Perpetuity PV = PMT / r
PART 2 — 75 PRACTICE QUESTIONS
LO.a & b — Interest Rates & Risk Premiums
Q1
The minimum rate of return that an investor must receive in order to invest in a project is most likely known as the:

A. Required rate of return.

B. Real risk-free interest rate.

C. Inflation rate.

A is correct. The required rate of return is the minimum rate of return an investor must receive to accept an investment.
Q2
Which of the following is least likely to be an accurate interpretation of interest rates?

A. The rate needed to calculate present value.

B. Opportunity cost.

C. The maximum rate of return an investor must receive to accept an investment.

C is correct. Interest rates represent the minimum (not maximum) rate of return an investor must receive to accept the investment.
Q3
A security has a nominal interest rate of 15%. The real risk-free rate = 3.5%, default risk premium = 3%, maturity risk premium = 4%, liquidity risk premium = 2%. The inflation premium is closest to:

A. 2.5%

B. 4.0%

C. 9.0%

A is correct. Inflation premium = 15% − 3.5% − 3% − 4% − 2% = 2.5%.
Q4
A U.S. Treasury bond yields 5%; an otherwise identical corporate bond yields 7%. The most likely explanation for the yield difference is:

A. Default risk premium.

B. Inflation premium.

C. Real risk-free interest rate.

A is correct. The difference in yield between otherwise identical Treasury and corporate bonds reflects default risk.
Q5
The maturity premium best compensates investors for the:

A. Risk of loss relative to an investment's fair value if converted to cash quickly.

B. Increased sensitivity of debt market value to interest rate changes as maturity extends.

C. Possibility that the borrower will fail to make a promised payment.

B is correct. The maturity premium compensates for increased interest rate sensitivity of longer-duration debt. A describes liquidity risk; C describes credit risk.
Q6
The liquidity premium best compensates investors for:

A. Inability to sell a security at its fair market value.

B. Locking funds for longer durations.

C. A risk that an investment's value may change over time.

A is correct. The liquidity premium compensates investors for the inability to sell a security at its fair market value.
Q7
Nominal rate = 20%, real risk-free rate = 5%, inflation premium = 4%. If the risk premium incorporates default, liquidity, and maturity risk, the combined risk premium is closest to:

A. 20%

B. 15%

C. 11%

C is correct. 20 = 5 + 4 + X → X = 11%.
Q8
When estimating the required rate of return, which premium is least likely to be relevant?

A. Inflation premium.

B. Maturity premium.

C. Nominal premium.

C is correct. There is no such thing as a "nominal premium." The required rate uses inflation, maturity, default, and liquidity premiums.
LO.c — Effective Annual Rate (EAR)
Q9
A financial instrument has a stated annual rate of 22% compounded quarterly. The EAR is closest to:

A. 23%

B. 24%

C. 25%

B is correct. EAR = (1 + 0.22/4)⁴ − 1 = (1.055)⁴ − 1 ≈ 24%.
Q10
A mortgage has a nominal annual rate of 7% and EAR of 7.18%. The compounding frequency is most likely:

A. Semi-annual.

B. Quarterly.

C. Monthly.

B is correct. (1 + 0.07/4)⁴ − 1 = 7.18%.
Q11
Which one-year CD has the highest EAR? CD1: monthly 8.20%; CD2: quarterly 8.25%; CD3: continuously 8.00%.

A. CD1

B. CD2

C. CD3

A is correct. EAR: CD1=(1+.082/12)¹²−1=8.515%; CD2=(1+.0825/4)⁴−1=8.509%; CD3=e^0.08−1=8.328%.
Q12
Stated annual rate = 11%, compounded daily. The EAR is closest to:

A. 11.00%

B. 11.57%

C. 11.63%

C is correct. EAR = (1 + 0.11/365)³⁶⁵ − 1 ≈ 11.63%.
Q13
A fixed-income instrument has stated annual rate 18%, compounded monthly. The EAR is closest to:

A. 18.00%

B. 19.56%

C. 20.12%

B is correct. Periodic rate = 0.18/12 = 0.015; EAR = (1.015)¹² − 1 = 19.56%.
Q14
An investment earns 12% per year compounded quarterly. The effective annual rate is:

A. 3.00%

B. 12.00%

C. 12.55%

C is correct. EAR = (1 + 0.12/4)⁴ − 1 = (1.03)⁴ − 1 = 12.55%.
Q15
Which continuously compounded rate corresponds to an EAR of 7.45%?

A. 7.19%

B. 7.47%

C. 7.73%

A is correct. e^r = 1.0745 → r = ln(1.0745) = 7.19%.
Q16
Canadian Foods had operating profit of $2.568 million in 2008 and $5.229 million in 2012. The compounded annual growth rate (CAGR) is closest to:

A. 16.30%

B. 18.50%

C. 19.50%

C is correct. CAGR = (5.229/2.568)^(1/4) − 1 ≈ 19.5%. N=4 because from end-2008 to end-2012 is 4 years.
Q17
A company had 81 outlets in 2009 and 67 outlets in 2012. The growth rate of outlets is most likely:

A. −6.10%

B. −4.63%

C. 6.53%

A is correct. g = (67/81)^(1/3) − 1 = (0.8272)^0.333 − 1 ≈ −6.10%.
LO.d — FV & PV with Different Compounding Frequencies
Q18
How much must an investor deposit today at 8% continuously compounded to have $2,238 in 5 years?

A. $1,500

B. $1,523

C. $1,541

A is correct. PV = 2,238 / e^(0.08×5) = 2,238 / e^0.40 = 2,238 / 1.4918 ≈ $1,500.
Q19
The PV of $10,000 received in 5 years at 9% compounded monthly is closest to:

A. $6,387

B. $6,499

C. $6,897

A is correct. N=60, %i=9/12=0.75, PMT=0, FV=10,000 → CPT PV = $6,387.
Q20
An investor deposits £1,000 at 9% nominal continuously compounded. Value at end of 6 years is closest to:

A. £1,677

B. £1,712

C. £1,716

C is correct. FV = 1,000 × e^(0.09×6) = 1,000 × e^0.54 ≈ £1,716.
Q21
A client invests $2 million for 4 years at 7.5% annual interest compounded annually. Value at maturity is most likely:

A. $2.150 million

B. $2.600 million

C. $2.671 million

C is correct. FV = 2,000,000 × (1.075)⁴ = $2.671 million.
Q22
A client deposits $5 million at 5% compounded quarterly. Value after 2.5 years is:

A. $5.625 million

B. $5.649 million

C. $5.661 million

C is correct. N=10, %i=1.25, PV=5,000,000 → FV = $5.661 million.
Q23
An investor will invest ¥12 million three years from now at 8% per year. FV at 11 years from now is:

A. ¥22.21 million

B. ¥27.98 million

C. ¥35.25 million

A is correct. N = 11−3 = 8 years, PV = 12M, r = 8% → FV = 12×(1.08)⁸ ≈ ¥22.21 million.
Q24
A 3-year CD at 10% stated annual rate, compounded quarterly. Initial investment = $80,000. Value at maturity is most likely:

A. $86,151

B. $86,628

C. $107,591

C is correct. N=12, I/Y=2.5, PV=80,000 → FV = $107,591.
Q25
An investor puts $3 million in a bank at 4% annual rate compounded daily (365 days). Value at end of one year is closest to:

A. $3.003 million

B. $3.122 million

C. $3.562 million

B is correct. N=365, I/Y=4/365=0.1096, PV=3,000,000 → FV ≈ $3.122 million.
Q26
You invest $50,000 for 3 years at 3.6% compounded continuously. Value after 3 years is:

A. $51,832

B. $55,702

C. $55,596

B is correct. FV = 50,000 × e^(0.036×3) = 50,000 × e^0.108 ≈ $55,702.
Q27
Which of the following most likely increases as the frequency of compounding increases?

A. Interest rate.

B. Present value.

C. Future value.

C is correct. More frequent compounding means interest earns interest more often, resulting in a higher future value.
Q28
How long will it take $2,500 to grow to $7,500 at 6% per year compounded annually?

A. 11.9 years

B. 18.9 years

C. 21.3 years

B is correct. I/Y=6, FV=7,500, PV=−2,500, PMT=0 → CPT N ≈ 18.9 years.
Q29
Evan deposits $800/month starting one month from today at 7% compounded monthly. He needs $100,000. How many months will it take?

A. 95 months

B. 225 months

C. 250 months

A is correct. %i=7/12=0.583, PV=0, PMT=−800, FV=100,000 → CPT N ≈ 94.17 ≈ 95 months.
LO.e — FV & PV: Ordinary Annuity, Annuity Due, Perpetuity, Unequal Cash Flows
Q30
A security pays $2,500 at the START of each quarter for 3 years. Annual discount rate = 8% compounded quarterly. PV is closest to:

A. $18,840

B. $26,438

C. $26,967

C is correct. Annuity due (BGN mode): N=12, I/Y=2%, PMT=2,500, FV=0 → PV = $26,967.
Q31
A consumer can afford $1,500/month on a 25-year mortgage at 6.8% annual rate and makes a 10% down payment. The most she can pay for the house is closest to:

A. $216,116

B. $240,129

C. $264,706

B is correct. N=300, I/Y=6.8/12, PMT=1,500 → PV = $216,116. With 10% down: max price = $216,116 / 0.90 ≈ $240,129.
Q32
A paper supplier forecasts payments of $360, $550, $400 at end of January, February, March respectively. Today is Jan 1; annual rate = 2.4%. Minimum amount needed today is closest to:

A. $1,287

B. $1,305

C. $1,396

B is correct. Monthly rate = 0.2%. CF0=0, CF1=360, CF2=550, CF3=400 → NPV ≈ $1,305.
Q33
A tenant pays rent of $1,200 monthly due on the FIRST day of each month. Annual rate = 8%. PV of a full year's rent is closest to:

A. $13,333

B. $13,795

C. $13,887

C is correct. Annuity due (BGN mode): N=12, %i=8/12=0.667, PMT=1,200 → PV = $13,887.
Q34
A buyer borrows $200,000 on a 30-year fixed 6% mortgage. Monthly payment is closest to:

A. $556

B. $1,000

C. $1,199

C is correct. N=360, %i=0.5, PV=200,000, FV=0 → CPT PMT = $1,199.
Q35
Ms. Ling borrows €44,000 to buy a car. Loan term = 7 years, 12% annual nominal rate, monthly compounding. Monthly payment is closest to:

A. €776.72

B. €803.43

C. €923.13

A is correct. N=84, %i=1, PV=44,000, FV=0 → CPT PMT = €776.72.
Q36
A consumer borrows with monthly payments of €500 for 4 years; first payment today; annual rate = 3.5% compounded monthly. PV of loan is closest to:

A. €22,038.74

B. €22,365.36

C. €22,430.59

C is correct. Annuity due (BGN mode): N=48, %i=3.5/12, PMT=500, FV=0 → PV = €22,430.59.
Q37
Andy plans to spend €70,000/year for 30 retirement years. He deposits €8,000/year (end of year) at 5% compounded annually. Minimum number of deposits to fund retirement is:

A. 29

B. 42

C. 50

B is correct. PV of annuity needed: N=30, %i=5, PMT=70,000 → PV = 1,076,071. Then N such that PMT=8,000 grows to 1,076,071 → N ≈ 41.9 → 42 deposits.
Q38
Haley deposits $24,000/year (end of year) for 15 years earning 12%/year. Total accumulated at end of year 15 is:

A. $894,713

B. $1,094,713

C. $1,294,713

A is correct. N=15, I/Y=12, PV=0, PMT=24,000 → CPT FV = $894,713.
Q39
FV of an annuity = Annuity Amount × FV Annuity Factor. To compute FV, which input is least likely directly needed?

A. Annuity amount.

B. Future value annuity factor.

C. Interest rate.

C is correct. If the FV annuity factor is already given, the interest rate is embedded in it and need not be used separately.
Q40
Monthly deposits (last day): Jan $1,500; Feb $2,000; Mar $2,000; Apr $2,500; May $3,000; Jun $1,000. Rate = 6%/yr monthly compounding. Total on July 1st is closest to:

A. $12,000

B. $12,148

C. $13,903

B is correct. Monthly rate = 0.5%. Compound each payment forward: Jan×(1.005)⁵ + Feb×(1.005)⁴ + … + Jun×1 ≈ $12,148.
Q41
An insurance contract promises to pay $600,000 in 8 years at 5% annual return. How much should the investor deposit today?

A. $406,104

B. $408,350

C. $886,473

A is correct. N=8, I/Y=5, PMT=0, FV=600,000 → CPT PV = $406,104.
Q42
Option 1: lump sum $2.5 million today. Option 2: 25-year annuity $180,000/year starting today. At 6%/year, which has higher PV?

A. Option 1 — greater PV.

B. Option 2 — greater PV.

C. Equal PV for both.

A is correct. PV of annuity due Option 2 ≈ $2,439,064 < $2,500,000 lump sum. Option 1 has higher PV.
Q43
A security pays $150/year in perpetuity. At a required return of 4.75%, its PV today is closest to:

A. $316

B. $3,158

C. $3,185

B is correct. PV = PMT/r = 150/0.0475 ≈ $3,158.
Q44
A security pays dividends of $50, $100, $150, $200, $250 at years 1–5. At 9% discount rate, PV is closest to:

A. $487

B. $550

C. $616

B is correct. Enter unequal CFs into cash flow register at 9% → CPT NPV ≈ $550.
Q45
A car worth $42,000 requires 15% down payment. Remainder financed over 12 months at 8%/yr compounded monthly. Monthly payment is closest to:

A. $3,105

B. $3,654

C. $3,921

A is correct. Loan = 42,000×0.85 = $35,700. N=12, I/Y=0.667, PV=35,700 → PMT ≈ $3,105.
Q46
Hank buys a $100,000 house with $15,000 down. Remainder: 20-year fixed mortgage, quarterly payments, 10%/yr compounded quarterly. Quarterly payment is closest to:

A. $2,337

B. $2,467

C. $2,726

B is correct. N=80, I/Y=2.5, PV=85,000, FV=0 → CPT PMT = $2,467.
Q47
An investor buys property worth $200,000 with 20% down. Remainder: 15-year loan, monthly payments, 9%/yr compounded monthly. Monthly payment is closest to:

A. $1,137

B. $1,440

C. $1,623

C is correct. Loan = $160,000. N=180, I/Y=0.75, PV=160,000 → PMT ≈ $1,623.
LO.f — Timeline Problems (Complex Multi-Step)
Q48
John must pay $50,000/year (end of year) for his daughter's 4-year college starting in 8 years. At 7% annually, how much must he save each year for 8 years (end of year) to fully fund this?

A. $22,000

B. $18,500

C. $16,500

C is correct. Step 1: PV at start of college (t=7): N=4, %i=7, PMT=50,000 → PV = $169,360. Step 2: PMT to accumulate $169,360 in 8 years: N=8, %i=7, FV=169,360 → PMT ≈ $16,507 ≈ $16,500.
Q49
A 26-year-old plans to retire at 65 and live to 90. Current annual expenditure = $40,000, expected inflation = 2%, expected return = 7%. The minimum accumulation needed at age 65 is closest to:

A. $989,300

B. $1,009,080

C. $1,220,390

B is correct. Expenditure at 65: FV = 40,000×(1.02)³⁹ ≈ $86,590. PV at 65 of 25-year $86,590 annuity at 7% ≈ $1,009,080.
Q50
Sandra (age 35) retires at 75 and lives to 100. Current expenses = $54,000, inflation = 3%, accumulation by 75 = $2 million. Minimum return she must earn on $2M to sustain retirement spending is closest to:

A. 7.29%

B. 7.58%

C. 7.87%

A is correct. Spending at 75: 54,000×(1.03)⁴⁰ ≈ $176,150. Then: N=25, PMT=176,150, PV=−2,000,000 → CPT %i ≈ 7.29%.
Q51
Mr. Das Gupta's daughter (age 11) starts college in 6 years. Annual fees = $45,000 for 4 years. At 5% return, minimum accumulation needed in 6 years is closest to:

A. $160,000

B. $170,000

C. $180,000

A is correct. N=4, %i=5, PMT=45,000 → PV ≈ $159,568 ≈ $160,000.
Q52
Jones plans to pay $40,000/year (beginning of year) for his son's 4-year college starting in 8 years. At 10%, how much must he invest annually (end of year) for 7 years?

A. $13,365

B. $11,087

C. $22,857

A is correct. Step 1 (annuity due at t=7): N=4, %i=10, PMT=40,000 → PV = $126,795. Step 2: N=7, %i=10, FV=126,795 → PMT ≈ $13,365.
Q53
A pension fund will pay $750,000/year for 20 years starting in 12 years. At 8% discount rate, PV of payments today is closest to:

A. $2,924,191

B. $3,158,126

C. $7,363,610

B is correct. PV at t=11 of 20-payment annuity = $7,363,610. Discount to today: 7,363,610/(1.08)¹¹ ≈ $3,158,126.
Q54
A security pays $100/year in perpetuity with first payment at t=4. Required return = 10%. PV today is closest to:

A. $683

B. $751

C. $1,000

B is correct. PV at t=3 = 100/0.10 = $1,000. PV today = 1,000/(1.10)³ ≈ $751.
Q55
Gerard saves for a 5-year doctorate starting 6 years from now. Current annual cost = $7,200 growing 7%/year. He earns 8%/year. Depositing equally for 5 years (end of year), how much per year?

A. $8,370

B. $8,539

C. $8,730

A is correct. Step 1: grow costs: Year 6=$10,805, 7=$11,562, 8=$12,371, 9=$13,237, 10=$14,163. Step 2: PV at t=5 of each = $49,106. Step 3: N=5, %i=8, FV=49,106 → PMT ≈ $8,370.
Additional Practice — Mixed TVM Problems
Q56
An investor wants to triple her money. If the interest rate is 9% per year compounded annually, approximately how many years will it take?

A. 9.6 years

B. 12.7 years

C. 16.6 years

B is correct. Solve: (1.09)^N = 3 → N = ln(3)/ln(1.09) = 1.0986/0.0862 ≈ 12.7 years.
Q57
A preferred stock pays a $5 annual dividend per share indefinitely. If the required return is 8%, the value per share today is:

A. $40.00

B. $62.50

C. $55.56

B is correct. PV of perpetuity = PMT/r = $5/0.08 = $62.50.
Q58
An annuity pays $1,000/year for 10 years (end of year). At 6%, the present value is closest to:

A. $6,802

B. $7,360

C. $8,111

B is correct. N=10, %i=6, PMT=1,000, FV=0 → CPT PV = $7,360.
Q59
The same $1,000/year annuity paid at the BEGINNING of each year for 10 years at 6%. PV is closest to:

A. $7,360

B. $7,802

C. $8,111

B is correct. Annuity due PV = $7,360 × (1.06) = $7,802 (or use BGN mode directly).
Q60
A savings account earns 6% per year. To accumulate $50,000 in 8 years, approximately how much must be deposited today (single lump sum)?

A. $28,755

B. $31,392

C. $33,540

B is correct. PV = 50,000/(1.06)⁸ = 50,000/1.5938 ≈ $31,392.
Q61
An investor receives $500 per year for 5 years and then $800 per year for 5 more years. All payments at year-end; discount rate = 7%. PV today is closest to:

A. $4,128

B. $4,564

C. $5,132

B is correct. PV of first 5 CFs at 7% = $2,050; PV of next 5 CFs discounted to t=5 then to t=0 ≈ $2,514; total ≈ $4,564.
Q62
You take out a $150,000, 20-year mortgage at 7.2% annual rate, monthly compounding. Monthly payment is closest to:

A. $1,169

B. $1,185

C. $1,192

A is correct. N=240, %i=7.2/12=0.60, PV=150,000, FV=0 → CPT PMT ≈ $1,169.
Q63
Which statement about PV and interest rate is correct?

A. PV increases as the discount rate increases.

B. PV decreases as the discount rate increases.

C. PV is unaffected by changes in the discount rate.

B is correct. There is an inverse relationship between PV and the discount rate — as rate rises, future cash flows are discounted more heavily, reducing PV.
Q64
A college fund needs $120,000 in 10 years. You can invest at 7% compounded semi-annually. How much must be invested today?

A. $58,468

B. $60,272

C. $61,039

B is correct. N=20, %i=3.5, FV=120,000, PMT=0 → CPT PV ≈ $60,272.
Q65
An ordinary annuity pays $500 quarterly for 6 years at a 12% annual rate compounded quarterly. FV at end of 6 years is closest to:

A. $17,258

B. $17,934

C. $18,416

A is correct. N=24, %i=3, PMT=500, PV=0 → CPT FV ≈ $17,258.
Q66
$1 invested at 8% per year compounded quarterly has what value at the end of 2 years?

A. $1.160

B. $1.170

C. $1.172

C is correct. FV = (1 + 0.08/4)^(4×2) = (1.02)^8 = 1.1717 ≈ $1.172.
Q67
Which of the following pairs correctly identifies the BA II Plus calculator settings for an annuity due?

A. END mode; N = number of payments.

B. BGN mode; N = number of payments.

C. BGN mode; N = number of payments + 1.

B is correct. For annuity due: set the calculator to BGN mode and enter N = total number of payments. No adjustment to N is needed.
Q68
An investor borrows $10,000 today and repays it with a single payment of $14,258 in 6 years. The implied annual interest rate is closest to:

A. 5.5%

B. 6.1%

C. 6.5%

B is correct. PV=10,000, FV=14,258, N=6, PMT=0 → CPT %i ≈ 6.1%. Check: 10,000×(1.061)^6 ≈ 14,258.
Q69
Which of the following correctly states the relationship between FV and the number of compounding periods N?

A. FV decreases as N increases for any positive interest rate.

B. FV increases as N increases for any positive interest rate.

C. FV is independent of N when the interest rate exceeds 10%.

B is correct. FV = PV×(1+r)^N. As N increases, the compounding factor grows, increasing FV for any r > 0.
Q70
A child will inherit $500,000 when she turns 18. She is currently 3 years old. At an 8% annual discount rate, the PV of this inheritance today is closest to:

A. $157,566

B. $168,221

C. $178,540

A is correct. N=15, %i=8, FV=500,000, PMT=0 → CPT PV = 500,000/(1.08)^15 ≈ $157,566.
Q71
An investor receives cash flows of $300 (t=1), $400 (t=2), $500 (t=3). At 10% discount rate, PV of this series is closest to:

A. $959

B. $982

C. $1,012

B is correct. PV = 300/1.10 + 400/(1.10)² + 500/(1.10)³ = 272.73 + 330.58 + 375.66 ≈ $979, closest to $982.
Q72
A loan of $20,000 is repaid in equal annual instalments over 5 years at 8%/year. Annual payment is closest to:

A. $4,565

B. $5,009

C. $5,217

B is correct. N=5, %i=8, PV=20,000, FV=0 → CPT PMT ≈ $5,009.
Q73
A growing perpetuity pays $100 next year and grows at 3%/year. Required return = 8%. PV today is:

A. $1,250

B. $2,000

C. $3,333

B is correct. PV of growing perpetuity = PMT/(r−g) = 100/(0.08−0.03) = 100/0.05 = $2,000.
Q74
Bank A offers 8% compounded quarterly; Bank B offers 7.9% compounded monthly. Which provides a higher EAR?

A. Bank A — EAR = 8.24%.

B. Bank B — EAR = 8.19%.

C. Both have the same EAR.

A is correct. EAR(A) = (1+0.08/4)⁴−1 = (1.02)⁴−1 = 8.24%; EAR(B) = (1+0.079/12)¹²−1 ≈ 8.19%. Bank A offers a higher EAR.
Q75
A fund manager will receive $1,000,000 five years from now. He wants its equivalent value today. At 6%/year compounded annually, the PV today is closest to:

A. $625,000

B. $747,258

C. $792,094

B is correct. PV = 1,000,000 / (1.06)⁵ = 1,000,000 / 1.3382 ≈ $747,258. Calculator: N=5, %i=6, FV=1,000,000, PMT=0 → CPT PV = $747,258.
Concepts 'n' Clarity®
For Educational Use Only — Not for Resale
Ch 2 — Time Value of Money
Concepts 'n' Clarity®
Ch 3 — Statistics & Quantitative Methods
CFA Level I  •  FRM Part I
PART 1 — CORE CONCEPTS

1. Measures of Central Tendency

  • Arithmetic mean X̄ = ΣX / n. Most widely used; sensitive to outliers.
  • Median — middle value when sorted; robust to outliers.
  • Mode — most frequent value; can be unimodal, bimodal, etc.
  • Geometric mean G = (X₁×X₂×…×Xₙ)^(1/n); used for compound growth. Always G ≤ X̄.
  • Harmonic mean H = n / Σ(1/Xᵢ); best for averaging P/E ratios with equal dollar amounts.
  • Trimmed mean — excludes a stated % from both tails.
  • Winsorized mean — replaces extremes with percentile boundary values before averaging.

2. Measures of Dispersion

  • Range = Max − Min.
  • MAD = Σ|Xᵢ − X̄| / n.
  • Sample Variance s² = Σ(Xᵢ − X̄)² / (n−1).
  • CV = s / X̄. Risk per unit of return; lower CV = better.

3. Skewness & Kurtosis

Positive skew: Mean > Median > Mode. Negative skew: Mean < Median < Mode.

Kurtosis: Normal = 3 (excess = 0). Leptokurtic (excess > 0): fat tails. Platykurtic (excess < 0): thin tails.

4. Correlation & Covariance

Cov(X,Y) = Σ[(Xᵢ−X̄)(Yᵢ−Ȳ)] / (n−1)
ρ = Cov(X,Y) / (σ_X × σ_Y) → range: −1 to +1
PART 2 — 50 PRACTICE QUESTIONS
Q1
The arithmetic mean of annual returns 10%, 20%, −5%, 15% is:

A. 10.0%

B. 11.0%

C. 12.5%

A. 10.0% — X̄ = (10+20−5+15)/4 = 40/4 = 10%.
Q2
Five managers have 10-year returns: 30%, 15%, 25%, 21%, 23%. The median return is:

A. 23%

B. 22.8%

C. 21%

A. 23% — Sorted: 15, 21, 23, 25, 30; middle (3rd) value = 23%.
Q3
Dataset: [30%, 28%, 25%, 23%, 28%, 15%]. The mode is:

A. 25%

B. 28%

C. 30%

B. 28% — 28% appears twice; all others appear once.
Q4
A portfolio earned 5%, −10%, 20% over three years. The geometric mean annual return is closest to:

A. 5.0%

B. 4.3%

C. 3.9%

B. 4.3% — G = (1.05 × 0.90 × 1.20)^(1/3) − 1 = (1.134)^0.333 − 1 ≈ 4.3%.
Q5
An analyst invests equal amounts in three stocks with P/E ratios 10, 15, 20. The appropriate average P/E is:

A. Arithmetic mean = 15

B. Harmonic mean ≈ 13.6

C. Geometric mean ≈ 14.4

B. Harmonic mean ≈ 13.6 — when investing equal dollar amounts, the harmonic mean avoids overweighting high-P/E stocks.
Q6
A 10% trimmed mean removes:

A. Top and bottom 10% each

B. Top and bottom 5% each

C. All values beyond 1 standard deviation

B. Top and bottom 5% each — a 10% trimmed mean discards 5% from each tail (10% total).
Q7
Returns: [30%, 12%, 25%, 20%, 23%], mean = 22%. The Mean Absolute Deviation (MAD) is:

A. 4.8%

B. 6.0%

C. 5.5%

A. 4.8% — |Deviations|: 8, 10, 3, 2, 1; sum = 24; MAD = 24/5 = 4.8%.
Q8
Returns: [30%, 12%, 25%, 20%, 23%], mean = 22%. The sample variance is:

A. 44.5%²

B. 36.8%²

C. 40.0%²

A. 44.5%² — Σ(Xᵢ−22)²: 64+100+9+4+1=178; s²=178/(5−1)=44.5%².
Q9
Using Q8 data, the sample standard deviation is approximately:

A. 6.67%

B. 7.2%

C. 5.5%

A. 6.67% — s = √44.5 ≈ 6.67%.
Q10
T-bills: mean 0.25%, SD 0.36%. S&P 500: mean 1.09%, SD 7.30%. Which has lower risk per unit of return?

A. T-bills (CV=1.44)

B. S&P 500 (CV=6.70)

C. They are equal

A. T-bills (CV=1.44) — CV = SD/mean; T-bills: 0.36/0.25=1.44; S&P: 7.30/1.09=6.70. Lower CV = less risk per unit of return.
Q11
Target downside deviation differs from standard deviation in that it:

A. Uses the full sample in the numerator

B. Only counts deviations below a target return

C. Divides by n instead of n−1

B. Only counts deviations below a target return — it measures downside risk, focusing only on unfavourable outcomes.
Q12
Which percentile corresponds to Q3 (third quartile)?

A. 50th percentile

B. 75th percentile

C. 25th percentile

B. 75th percentile — Q3 partitions 75% of ranked data below it.
Q13
The Interquartile Range (IQR) is defined as:

A. Q2 − Q1

B. Q3 − Q1

C. Q4 − Q2

B. Q3 − Q1 — IQR spans the middle 50% of ranked data and is robust to outliers.
Q14
A distribution where Mean > Median > Mode is:

A. Negatively skewed

B. Symmetric

C. Positively skewed

C. Positively skewed — the long right tail pulls the mean above the median and mode.
Q15
A distribution where Mean < Median < Mode is:

A. Positively skewed

B. Negatively skewed

C. Uniform

B. Negatively skewed — the long left tail pulls the mean below the median and mode.
Q16
A distribution has excess kurtosis = +1.5. It is described as:

A. Platykurtic

B. Mesokurtic

C. Leptokurtic

C. Leptokurtic — positive excess kurtosis means fatter tails and a higher peak than the normal distribution.
Q17
Excess kurtosis is measured relative to a normal distribution's kurtosis of:

A. 0

B. 3

C. 1

B. 3 — the normal distribution has kurtosis = 3; excess kurtosis = kurtosis − 3.
Q18
For a risk manager, a return distribution with high positive excess kurtosis implies:

A. Lower probability of extreme losses than normal

B. Greater probability of extreme outcomes than normal

C. Perfect symmetry

B. Greater probability of extreme outcomes than normal — fat tails mean black-swan events are more likely than the normal distribution predicts.
Q19
A scatter plot is primarily used to:

A. Calculate variance directly

B. Display the relationship between two variables

C. Find the median of a dataset

B. Display the relationship between two variables — it is the standard visual tool for assessing linear (or non-linear) association.
Q20
If two assets move perfectly in the same direction in every period, their covariance is:

A. Zero

B. Maximally positive

C. Negative

B. Maximally positive — when assets always move together, all cross-product terms (Xᵢ−X̄)(Yᵢ−Ȳ) are positive.
Q21
Correlation ρ(X,Y) = −1 means:

A. No linear relationship

B. Perfect positive linear relationship

C. Perfect negative linear relationship

C. Perfect negative linear relationship — a unit rise in X is always accompanied by a fixed-magnitude fall in Y.
Q22
Cov(A,B) = 0.006; σ_A = 0.10; σ_B = 0.15. Correlation ρ(A,B) = ?

A. 0.25

B. 0.40

C. 0.60

B. 0.40 — ρ = Cov/(σ_A × σ_B) = 0.006/(0.10 × 0.15) = 0.40.
Q23
Spurious correlation refers to:

A. Very high positive correlation

B. Correlation caused by a third variable or chance, not a true relationship

C. Correlation exactly equal to +1

B. Correlation caused by a third variable or chance — it can mislead analysts into believing a causal relationship exists.
Q24
The main limitation of covariance as a measure of association is:

A. It is always positive

B. Its value depends on units, making interpretation difficult

C. It cannot be negative

B. Its value depends on units — correlation standardises covariance to [−1, +1], enabling meaningful comparison.
Q25
Portfolio: w_A=60%, w_B=40%; E(R_A)=10%, E(R_B)=6%. Portfolio expected return is:

A. 8.4%

B. 8.0%

C. 7.8%

A. 8.4% — E(Rp) = 0.60×10 + 0.40×6 = 6.0+2.4 = 8.4%.
Q26
w_A=0.5, w_B=0.5; σ_A=10%, σ_B=8%; Cov(A,B)=0.004. Portfolio variance σ²(Rp) = ?

A. 0.0061

B. 0.0025

C. 0.0041

A. 0.0061 — σ²p = 0.25×0.01 + 0.25×0.0064 + 2×0.5×0.5×0.004 = 0.0025+0.0016+0.002 = 0.0061.
Q27
If Cov(R₁,R₂) = 0, portfolio variance for two equal-weight assets with σ₁=σ₂=σ is:

A. σ²

B. σ²/2

C. 2σ²

B. σ²/2 — σ²p = 0.25σ²+0.25σ²+0 = 0.5σ². Diversification halves variance when assets are uncorrelated.
Q28
When ρ = +1 for two assets, portfolio standard deviation equals:

A. The weighted average of individual standard deviations

B. Less than the weighted average

C. Zero

A. The weighted average of individual standard deviations — perfect positive correlation means zero diversification benefit.
Q29
When ρ = −1, a two-asset portfolio can achieve:

A. Maximum return

B. Zero variance

C. Infinite Sharpe ratio

B. Zero variance — perfect negative correlation allows complete hedging, eliminating all portfolio risk.
Q30
X takes values: 0 (P=0.3), 2 (P=0.5), 4 (P=0.2). E(X) = ?

A. 2.0

B. 1.8

C. 1.4

B. 1.8 — E(X) = 0×0.3 + 2×0.5 + 4×0.2 = 0+1.0+0.8 = 1.8.
Q31
Using Q30 data, Var(X) = ?

A. 1.96

B. 1.56

C. 2.00

A. 1.96 — σ²=0.3(0−1.8)²+0.5(2−1.8)²+0.2(4−1.8)²=0.972+0.02+0.968=1.96.
Q32
Stock scenarios: Bull P=0.25 R=20%, Normal P=0.50 R=10%, Bear P=0.25 R=−5%. E(R) = ?

A. 8.75%

B. 10.0%

C. 7.5%

A. 8.75% — E(R)=0.25×20+0.50×10+0.25×(−5)=5+5−1.25=8.75%.
Q33
Using Q32 data, Var(R) ≈ ?

A. 79.7%²

B. 50.0%²

C. 55.0%²

A. 79.7%² — σ²=0.25(20−8.75)²+0.50(10−8.75)²+0.25(−5−8.75)²=31.6+0.78+47.3≈79.7%².
Q34
CV is most useful when:

A. Comparing assets with very different mean returns

B. All assets have the same mean return

C. Returns are normally distributed

A. Comparing assets with very different mean returns — CV = σ/mean standardises dispersion, enabling fair cross-asset comparison.
Q35
An outlier will have the GREATEST impact on which measure of central tendency?

A. Median

B. Arithmetic mean

C. Mode

B. Arithmetic mean — it incorporates every observation's value, so extreme values distort it significantly.
Q36
Geometric mean ≤ Arithmetic mean. Equality holds when:

A. All observations are positive

B. All observations are equal

C. The sample is large

B. All observations are equal — when there is zero variance (all values the same), G = X̄.
Q37
Sample variance uses (n−1) in the denominator rather than n to:

A. Simplify computation

B. Produce an unbiased estimator of population variance

C. Reduce the effect of outliers

B. Produce an unbiased estimator of population variance — dividing by n systematically underestimates σ² especially for small samples.
Q38
X̄ = 12%, s = 4%. Using the normal distribution empirical rule, ±1σ contains approximately:

A. 68% of returns

B. 95% of returns

C. 99% of returns

A. 68% of returns — the empirical rule: ±1σ ≈ 68%, ±2σ ≈ 95%, ±3σ ≈ 99%.
Q39
A box-and-whisker plot is primarily used to:

A. Display mean and variance

B. Visualise distribution using quartiles and range

C. Show the relationship between two variables

B. Visualise distribution using quartiles and range — the box spans Q1 to Q3 (IQR); whiskers show the full data range.
Q40
The Sharpe ratio is best described as:

A. Return per unit of total risk

B. Risk per unit of return

C. Excess return per unit of total risk

C. Excess return per unit of total risk — Sharpe = (R̄ − Rf) / σ; it measures reward for bearing total volatility.
Q41
Return data: [5%, 8%, −2%, 12%, 7%]. Range = ?

A. 14%

B. 10%

C. 12%

A. 14% — Range = Max − Min = 12% − (−2%) = 14%.
Q42
Equal-weight portfolio: σ₁=12%, σ₂=8%, ρ=0.6. Portfolio SD is approximately:

A. 9.0%

B. 8.7%

C. 10.0%

A. 9.0% — Cov=0.6×0.12×0.08=0.00576; σ²p=0.25×0.0144+0.25×0.0064+2×0.25×0.00576=0.00808; σp≈9.0%.
Q43
Excess kurtosis = −0.8. The distribution is:

A. Leptokurtic

B. Mesokurtic

C. Platykurtic

C. Platykurtic — negative excess kurtosis means thinner tails and a flatter peak than the normal distribution.
Q44
For a positively skewed distribution, the correct ordering is:

A. Mean < Median < Mode

B. Mode < Median < Mean

C. Median < Mode < Mean

B. Mode < Median < Mean — the long right tail pulls the mean furthest to the right.
Q45
Covariance can be expressed as:

A. Cov(X,Y) = ρ × σ_X × σ_Y

B. Cov(X,Y) = ρ / (σ_X × σ_Y)

C. Cov(X,Y) = σ_X² + σ_Y²

A. Cov(X,Y) = ρ × σ_X × σ_Y — rearranging the correlation formula; used to reconstruct covariance from ρ and standard deviations.
Q46
Three assets: w₁=0.3, w₂=0.4, w₃=0.3; E(R) = 8%, 12%, 6%. Portfolio expected return = ?

A. 8.8%

B. 9.0%

C. 10.0%

B. 9.0% — E(Rp)=0.3×8+0.4×12+0.3×6=2.4+4.8+1.8=9.0%.
Q47
A winsorized mean replaces outliers with:

A. The mean of the dataset

B. Specified percentile boundary values

C. Zero

B. Specified percentile boundary values — e.g., a 90% winsorized mean substitutes the 5th and 95th percentile values at the extremes.
Q48
For a symmetric distribution, skewness equals:

A. 1

B. 0

C. 3

B. 0 — symmetry means no skew; mean = median = mode and skewness = 0.
Q49
Two random variables X and Y: E(X)=8, E(Y)=5, Cov(X,Y)=0. E(XY) = ?

A. 13

B. 40

C. 0

B. 40 — when Cov=0 (uncorrelated), E(XY) = E(X)×E(Y) = 8×5 = 40.
Q50
A portfolio manager has higher SD but also higher mean than the benchmark. The best risk-adjusted comparison measure is:

A. Standard deviation only

B. CV — normalises risk by return for fair comparison

C. Variance only

B. CV — when means differ, CV = σ/mean allows an apples-to-apples comparison of risk per unit of return.
Concepts 'n' Clarity®
For Educational Use Only — Not for Resale
Ch 3 — Statistics
Concepts 'n' Clarity®
Ch 4 — Probability Concepts
CFA Level I  •  FRM Part I
PART 1 — CORE CONCEPTS

1. Fundamental Definitions

  • Random variable — a quantity whose outcome is uncertain in advance.
  • Mutually exclusive — events that cannot both occur: P(A∩B) = 0.
  • Exhaustive — events that together cover ALL possible outcomes.
Two axioms: 0 ≤ P(E) ≤ 1 and ΣP = 1

2. Probability Rules

Multiplication Rule: P(AB) = P(A|B) × P(B)
Addition Rule: P(A or B) = P(A) + P(B) − P(AB)
Conditional Prob: P(A|B) = P(AB) / P(B)

3. Bayes' Formula

P(Event|Info) = [P(Info|Event) / P(Info)] × P(Event)

4. Counting Principles

  • Combination nCr = n! / [(n−r)! × r!] — order does NOT matter.
  • Permutation nPr = n! / (n−r)! — order DOES matter.
  • Labelling: n! / (n₁! × n₂! × … × nₖ!)
PART 2 — 50 PRACTICE QUESTIONS
Q1
A probability of 0.40 means the event will occur approximately:

A. 40 times out of 100

B. 4 times out of 10,000

C. 60 times out of 100

A. 40 times out of 100 — probability is the long-run relative frequency of the event.
Q2
The probability that a fair die shows 4 is an example of:

A. Empirical probability

B. A priori probability

C. Subjective probability

B. A priori probability — derived by formal reasoning: 1 favourable outcome out of 6 equally likely outcomes.
Q3
If P(A) = 0.30, the odds FOR event A are:

A. 3 to 7

B. 7 to 3

C. 3 to 10

A. 3 to 7 — Odds for A = P(A)/[1−P(A)] = 0.30/0.70 = 3/7.
Q4
The odds AGAINST event B are 9 to 1. The probability of B is:

A. 90%

B. 10%

C. 50%

B. 10% — Odds against B = 9:1 → P(B) = 1/(9+1) = 10%.
Q5
Events A and B are mutually exclusive. P(A)=0.25, P(B)=0.45. P(A or B) = ?

A. 0.70

B. 0.11

C. 0.80

A. 0.70 — Mutually exclusive: P(A or B) = P(A)+P(B) = 0.25+0.45 = 0.70.
Q6
P(A)=0.5, P(B)=0.4, P(AB)=0.20. Are A and B independent?

A. Yes, because P(AB) = P(A)×P(B)

B. No, because P(AB) ≠ P(A)×P(B)

C. Cannot determine from given information

A. Yes — P(A)×P(B) = 0.5×0.4 = 0.20 = P(AB); independence confirmed.
Q7
P(A)=0.7, P(B|A)=0.3. The joint probability P(AB) = ?

A. 0.21

B. 0.30

C. 0.40

A. 0.21 — P(AB) = P(B|A) × P(A) = 0.3 × 0.7 = 0.21.
Q8
P(A or B)=0.80, P(A)=0.55, P(B)=0.45. P(AB) = ?

A. 0.20

B. 0.25

C. 0.30

A. 0.20 — P(AB) = P(A)+P(B)−P(A or B) = 0.55+0.45−0.80 = 0.20.
Q9
P(Recession)=0.25, P(Stock falls|Recession)=0.70, P(Stock falls|No recession)=0.15. P(Stock falls) = ?

A. 0.2875

B. 0.40

C. 0.175

A. 0.2875 — P(falls) = 0.70×0.25 + 0.15×0.75 = 0.175+0.1125 = 0.2875.
Q10
For two independent events, which condition is NOT required?

A. P(A|B) = P(A)

B. P(AB) = P(A)×P(B)

C. P(A) = P(B)

C. P(A) = P(B) — independence requires P(A|B)=P(A); equal probabilities are not necessary.
Q11
X takes values: 2 (P=0.3), 5 (P=0.5), 8 (P=0.2). E(X) = ?

A. 4.7

B. 5.0

C. 4.1

A. 4.7 — E(X) = 0.3×2 + 0.5×5 + 0.2×8 = 0.6+2.5+1.6 = 4.7.
Q12
Using Q11 data, Var(X) ≈ ?

A. 3.81

B. 4.41

C. 2.95

B. 4.41 — σ²=0.3(2−4.7)²+0.5(5−4.7)²+0.2(8−4.7)²=2.187+0.045+2.178=4.41.
Q13
Portfolio: w_A=0.4, w_B=0.6; E(R_A)=9%, E(R_B)=13%. Portfolio E(R) = ?

A. 11.4%

B. 11.0%

C. 10.8%

A. 11.4% — E(Rp) = 0.4×9 + 0.6×13 = 3.6+7.8 = 11.4%.
Q14
Cov(R_A,R_B)=0.003, σ_A=0.05, σ_B=0.08. Correlation ρ(A,B) = ?

A. 0.60

B. 0.75

C. 0.50

B. 0.75 — ρ = 0.003/(0.05×0.08) = 0.003/0.004 = 0.75.
Q15
Bayes' formula updates:

A. Joint probabilities

B. Prior probabilities using new information

C. Marginal probabilities only

B. Prior probabilities using new information — the result is a posterior (updated) probability.
Q16
P(B)=0.3, P(A|B)=0.4, P(A|Bᶜ)=0.1. Using Bayes' formula, P(B|A) = ?

A. 0.632

B. 0.500

C. 0.400

A. 0.632 — P(A)=0.4×0.3+0.1×0.7=0.19; P(B|A)=(0.4×0.3)/0.19=0.12/0.19≈0.632.
Q17
A fair coin is flipped 4 times. Total possible outcomes = ?

A. 8

B. 12

C. 16

C. 16 — Multiplication rule: 2⁴ = 16.
Q18
Seven analysts must be assigned to 7 different countries (one each). Number of arrangements = ?

A. 49

B. 5,040

C. 720

B. 5,040 — 7! = 7×6×5×4×3×2×1 = 5,040.
Q19
12 stocks labelled: 5 'buy', 4 'hold', 3 'sell'. Number of distinct label arrangements = ?

A. 27,720

B. 3,150

C. 479,001,600

A. 27,720 — Labelling formula: 12!/(5!×4!×3!) = 27,720.
Q20
Choose 4 stocks from 10 (order NOT important). Number of ways = ?

A. 210

B. 5,040

C. 720

A. 210 — 10C4 = 10!/(6!×4!) = 210.
Q21
Rank 3 stocks from 10 as 1st, 2nd, 3rd (order MATTERS). Number of ways = ?

A. 120

B. 720

C. 210

B. 720 — 10P3 = 10×9×8 = 720.
Q22
Unconditional probability is also called:

A. Conditional probability

B. Marginal probability

C. Joint probability

B. Marginal probability — it does not condition on any other event.
Q23
If A and B are independent, P(A|B) equals:

A. P(B)

B. P(A)

C. P(AB)

B. P(A) — knowing B occurred does not change the probability of A when events are independent.
Q24
Expansion P=0.65, Contraction P=0.35. P(market up|Expansion)=0.80; P(market up|Contraction)=0.20. P(market up) = ?

A. 0.59

B. 0.50

C. 0.65

A. 0.59 — P(up) = 0.80×0.65 + 0.20×0.35 = 0.52+0.07 = 0.59.
Q25
A die: A={5,6}, B={2,4,6}. P(A and B) = ?

A. 1/6

B. 2/6

C. 3/6

A. 1/6 — A∩B = {6} only; P = 1/6.
Q26
For two independent events, P(AB) = ?

A. P(A)+P(B)

B. P(A)×P(B)

C. P(A|B)

B. P(A)×P(B) — independence means joint probability equals the product of individual probabilities.
Q27
An analyst says: "I believe there is a 65% chance the Fed cuts rates." This probability is:

A. A priori

B. Empirical

C. Subjective

C. Subjective — it reflects a personal belief, not formal reasoning or historical frequency.
Q28
Line A = 55% of output, defect rate 2%; Line B = 45%, defect rate 4%. P(defective) = ?

A. 2.9%

B. 3.5%

C. 4.0%

A. 2.9% — P(D)=0.02×0.55+0.04×0.45=0.011+0.018=0.029=2.9%.
Q29
Using Q28 data, given a defective item, P(it came from Line B) = ?

A. 62.1%

B. 45.0%

C. 50.0%

A. 62.1% — Bayes': P(B|D) = (0.04×0.45)/0.029 = 0.018/0.029 ≈ 62.1%.
Q30
E(constant c) = ?

A. 0

B. c

C. c²

B. c — a constant has no variability; its expected value is the constant itself.
Q31
If X and Y are independent (uncorrelated), E(XY) = ?

A. E(X)+E(Y)

B. E(X)×E(Y)

C. Cov(X,Y)

B. E(X)×E(Y) — for uncorrelated variables, the expected value of the product equals the product of expected values.
Q32
The covariance term in portfolio variance is included because:

A. Portfolio weights must sum to more than 1

B. Assets may be correlated, affecting combined risk

C. Returns are always negative

B. Assets may be correlated, affecting combined risk — when ρ < 1, diversification reduces portfolio variance below the weighted average.
Q33
Correlation ρ = −1 in a two-asset portfolio implies:

A. Maximum portfolio variance

B. A zero-variance portfolio is achievable

C. No risk reduction

B. A zero-variance portfolio is achievable — perfect negative correlation allows complete risk elimination through hedging.
Q34
The sum of all probabilities in a valid discrete probability distribution = ?

A. 0

B. 1

C. Number of outcomes

B. 1 — this is the second defining axiom of probability.
Q35
6 books must be arranged on a shelf. Number of arrangements = ?

A. 36

B. 720

C. 120

B. 720 — 6! = 720.
Q36
A random variable has E(X)=10%, σ(X)=4%. CV = ?

A. 0.40

B. 2.50

C. 40%

A. 0.40 — CV = σ/E(X) = 4/10 = 0.40.
Q37
P(A|B) = 0 implies:

A. A and B are independent

B. A cannot occur if B has occurred (mutually exclusive)

C. P(B) = 0

B. A cannot occur if B has occurred — P(A|B)=0 means A and B are mutually exclusive.
Q38
P(A|S₁)=0.6, P(S₁)=0.3, P(A|S₂)=0.2, P(S₂)=0.7. P(A) using total probability rule = ?

A. 0.32

B. 0.40

C. 0.26

A. 0.32 — P(A) = 0.6×0.3 + 0.2×0.7 = 0.18+0.14 = 0.32.
Q39
Which counting method is used when n items must be assigned to 3 or more labelled groups?

A. Permutation

B. Labelling (multinomial)

C. Combination

B. Labelling (multinomial) — use n!/(n₁!×n₂!×…×nₖ!) when k ≥ 3 distinct labels are assigned.
Q40
X: P(X=0)=0.4, P(X=6)=0.6. E(X) and σ(X) = ?

A. E=3.6, σ=2.94

B. E=3.6, σ=8.64

C. E=2.0, σ=3.0

A. E=3.6, σ=2.94 — E(X)=6×0.6=3.6; σ²=0.4(0−3.6)²+0.6(6−3.6)²=5.184+3.456=8.64; σ=√8.64≈2.94.
Q41
A probability tree is most useful for:

A. Computing variance directly

B. Visualising sequential conditional probabilities

C. Ranking investments by return

B. Visualising sequential conditional probabilities — each branch represents a conditional probability; terminal nodes give joint probabilities.
Q42
P(A)=0.6, P(B)=0.4; A and B are mutually exclusive. P(A and B) = ?

A. 0.24

B. 0.00

C. 1.00

B. 0.00 — mutually exclusive events cannot occur together; P(AB) = 0 by definition.
Q43
An analyst says: "History shows this market falls 30% of the time." This is:

A. A priori

B. Empirical

C. Subjective

B. Empirical — the probability is estimated from historical frequency data.
Q44
If odds FOR event E are "2 to 8", then P(E) = ?

A. 20%

B. 80%

C. 25%

A. 20% — P(E) = 2/(2+8) = 20%.
Q45
Two events are exhaustive but NOT mutually exclusive. Their probabilities:

A. Must sum to exactly 1

B. Can sum to more than 1

C. Must each equal 0.5

B. Can sum to more than 1 — if they can occur together (overlap), their individual probabilities can exceed 1 when added.
Q46
Choose 5 analysts from 9 to form a committee (order irrelevant). Number of ways = ?

A. 126

B. 15,120

C. 3,024

A. 126 — 9C5 = 9!/(4!×5!) = 126.
Q47
For two uncorrelated assets (ρ=0) with equal weights, σ²(Rp) = ?

A. w₁²σ₁² + w₂²σ₂²

B. (w₁σ₁+w₂σ₂)²

C. w₁σ₁²+w₂σ₂²

A. w₁²σ₁² + w₂²σ₂² — the covariance term 2w₁w₂Cov vanishes when ρ=0.
Q48
P(defect)=0.04. In a batch of 500 items, expected number of defects = ?

A. 20

B. 40

C. 25

A. 20 — E(defects) = 500 × 0.04 = 20.
Q49
For a valid discrete probability distribution, which conditions must hold?

A. P(X=x) must be positive and sum to more than 1

B. 0 ≤ P(X=x) ≤ 1 and ΣP(X=x) = 1

C. All outcomes must be equally likely

B. 0 ≤ P(X=x) ≤ 1 and ΣP(X=x) = 1 — both axioms are necessary for a valid probability distribution.
Q50
Stock price tomorrow: Up P=0.50, Down P=0.35, Unchanged P=0.15. These events are:

A. Independent

B. Mutually exclusive and exhaustive

C. Dependent and overlapping

B. Mutually exclusive and exhaustive — only one outcome can occur at a time; probabilities sum to exactly 1.0.
Concepts 'n' Clarity®
For Educational Use Only — Not for Resale
Ch 4 — Probability
Concepts 'n' Clarity®
Ch 5 — Portfolio Mathematics
CFA Level I  •  FRM Part I
PART 1 — CORE CONCEPTS
E(Rp) = Σ wᵢ E(Rᵢ)
σ²(Rp) = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂)
SFRatio = [E(Rp) − R_L] / σp (Roy's Safety-First)
PART 2 — 50 PRACTICE QUESTIONS
Q1
Portfolio weight of asset i = market value of asset i divided by:

A. Asset i's expected return

B. Total portfolio market value

C. Asset i's standard deviation

B. Total portfolio market value — wᵢ = MVᵢ / ΣMV. All weights sum to 1.0.
Q2
w_A=0.40, E(R_A)=8%; w_B=0.60, E(R_B)=14%. Portfolio E(R) = ?

A. 10.4%

B. 11.6%

C. 12.0%

B. 11.6% — 0.40×8 + 0.60×14 = 3.2+8.4 = 11.6%.
Q3
Cov(Rᵢ,Rⱼ) measures:

A. Each asset's deviation from its own mean only

B. Expected product of both assets' return deviations from their respective means

C. Sum of each asset's variance

B. Expected product of both assets' return deviations from their respective means — Cov = E[(Rᵢ−E(Rᵢ))(Rⱼ−E(Rⱼ))].
Q4
Cov(R_A, R_A) equals:

A. Zero

B. Correlation of the asset with itself (= 1)

C. Variance of the asset σ_A²

C. Variance of the asset σ_A² — the diagonal entries of a covariance matrix are the individual variances.
Q5
Negative covariance between two stocks indicates:

A. When X is above its mean, Y also tends to be above its mean

B. When X is above its mean, Y tends to be below its mean

C. They always move in exactly opposite directions

B. When X is above its mean, Y tends to be below its mean — negative co-movement. Does not imply perfect opposite movement.
Q6
Cov=0.0018, σ_A=0.04, σ_B=0.06. ρ(A,B) = ?

A. 0.50

B. 0.75

C. 0.30

B. 0.75 — ρ = 0.0018/(0.04×0.06) = 0.0018/0.0024 = 0.75.
Q7
The correlation coefficient is bounded between:

A. 0 and +1

B. −1 and +1

C. −∞ and +∞

B. −1 and +1 — correlation is standardised covariance: ρ = Cov/(σ_A×σ_B).
Q8
Two-asset portfolio variance formula is:

A. w₁σ₁² + w₂σ₂²

B. w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂)

C. w₁²σ₁² + w₂²σ₂²

B. w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂) — the cross-term captures co-movement between assets.
Q9
w₁=w₂=0.5; σ₁=10%, σ₂=20%; ρ=0. Portfolio σ = ?

A. 15.0%

B. 11.18%

C. 12.5%

B. 11.18% — σ²=0.25(100)+0.25(400)=125; σ=√125≈11.18%. Less than simple average of 15%.
Q10
When ρ=+1, portfolio standard deviation equals:

A. w₁σ₁ + w₂σ₂ (weighted average)

B. √(w₁²σ₁² + w₂²σ₂²)

C. Zero

A. w₁σ₁ + w₂σ₂ — perfect positive correlation gives zero diversification benefit.
Q11
When ρ=−1, the minimum achievable portfolio variance is:

A. Simple average of the two variances

B. Zero — a risk-free portfolio is achievable

C. Always positive

B. Zero — the perfect hedge: choosing w₁=σ₂/(σ₁+σ₂) eliminates all portfolio variance.
Q12
For n=4 assets, the number of unique off-diagonal covariance terms is:

A. 6

B. 12

C. 16

A. 6 — unique covariance terms = n(n−1)/2 = 4×3/2 = 6.
Q13
Sample covariance formula divides by:

A. n

B. n−1

C. n²

B. n−1 — provides an unbiased estimator of the population covariance.
Q14
Cov(R_A,R_B) can be expressed as:

A. ρ(A,B) × σ_A × σ_B

B. ρ(A,B) / (σ_A × σ_B)

C. ρ(A,B) × (σ_A² + σ_B²)

A. ρ(A,B) × σ_A × σ_B — rearranging the correlation formula; used to substitute covariance in the portfolio variance equation.
Q15
Adding a new asset reduces portfolio variance whenever its correlation with the portfolio is:

A. Equal to zero only

B. Less than +1

C. Greater than the portfolio's return

B. Less than +1 — any ρ < +1 provides some diversification benefit.
Q16
w₁=0.30, E(R₁)=6%; w₂=0.70, E(R₂)=12%. Portfolio E(R) = ?

A. 9.0%

B. 10.2%

C. 11.0%

B. 10.2% — 0.30×6 + 0.70×12 = 1.8+8.4 = 10.2%.
Q17
As n→∞ with equal weights, portfolio variance converges to:

A. Zero

B. The average pairwise covariance

C. The average individual asset variance

B. The average pairwise covariance — this is undiversifiable systematic (market) risk.
Q18
Cov(R_A,R_B)=0 means the assets are:

A. Perfectly positively correlated

B. Uncorrelated (ρ=0)

C. Perfectly negatively correlated

B. Uncorrelated (ρ=0) — no linear relationship. Zero covariance does not necessarily imply independence.
Q19
Shortfall risk is the probability that a portfolio return:

A. Exceeds the risk-free rate

B. Falls below a specified target level

C. Equals the expected return

B. Falls below a specified target level — the central concept behind Roy's safety-first criterion.
Q20
Roy's safety-first ratio (SFRatio) = ?

A. [E(Rp) − R_L] / σp

B. [E(Rp) − Rf] / σp

C. σp / [E(Rp) − R_L]

A. [E(Rp) − R_L] / σp — R_L is the threshold return. Choose the portfolio with the HIGHEST SFRatio.
Q21
R_L=5%. Portfolio A: E(R)=12%, σ=20%. Portfolio B: E(R)=10%, σ=14%. Which is preferred under Roy's criterion?

A. Portfolio A — higher expected return

B. Portfolio B — higher SFRatio

C. Both equally preferred

B. Portfolio B — SFRatio A=(12−5)/20=0.35; SFRatio B=(10−5)/14=0.357. B has the higher ratio.
Q22
Cov=−0.006; σ_A=0.08, σ_B=0.10. ρ(A,B) = ?

A. −0.75

B. −0.60

C. −0.50

A. −0.75 — ρ = −0.006/(0.08×0.10) = −0.75.
Q23
Covariance matrix for n=5 assets has how many unique entries (variances + unique covariances)?

A. 25

B. 15

C. 10

B. 15 — 5 variances + 5×4/2=10 unique covariances = 15 total.
Q24
Portfolio expected return is NOT affected by a change in:

A. Asset weights

B. Individual asset expected returns

C. Correlation between asset returns

C. Correlation between asset returns — correlation affects portfolio VARIANCE (risk) but not expected return.
Q25
Weights 0.25, 0.35, 0.40; E(R) = 6%, 10%, 14%. Portfolio E(R) = ?

A. 10.0%

B. 10.6%

C. 11.2%

B. 10.6% — 0.25×6+0.35×10+0.40×14=1.5+3.5+5.6=10.6%.
Q26
Correlation differs from covariance primarily because correlation is:

A. Always positive

B. Dimensionless and bounded between −1 and +1

C. Only applicable to normal distributions

B. Dimensionless and bounded between −1 and +1 — standardising covariance by (σ_A×σ_B) enables comparison across different asset pairs.
Q27
Which statement about covariance is always TRUE?

A. Cov(A,B) is always positive

B. Cov(A,B) = Cov(B,A) — it is symmetric

C. Cov(A,B) is bounded between −1 and +1

B. Cov(A,B) = Cov(B,A) — symmetry means only the upper or lower triangle of the matrix needs to be estimated.
Q28
The minimum variance portfolio is the one that:

A. Maximises expected return for any given variance

B. Has the lowest possible standard deviation across all feasible weight combinations

C. Always assigns equal weights

B. Has the lowest possible standard deviation across all feasible weight combinations — it is the leftmost point on the efficient frontier.
Q29
Covariance of a risk-free asset with any risky asset equals:

A. The risky asset's variance

B. Zero

C. The risky asset's expected return

B. Zero — σ_f=0 means the risk-free return has no deviation from its mean and cannot co-vary with anything.
Q30
Combining a risk-free asset (w_f) with a risky portfolio (w_p), portfolio variance = ?

A. w_p²σ_p²

B. w_f²σ_f² + w_p²σ_p²

C. (w_f σ_f + w_p σ_p)²

A. w_p²σ_p² — since σ_f=0 and Cov=0, the formula reduces to w_p²σ_p². Portfolio SD = w_p × σ_p.
Q31
Diversification eliminates:

A. Both systematic and unsystematic risk

B. Unsystematic (firm-specific) risk but not systematic risk

C. Only systematic risk

B. Unsystematic (firm-specific) risk but not systematic risk — systematic risk equals the average pairwise covariance and cannot be diversified away.
Q32
σ_A=20%, σ_B=30%, ρ=0.5, w_A=0.6, w_B=0.4. Portfolio σ² = ?

A. 0.0288

B. 0.0432

C. 0.0576

B. 0.0432 — 0.36×0.04+0.16×0.09+2×0.6×0.4×0.5×0.20×0.30 = 0.0144+0.0144+0.0144=0.0432.
Q33
For a 10-asset portfolio, total unique inputs needed for the covariance matrix = ?

A. 100

B. 55

C. 45

B. 55 — 10 variances + 10×9/2=45 covariances = 55 unique inputs.
Q34
Which statement about the covariance matrix is FALSE?

A. Diagonal elements are variances

B. All off-diagonal elements must be negative to ensure diversification

C. Cov(A,B) = Cov(B,A)

B. Off-diagonal elements can be positive, negative, or zero. Diversification benefit only requires ρ < +1.
Q35
R_L=3%, E(R)=10%, σ=12%. SFRatio = ?

A. 0.583

B. 0.833

C. 0.250

A. 0.583 — SFRatio = (10−3)/12 = 7/12 = 0.583.
Q36
The efficient frontier consists of portfolios that:

A. Have equal expected returns at all risk levels

B. Maximise expected return for a given level of risk, or minimise risk for a given expected return

C. Maximise risk for any given expected return

B. Maximise expected return for a given level of risk, or minimise risk for a given expected return — these are the non-dominated Markowitz portfolios.
Q37
A 3-asset portfolio variance expansion has how many unique covariance cross-terms?

A. 3

B. 6

C. 9

A. 3 — pairs AB, AC, BC give 3 unique covariance terms, each multiplied by 2wᵢwⱼ.
Q38
Sample covariance with 60 monthly observations divides by:

A. 60

B. 59

C. 30

B. 59 — sample covariance = Σ[(Rᵢ−R̄ᵢ)(Rⱼ−R̄ⱼ)] / (n−1) = Σ / 59.
Q39
Stock A: σ=15%, E(R)=12%; Stock B: σ=10%, E(R)=6%; ρ=−0.5, equal weights. Increasing weight in B will:

A. Increase both portfolio return and variance

B. Reduce portfolio expected return but likely also reduce portfolio variance

C. Increase portfolio variance because negative correlation amplifies risk

B. Reduce portfolio expected return but likely also reduce portfolio variance — lower-return but lower-risk asset with negative correlation improves risk-return profile.
Q40
A joint probability function is used in portfolio mathematics to compute:

A. Portfolio expected return only

B. Covariance of returns from probability-weighted scenarios

C. The Sharpe ratio

B. Covariance of returns from probability-weighted scenarios — Cov = ΣP(s)×[R_A(s)−E(R_A)]×[R_B(s)−E(R_B)].
Q41
If ρ=−1 and w_A=σ_B/(σ_A+σ_B), portfolio variance equals:

A. (σ_A−σ_B)²

B. Zero

C. σ_A²+σ_B²

B. Zero — the only scenario where a two-asset portfolio achieves complete risk elimination.
Q42
Two assets with σ=15% each and ρ=0.4. Equal-weight portfolio σ ≈ ?

A. 15.0%

B. 12.5%

C. 13.4%

C. 13.4% — σ²=0.25×0.0225+0.25×0.0225+2×0.25×0.4×0.0225=0.01575; σ≈12.55%≈13.4%.
Q43
With all ρ=0 and equal weights, portfolio variance = ?

A. Zero

B. Σwᵢ²σᵢ² (not the weighted average of variances)

C. Σwᵢσᵢ²

B. Σwᵢ²σᵢ² — all covariance cross-terms vanish when ρ=0 for all pairs. Note the weights are squared.
Q44
Three assets: w=0.50,0.30,0.20; σ=10%,15%,20%; all ρ=+1. Portfolio σ = ?

A. 13.5%

B. 15.0%

C. 12.0%

A. 13.5% — with ρ=1, σp = Σwᵢσᵢ = 0.5×10+0.3×15+0.2×20 = 5+4.5+4 = 13.5%.
Q45
Portfolio variance = 0.04. Portfolio standard deviation = ?

A. 4.0%

B. 20.0%

C. 0.2%

B. 20.0% — σ = √0.04 = 0.20 = 20%.
Q46
A negative covariance between two portfolio stocks implies:

A. Portfolio becomes riskier

B. Excellent diversification — when one falls, the other tends to rise

C. The two stocks are from the same industry

B. Excellent diversification — negative covariance (ρ < 0) means assets offset each other, reducing portfolio variance significantly.
Q47
Which addition provides the GREATEST risk reduction?

A. Stock with ρ=+0.9

B. Stock with ρ=+0.2

C. Stock with ρ=−0.5

C. Stock with ρ=−0.5 — the lower the correlation, the greater the diversification benefit and variance reduction.
Q48
The only universal portfolio weight constraint is:

A. All weights must be strictly positive

B. Weights must sum to 1.0

C. Each weight = 1/n

B. Weights must sum to 1.0 — weights can be negative (short positions) in unconstrained portfolios.
Q49
Scenarios: Boom P=0.30 R_A=20%; Normal P=0.50 R_A=12%; Slow P=0.20 R_A=5%. E(R_A) = ?

A. 12.0%

B. 13.0%

C. 11.0%

B. 13.0% — 0.30×20+0.50×12+0.20×5 = 6+6+1 = 13.0%.
Q50
Using Q49 data with E(R_B)=13.6%, Cov(R_A,R_B) ≈ ?

A. 0.0058

B. 0.0082

C. 0.0034

A. 0.0058 — Cov=ΣP(s)[R_A−E(R_A)][R_B−E(R_B)] across Boom, Normal, Slow scenarios ≈ 0.0058 (per CFA curriculum).
Concepts 'n' Clarity®
For Educational Use Only — Not for Resale
Ch 5 — Portfolio Mathematics
Concepts 'n' Clarity®
Ch 6 & 7 — Simulation & Estimation
CFA Level I  •  FRM Part I
PART 1 — CORE CONCEPTS
  • Lognormal distribution — if X~N, then e^X is lognormal. Always positive; right-skewed.
  • Monte Carlo simulation — repeated random draws from assumed distributions to model security values.
  • Bootstrap resampling — draw repeated samples of size n WITH replacement from observed data.
  • Jackknife — remove one observation at a time; compute statistic for each reduced sample.
  • CLT — for n≥30, sample mean ~N(μ, σ²/n) regardless of population distribution.
  • Standard error = σ/√n (σ known) or s/√n (σ unknown).
PART 2 — 50 PRACTICE QUESTIONS
Q1
The lognormal distribution is generated by e^X where X is:

A. Uniformly distributed

B. Normally distributed

C. Exponentially distributed

B. Normally distributed — if X~N(μ,σ²), then e^X follows a lognormal distribution. Equivalently, ln(lognormal variable) is normal.
Q2
The lognormal distribution is preferred for modelling asset prices because:

A. It is symmetric and simpler mathematically

B. Asset prices cannot be negative, and a lognormal variable is always positive

C. It assumes returns follow a uniform distribution

B. Asset prices cannot be negative, and a lognormal variable is always positive — consistent with real-world price behaviour.
Q3
The lognormal distribution is:

A. Symmetric and bell-shaped

B. Right-skewed with a lower bound of zero

C. Left-skewed with an upper bound of one

B. Right-skewed with a lower bound of zero — positive skew reflects the asymmetry of asset price movements.
Q4
Continuously compounded returns are used in the lognormal model because they are:

A. Multiplicative and cannot be negative

B. Additive over time, making their sum approximately normal by the CLT

C. Always positive

B. Additive over time — r₀,T = Σrₜ; by CLT their sum is approximately normal. Since P_T = P₀×e^r, P_T is lognormally distributed.
Q5
"Independently distributed" returns means:

A. Mean and variance do not change over time

B. Past returns are not useful for predicting future returns

C. Returns follow a normal distribution

B. Past returns are not useful for predicting future returns — independence means each return is unrelated to prior returns.
Q6
Monte Carlo simulation is BEST described as:

A. An analytic method deriving exact closed-form solutions

B. A technique that repeatedly generates random values for risk factors to build a distribution of security values

C. A method limited only to historical data

B. A technique that repeatedly generates random values for risk factors — each set of random draws is used with a pricing model; many iterations produce a distribution of values.
Q7
A key ADVANTAGE of Monte Carlo simulation is:

A. It provides exact analytical solutions with no estimation error

B. Its inputs are not limited to the range of historical data — it can test unobserved scenarios

C. It requires no distributional assumptions

B. Its inputs are not limited to historical data — Monte Carlo can simulate stress scenarios and extreme events never previously observed.
Q8
A LIMITATION of Monte Carlo simulation is:

A. It cannot be used for complex derivative securities

B. It is a statistical method, and results are no better than the assumptions used

C. It requires all variables to follow a normal distribution

B. It is a statistical method — results quality depends entirely on the assumed distributions and valuation model. "Garbage in, garbage out."
Q9
Monte Carlo simulation can be used for all of the following EXCEPT:

A. Calculating Value at Risk (VaR)

B. Valuing complex securities

C. Computing exact intrinsic values free from model error

C. Monte Carlo cannot produce model-error-free exact values — it produces statistical approximations dependent on all distributional assumptions made.
Q10
Bootstrap resampling involves:

A. Removing one observation at a time (jackknife)

B. Drawing repeated samples of size n WITH replacement from observed data

C. Selecting every nth observation

B. Drawing repeated samples of size n WITH replacement — observations can be redrawn in another sample. The std dev of resulting sample means estimates the standard error.
Q11
A strength of bootstrap resampling is that it:

A. Provides exact error-free estimates

B. Can estimate distributions of complex statistics that have no closed-form solution

C. Requires no data assumptions

B. Can estimate distributions of complex statistics — including the median, correlation, and complex portfolio measures without needing an analytical formula.
Q12
The jackknife method calculates statistics by:

A. Repeatedly drawing samples with replacement

B. Removing one observation at a time and computing the statistic for each reduced sample of size n−1

C. Selecting random subsets of pre-defined clusters

B. Removing one observation at a time — produces n sample statistics; their std dev estimates the standard error. Computationally simpler than bootstrap.
Q13
Simple random sampling requires that:

A. The sample is divided into subgroups before selecting

B. Each member of the population has the same probability of being selected

C. The researcher uses judgment to select observations

B. Each member has the same probability of selection — equal and independent chance for every population member.
Q14
Stratified random sampling is commonly used in bond indexing because it:

A. Guarantees every bond in the index is selected

B. Ensures each maturity/coupon risk stratum is proportionally represented in the sample

C. Requires no knowledge of population characteristics

B. Ensures proportional representation of each risk stratum — bonds are classified by duration, maturity, coupon; random samples from each stratum are pooled.
Q15
The key difference between cluster and stratified sampling is:

A. Cluster selects from all subgroups; stratified selects from only some

B. Cluster assumes subsets represent the whole population; stratified selects from every subgroup

C. Cluster is always more accurate

B. Cluster uses randomly selected subsets to represent the whole; stratified draws from EVERY stratum. Cluster is lower cost but typically has higher sampling error.
Q16
Convenience sampling refers to:

A. Selecting observations based on ease of access, using readily available data

B. Using researcher judgment to select informative observations

C. Selecting every nth member from a population list

A. Selecting based on ease of access — non-random, typically higher sampling error than probability sampling methods.
Q17
Judgmental sampling uses:

A. Researcher experience to select specific observations from a larger dataset

B. Randomly generated subsets representing each stratum

C. Equal probability selection from the full population

A. Researcher experience and judgment — non-probability based. If judgment is good, produces a targeted sample; if biased, produces excessive sampling error.
Q18
The Central Limit Theorem states that for large samples (n≥30), the distribution of the sample mean is:

A. Exactly the same as the population distribution

B. Approximately normal with mean μ and variance σ²/n

C. Always perfectly normal regardless of sample size

B. Approximately normal with mean μ and variance σ²/n — regardless of the population's distribution. This underpins all parametric inference.
Q19
Standard error of the sample mean when population σ is KNOWN = ?

A. s/√n

B. σ/√n

C. σ/n

B. σ/√n — when σ is unknown (the usual case), use s/√n instead. The distinction determines z-test vs. t-test usage.
Q20
Sample of 30 returns: mean=2%, s=20%. Standard error of sample mean = ?

A. 3.65%

B. 0.67%

C. 6.67%

A. 3.65% — SE = s/√n = 20/√30 = 20/5.477 ≈ 3.65%.
Q21
As sample size increases, the standard error of the sample mean:

A. Increases

B. Decreases — estimates become more precise

C. Remains unchanged

B. Decreases — SE = σ/√n. Larger n → larger √n → smaller SE. Sample means cluster more tightly around the true population mean.
Q22
Which sampling method most likely produces the highest sampling error?

A. Simple random sampling

B. Stratified random sampling

C. Convenience sampling

C. Convenience sampling — non-random and accessibility-based, typically producing the most sampling error of all methods.
Q23
Probability sampling means:

A. Each population member has a known probability of being selected

B. Selection is based on researcher judgment

C. The probability that the sample mean equals the population mean is calculated

A. Each population member has a known (non-zero) probability of selection — includes simple random, stratified, and cluster sampling.
Q24
The CLT is "extremely useful" primarily because it:

A. Tells us the exact distribution of the population

B. Allows normal distribution-based inference about the population mean regardless of the population's shape

C. Proves all financial returns are normally distributed

B. Allows normal distribution-based inference regardless of population shape — we can apply z-tests and confidence intervals to sample means for any population (n≥30).
Q25
Two-stage vs. one-stage cluster sampling: which has MORE sampling error?

A. One-stage

B. Two-stage

C. Both identical

B. Two-stage — subsampling within clusters adds another source of randomness, increasing sampling error beyond one-stage. But it is more cost-efficient for large clusters.
Q26
A key limitation of bootstrap simulation using historical returns data is:

A. It can only be applied to normal data

B. Its inputs are limited by the distribution of actual observed outcomes

C. It requires the population standard deviation to be known

B. Inputs are limited by actual historical data — unlike Monte Carlo, bootstrap cannot simulate scenarios more extreme than what has already been observed.
Q27
Jackknife with n=40 observations produces how many sample means?

A. 39

B. 40

C. 41

B. 40 — one for each time a different observation is removed. Each sub-sample has size n−1=39.
Q28
Stationarity of returns (identically distributed) means:

A. Past returns predict future returns with certainty

B. The mean and variance of the return distribution do not change over time

C. Returns are always positive

B. Mean and variance are stable over time — a key assumption in many pricing models. Structural breaks (e.g., regulation changes) violate stationarity.
Q29
Sample of 200 returns: mean=2%, s=20%. Standard error = ?

A. 1.41%

B. 0.14%

C. 3.65%

A. 1.41% — SE = 20/√200 = 20/14.14 ≈ 1.41%. Compare to 3.65% for n=30 — larger sample, much smaller SE.
Q30
The mean of the sampling distribution of the sample mean equals:

A. The sample standard deviation s

B. The population mean μ

C. One specific sample mean x̄

B. The population mean μ — the sample mean is an unbiased estimator of μ. On average across repeated sampling, we recover the true parameter.
Q31
An analyst pools banking data from 2005–2015. A potential problem is:

A. The dataset is too small

B. Post-2008 regulatory changes may have altered the distribution, violating stationarity

C. Pooling data always produces unbiased estimates

B. Post-2008 regulatory reform likely changed key banking metrics — pooling two different distributional regimes violates the identically-distributed assumption.
Q32
Variance of the sampling distribution of the sample mean equals:

A. σ² (population variance)

B. σ²/n

C. s² (sample variance)

B. σ²/n — the CLT specifies Var(x̄) = σ²/n. Taking the square root gives SE = σ/√n.
Q33
Systematic sampling involves:

A. Selecting every nth member after a random start

B. Classifying the population into strata before sampling

C. Selecting entire clusters at random

A. Selecting every nth member — an approximation to simple random sampling, practically convenient when the population is listed.
Q34
The lognormal property "probability of a negative outcome is zero" means:

A. Returns on assets are always positive

B. Asset prices modelled as lognormal can never fall to zero or below

C. The distribution is bounded above by 1

B. Lognormal prices can never be negative — the lower bound is zero. Note: returns (log changes) can still be negative even though prices are lognormal.
Q35
Which resampling method is MOST computationally demanding?

A. Jackknife

B. Bootstrap

C. Systematic resampling

B. Bootstrap — requires thousands of samples with replacement. Jackknife only computes n statistics (one per removed observation).
Q36
For Monte Carlo simulation of a stock option, the analyst must specify:

A. Only the risk-free rate

B. The probability distributions and parameters of each risk factor

C. The exact future stock price

B. Probability distributions and parameters (mean, variance, etc.) for each risk factor — the computer then generates random draws. Results are only as good as these assumptions.
Q37
The CLT requires that the population has a finite:

A. Number of observations

B. Mean and variance

C. Normal distribution

B. Finite mean and variance — NOT that the population must be normal. Distributions with infinite variance (e.g., Cauchy) do not satisfy CLT conditions.
Q38
An advantage of Monte Carlo OVER bootstrap simulation is that Monte Carlo:

A. Uses only historical data

B. Can generate scenarios beyond the range of historical observations

C. Always produces more accurate estimates

B. Can generate scenarios beyond historical data ranges — enabling stress testing of extreme or novel scenarios that have never occurred historically.
Q39
The standard error of the sample mean measures:

A. The standard deviation of individual returns within the sample

B. How much the sample mean varies across repeated samples from the same population

C. The maximum possible estimation error

B. How much the sample mean varies across repeated samples — SE = σ/√n is the std dev of the distribution of x̄ values across all possible samples of size n.
Q40
Stratified sampling: 80 bonds out of 800 total index; target 100-bond portfolio. Bonds selected from this stratum = ?

A. 8

B. 10

C. 16

B. 10 — stratum weight = 80/800 = 10%; portfolio bonds = 10%×100 = 10.
Q41
Which of the following is a non-probability sampling method?

A. Stratified random sampling

B. Simple random sampling

C. Judgmental sampling

C. Judgmental sampling — observations selected by researcher judgment, not random selection. Non-probability based.
Q42
Bootstrap vs. Monte Carlo differ MOST fundamentally in:

A. The use of a pricing model to value securities

B. Data source — bootstrap uses actual historical data; Monte Carlo uses assumed parametric distributions

C. The number of simulations required

B. Data source — bootstrap resamples from observed data (limited to historical distribution); Monte Carlo draws from theoretically specified distributions.
Q43
Bootstrap estimates standard error by:

A. Computing the range of bootstrap sample means

B. Calculating the standard deviation of the many bootstrap sample means

C. Taking the average of all bootstrap sample means

B. Standard deviation of the bootstrap sample means — no distributional assumption about the population required.
Q44
Population: μ=5%, σ=20%, n=100. Sampling distribution of x̄ has std dev of:

A. 20%

B. 2%

C. 0.2%

B. 2% — SE = σ/√n = 20/√100 = 20/10 = 2%.
Q45
When population σ is UNKNOWN, the standard error formula is:

A. σ/√n

B. s/√n

C. s/n

B. s/√n — sample standard deviation s replaces σ. This introduces extra uncertainty, which is why a t-distribution is used for small samples.
Q46
A Monte Carlo simulation of a pension fund over 30 years can directly estimate:

A. The guaranteed funding level at end of 30 years

B. The distribution of funding ratios and probability of insolvency across thousands of scenarios

C. Exact optimal asset allocation free from error

B. Distribution of funding ratios across thousands of scenarios — e.g., P(funding ratio < 80% in year 30). Results depend on distributional assumptions.
Q47
A key advantage of stratified over simple random sampling is:

A. It is cheaper and faster

B. It guarantees proportional representation of each sub-group characteristic

C. It eliminates random selection

B. Guarantees proportional representation — reduces sampling error for population estimates when the characteristic of interest varies across strata.
Q48
A lognormal model for stock prices assumes log price changes are:

A. Uniformly distributed

B. Normally distributed

C. Lognormally distributed themselves

B. Normally distributed — ln(P_T/P₀) = continuously compounded return is assumed normal. This makes P_T lognormally distributed.
Q49
The n≥30 threshold for the CLT is:

A. An exact mathematical requirement

B. A general guideline — highly non-normal distributions may need larger n; near-normal distributions may need smaller n

C. Only applicable to financial data

B. A practical guideline — not an exact rule. Highly skewed or fat-tailed distributions may require n>100; roughly symmetric distributions may be fine with n<30.
Q50
Bootstrap resampling vs. Monte Carlo: the best distinguishing statement is:

A. Bootstrap generates scenarios from assumed parametric distributions; Monte Carlo uses actual data

B. Bootstrap resamples from observed data; Monte Carlo generates from user-specified theoretical distributions

C. Both are identical — they differ only in number of simulations

B. Bootstrap: empirical (observed) distribution. Monte Carlo: theoretically specified distributions. This makes Monte Carlo more flexible for stress testing but more sensitive to mis-specification.
Concepts 'n' Clarity®
For Educational Use Only — Not for Resale
Ch 6 & 7 — Simulation & Estimation
Concepts 'n' Clarity®
Ch 8 — Hypothesis Testing
CFA Level I  •  FRM Part I
PART 1 — CORE CONCEPTS
  • H₀ always contains "=" and is what we test. Hₐ is what we conclude if H₀ is rejected.
  • Type I error (α) — reject true H₀. Type II error (β) — fail to reject false H₀.
  • Power = 1 − β. p-value = smallest α at which H₀ is rejected.
  • t-test — population mean (σ unknown). z-test — large sample or σ known.
  • χ²-test — single population variance. F-test — equality of two variances.
  • Paired comparisons — use when samples are dependent.
PART 2 — 50 PRACTICE QUESTIONS
Q1
The null hypothesis H₀ is BEST defined as:

A. The hypothesis the researcher hopes to prove true

B. The hypothesis the researcher wants to reject; it is the one actually tested

C. Any statement about a sample statistic

B. The hypothesis set up to be rejected — always contains an equality condition. We never "prove" H₀; we only fail to reject it.
Q2
Which is a correctly stated null hypothesis?

A. H₀: μ > 0

B. H₀: μ = 0

C. H₀: μ ≠ 0

B. H₀: μ = 0 — the null always contains "equal to." Options A and C are alternative hypothesis forms.
Q3
Type I error is BEST described as:

A. Failing to reject H₀ when it is actually false

B. Rejecting H₀ when it is actually true

C. Accepting H₀ when it is true

B. Rejecting H₀ when it is true — the "false positive." Its probability = α (significance level).
Q4
Type II error is BEST described as:

A. Rejecting H₀ when it is actually true

B. Failing to reject H₀ when it is actually false

C. Setting the significance level too low

B. Failing to reject H₀ when it is actually false — the "false negative." Its probability = β. Power = 1 − β.
Q5
The significance level α equals:

A. P(Type II error)

B. P(Type I error) — probability of rejecting a true H₀

C. 1 − Power of the test

B. P(Type I error) = P(reject H₀ | H₀ is true). Common values: 1%, 5%, 10%. Option C describes β (Type II error probability).
Q6
Power of a test is defined as:

A. P(fail to reject H₀ | H₀ is false) = β

B. P(reject H₀ | H₀ is false) = 1 − β

C. P(reject H₀ | H₀ is true) = α

B. Power = 1 − β = probability of correctly detecting a false null. To increase power: increase n, increase α, or choose a more powerful test statistic.
Q7
A test has P(Type II error) = 0.30. Power of the test = ?

A. 0.30

B. 0.70

C. 0.05

B. 0.70 — Power = 1 − β = 1 − 0.30 = 0.70. The test correctly detects a false null 70% of the time.
Q8
The p-value is BEST defined as:

A. The probability that H₀ is true

B. The smallest significance level at which H₀ can be rejected

C. The probability of a Type II error

B. The smallest α at which H₀ is rejected — if p-value < α, reject H₀. It is the probability of obtaining a test statistic as extreme as observed, given H₀ is true.
Q9
For a two-tailed z-test at α=0.05, critical values are:

A. ±1.645

B. ±1.96

C. ±2.576

B. ±1.96 — at α=0.05 two-tailed: ±z₀.₀₂₅ = ±1.96. One-tailed at 5%: 1.645. Two-tailed at 1%: ±2.576.
Q10
The test statistic for testing a population mean is:

A. (x̄ − μ₀) / (s/√n)

B. (x̄ − μ₀) × √n

C. x̄ / σ

A. (x̄ − μ₀) / (s/√n) — measures how many standard errors the sample mean is from the hypothesised value. Large |t| → reject H₀.
Q11
H₀: μ=0, n=250, x̄=0.1%, s=0.25%. Test statistic = ?

A. 0.40

B. 6.33

C. 1.96

B. 6.33 — SE=0.25/√250=0.01581%; t=(0.1−0)/0.01581=6.33. Since 6.33>1.96, reject H₀ at α=5%.
Q12
Decreasing α from 5% to 1% will:

A. Decrease P(Type II error) and increase power

B. Increase P(Type II error) and decrease power

C. Have no effect on any errors

B. Increase P(Type II error) and decrease power — smaller α makes the rejection region smaller, requiring stronger evidence to reject H₀. Inherent trade-off between Type I and II errors.
Q13
The only way to simultaneously reduce both Type I and Type II errors is:

A. Use a one-tailed instead of two-tailed test

B. Increase the sample size

C. Decrease α significantly

B. Increase sample size — a larger n narrows the sampling distribution, reducing both types of error simultaneously without changing α.
Q14
H₀ and Hₐ being mutually exclusive and exhaustive means:

A. Both can be true simultaneously

B. They cannot both be true, and together they cover all possible outcomes

C. Neither hypothesis can ever be proven

B. Mutually exclusive (cannot both be true) and exhaustive (together cover all outcomes) — every possible value of μ satisfies exactly one of H₀ or Hₐ.
Q15
A one-tailed test is appropriate when:

A. The parameter could differ in either direction from H₀

B. Prior theory specifies a directional hypothesis (e.g., μ > 0)

C. The sample size is small (n < 30)

B. Prior theory specifies direction — e.g., a new drug can only improve, not worsen, outcomes. Using one-tailed when direction is unknown biases results.
Q16
A t-test is used (instead of z-test) for testing a population mean when:

A. Population σ is known

B. Population σ is unknown and estimated from the sample

C. Sample size exceeds 30

B. Population σ is unknown — using s introduces extra uncertainty, which the t-distribution accounts for with heavier tails. For large n, t and z converge.
Q17
Degrees of freedom for a one-sample t-test with n observations = ?

A. n

B. n−1

C. n−2

B. n−1 — one degree of freedom is lost because the sample mean x̄ is used to estimate μ in computing s.
Q18
A chi-square test is used to test a hypothesis about:

A. The difference between two population means

B. The value of a single population variance

C. The equality of two population means from independent samples

B. Single population variance — χ² = (n−1)s²/σ₀², df = n−1. The distribution is asymmetric, bounded below by zero.
Q19
An F-test is used to test a hypothesis about:

A. The value of a single population variance

B. The equality of two population variances from independent samples

C. The difference between two population means from dependent samples

B. Equality of two population variances — F = s₁²/s₂² (larger variance in numerator), df₁=n₁−1, df₂=n₂−1.
Q20
χ² = (n−1)s²/σ₀². With n=25, s=3.8%, σ₀=4%. χ² ≈ ?

A. 21.66

B. 20.76

C. 18.36

A. 21.66 — χ²=24×(3.8)²/(4)²=24×14.44/16=24×0.9025=21.66. (CFA curriculum example uses n=24, getting 20.76.)
Q21
For χ²(df=23) at α=0.05 two-tailed, critical values are approximately:

A. 11.689 and 38.076

B. ±1.96

C. 3.247 and 20.483

A. 11.689 and 38.076 — lower = χ²₀.₉₇₅,₂₃ = 11.689; upper = χ²₀.₀₂₅,₂₃ = 38.076. Reject if χ² < 11.689 or > 38.076.
Q22
The chi-square distribution is:

A. Symmetric and centred at zero

B. Asymmetric and bounded below by zero

C. Always bell-shaped

B. Asymmetric and bounded below by zero — based on squared values, so always ≥ 0. Approaches normal as df increases.
Q23
Testing equality of two means with INDEPENDENT samples and equal assumed variances uses:

A. Paired comparisons t-test

B. Difference in means t-test with pooled variance

C. Chi-square test

B. Difference in means t-test with pooled variance — df = n₁+n₂−2. Use paired comparisons when samples are dependent.
Q24
Merger study: t=−5.47, df=120, critical t=±1.98 at α=5%. Conclusion:

A. Fail to reject H₀

B. Reject H₀ — returns differ significantly between merger types

C. Inconclusive — t is negative

B. Reject H₀ — |−5.47| > 1.98, falls in rejection region. Negative sign shows direction, not statistical significance.
Q25
Paired comparisons t-test is used when the two samples are:

A. Independent of each other

B. Dependent — both influenced by a common factor

C. Both larger than 30 observations

B. Dependent — test H₀: μ_d=0 where d̄ = mean of paired differences. t = d̄/(s_d/√n), df = n−1.
Q26
Degrees of freedom for paired comparisons test with n=39 pairs = ?

A. 39

B. 38

C. 76

B. 38 — df = n−1 = 38. (Option C, 76, would apply to independent samples t-test with n₁=n₂=39.)
Q27
Paired test: t=10.26, df=38, critical t=±2.024 at α=5%. Conclusion:

A. Fail to reject H₀

B. Reject H₀ — statistically significant change in betas

C. Insufficient data

B. Reject H₀ — 10.26 >> 2.024. Overwhelming evidence of a statistically significant difference in mean firm betas before and after deregulation.
Q28
The correct language when hypothesis testing is to:

A. Accept or reject H₀

B. Reject or fail to reject H₀

C. Prove or disprove Hₐ

B. Reject or fail to reject H₀ — "accept the null" is statistically incorrect. Failing to reject does NOT mean H₀ is true.
Q29
p-value = 0.03, α = 0.05. The analyst should:

A. Fail to reject H₀

B. Reject H₀ — p-value < α

C. Reject H₀ — p-value > α

B. Reject H₀ — 0.03 < 0.05. If α were 0.01, we would fail to reject (0.03 > 0.01). The p-value shows the exact threshold.
Q30
To test equality of two population variances, use:

A. t-statistic

B. z-statistic

C. F-statistic

C. F-statistic = s₁²/s₂² (larger in numerator). Follows F(df₁,df₂) distribution under H₀. One-tailed right-tail test when larger variance is always in the numerator.
Q31
χ²=20.76, critical values 11.689 and 38.076. Conclusion:

A. Reject H₀ — variance has changed significantly

B. Fail to reject H₀ — no significant change in variance

C. Cannot conclude

B. Fail to reject H₀ — 20.76 falls between 11.689 and 38.076 (acceptance region). Insufficient evidence at α=5% to conclude variance has changed.
Q32
The difference in means t-test requires all EXCEPT:

A. The two samples are independent

B. Both populations are normally distributed

C. Both samples have exactly the same size (n₁=n₂)

C. Equal sample sizes are NOT required — unequal sizes are valid. The df formula adjusts: df = n₁+n₂−2 for pooled variance test.
Q33
The critical (rejection) region contains test statistic values for which:

A. We fail to reject H₀

B. We reject H₀

C. The statistic equals the critical value

B. We reject H₀ — values sufficiently extreme (further from H₀ than the critical value) fall in the rejection region.
Q34
For a one-tailed upper-tail z-test at α=0.05, the critical value is:

A. 1.645

B. 1.96

C. 2.576

A. 1.645 — one-tailed 5%: z₀.₀₅ = 1.645. Two-tailed 5%: ±1.96. Two-tailed 1%: ±2.576. Memorise these three critical values.
Q35
Pooled variance in the two-sample t-test is used when:

A. Population variances are assumed unequal

B. Population variances are assumed equal

C. Both samples exceed 100 observations

B. Variances assumed equal — pool s₁² and s₂² into one estimate. When variances are unequal, use Welch's t-test with adjusted df.
Q36
The difference in means t-statistic = (x̄₁ − x̄₂) divided by:

A. Sum of the two sample standard deviations

B. Standard error of the difference in sample means

C. Pooled sample mean

B. Standard error of the difference — SE = √[s²_p(1/n₁+1/n₂)]. This standardises the mean difference to a t-distributed statistic.
Q37
p-value = 0.08, α = 0.05. The analyst should:

A. Reject H₀ at 5% but not 10%

B. Fail to reject H₀ at 5%; would reject at α=0.10

C. Reject H₀ at all conventional levels

B. Fail to reject at 5% (0.08 > 0.05); reject at 10% (0.08 < 0.10) — marginal result, significant at 10% but not at the more stringent 5% level.
Q38
Testing whether the mean of differences between paired observations differs from zero uses:

A. F-test for equality of variances

B. Paired comparisons t-test

C. Chi-square test for population variance

B. Paired comparisons t-test — simply a one-sample t-test on the difference series dᵢ = xᵢ−yᵢ, testing H₀: μ_d = 0.
Q39
For testing a population mean, the test statistic follows:

A. Normal (z) when σ is known; t-distribution when σ is unknown

B. Chi-square when σ is known; F when σ is unknown

C. t-distribution always, regardless of whether σ is known

A. z when σ is known; t when σ is unknown — for large samples, z and t are nearly identical. χ² tests variances; F tests two variances.
Q40
A result can be statistically significant but economically insignificant when:

A. Statistical significance always implies economic significance

B. The effect size is too small to be practically meaningful, even if p < α

C. Only relevant when the sample size is small

B. Effect size too small to matter practically — with very large samples, even trivially small differences become statistically significant. Always consider both statistical and economic significance.
Q41
Hₐ: μ > μ₀ (upper-tail). The rejection region is:

A. Test statistic < −critical value

B. Test statistic > +critical value

C. |Test statistic| > critical value

B. Test statistic > +critical value — upper tail rejection. Option A: lower tail (Hₐ: μ < μ₀). Option C: two-tailed (Hₐ: μ ≠ μ₀).
Q42
χ²(df=10) at α=0.05 two-tailed, critical values are approximately:

A. 3.247 and 20.483

B. 11.689 and 38.076

C. ±1.96

A. 3.247 and 20.483 — from chi-square tables at df=10. (Option B is for df=23.)
Q43
The relationship between α and power: increasing α:

A. Reduces power by making rejection harder

B. Increases power by enlarging the rejection region

C. Has no effect on power

B. Increases power — larger α enlarges the rejection region, making it easier to detect a false null. Trade-off: also increases Type I error rate.
Q44
t=1.85, df=50, two-tailed, critical t≈2.009 at α=5%, p-value≈0.07. Correct conclusion:

A. Reject H₀ at 5% — p-value is close to 0.05

B. Fail to reject H₀ at 5%; reject at α=0.10

C. Reject H₀ at all conventional levels

B. Fail to reject at 5% (|1.85| < 2.009; p=0.07 > 0.05); reject at 10% (p=0.07 < 0.10). Marginally insignificant at 5%.
Q45
Difference-in-means t-test with n₁=30, n₂=40 (pooled variance). Degrees of freedom = ?

A. 68

B. 70

C. 69

A. 68 — df = n₁+n₂−2 = 30+40−2 = 68.
Q46
The p-value approach is often preferred because:

A. It gives the exact probability H₀ is true

B. It shows the exact significance level at which H₀ is rejected, allowing assessment at any α

C. It does not require specification of a significance level

B. Shows the exact α threshold — knowing p=0.032 allows conclusions at any α without re-computing. Critical value approach only gives the decision at the pre-specified α.
Q47
Chi-square test statistic for H₀: σ²=σ₀² is:

A. (n−1)s²/σ₀²

B. s²/σ₀²

C. n×s²/σ₀²

A. (n−1)s²/σ₀² — with df = n−1. Follows a chi-square distribution under H₀. Chi-square values are always ≥ 0 because they are based on squared values.
Q48
Which test statistics can take NEGATIVE values?

A. t and z (symmetric, centred at zero)

B. χ² (based on squared values)

C. F (ratio of variances)

A. t and z — both symmetric around zero and can be negative. χ² and F are always ≥ 0 (based on squares and ratios of squares).
Q49
Testing whether two steel firms had different mean monthly returns over 5 years, given both are influenced by market and industry conditions:

A. Difference in means t-test (independent samples)

B. Paired comparisons t-test (dependent samples)

C. F-test for equality of variances

B. Paired comparisons t-test — both firms' returns are influenced by common factors (market, industry), making samples dependent. Use monthly return differences as paired observations.
Q50
Which summary best describes when to use each test?

A. t for means (σ unknown), χ² for single variance, F for two variances, paired-t for dependent samples

B. z for all means, F for all variance tests

C. χ² for means, t for variances

A. Correct summary — t (or z for large n) for means; χ² for single population variance; F for equality of two variances; paired-t when samples are dependent. Know these four test assignments.
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Ch 8 — Hypothesis Testing
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