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Question 1Quantitative Methods and Statistics

Under a normal distribution, approximately what percentage of observations lies within two standard deviations of the mean?

  1. 99.7%
  2. 95% Correct answer.
  3. 68%
  4. 50%

Explanation

The empirical rule states that about 95% of a normal distribution lies within plus or minus two standard deviations. About 68% lies within one standard deviation and about 99.7% within three.

Question 2Quantitative Methods and Statistics

What is the possible range of the Pearson correlation coefficient?

  1. −2 to +2
  2. −1 to +1 Correct answer.
  3. −∞ to +∞
  4. 0 to +1

Explanation

Correlation is a standardized covariance, so it is bounded between −1 and +1. Its sign gives the direction of association and its magnitude gives the strength of linear association.

Question 3Quantitative Methods and Statistics

A Type I error occurs when a researcher does which of the following?

  1. Fails to reject a false null hypothesis
  2. Accepts the null hypothesis when it is true
  3. Rejects a true null hypothesis Correct answer.
  4. Rejects a false null hypothesis

Explanation

A Type I error is a false positive: rejecting the null hypothesis when it is actually true. Its probability is the significance level alpha.

Question 4Quantitative Methods and Statistics

An investment returns +10%, −10%, and +20% over three years. What is the geometric mean annual return?

  1. 6.00%
  2. 6.67%
  3. 5.00%
  4. 5.91% Correct answer.

Explanation

The geometric mean compounds the growth factors: [(1.10)(0.90)(1.20)]^(1/3) − 1 = 1.188^(1/3) − 1, which is about 5.91%. It falls below the arithmetic mean because of volatility drag.

Question 5Quantitative Methods and Statistics

A z-score of −1.5 indicates that the observation is:

  1. 0.15 standard deviations below the mean
  2. 15% below the mean
  3. 1.5 standard deviations above the mean
  4. 1.5 standard deviations below the mean Correct answer.

Explanation

A z-score counts standard deviations from the mean, and its sign gives the direction. A value of −1.5 therefore lies 1.5 standard deviations below the mean.

Question 6Quantitative Methods and Statistics

According to the law of large numbers, as sample size increases:

  1. the sample mean tends to approach the population mean Correct answer.
  2. the sample variance tends to zero
  3. the sample mean becomes more variable
  4. the population mean tends to approach the sample mean

Explanation

The law of large numbers states that the sample average converges to the population mean as the sample size grows, provided the population mean is finite.

Question 7Quantitative Methods and Statistics

For a right-skewed distribution, how does the mean typically compare with the median?

  1. The mean is less than the median
  2. The mean equals the median
  3. The mean is greater than the median Correct answer.
  4. No general relationship exists

Explanation

A long right tail pulls the mean upward relative to the median, so the mean typically exceeds the median for positive skew.

Question 8Quantitative Methods and Statistics

A survey interviews only weekday-afternoon shoppers at an upscale mall. The main threat to validity is:

  1. Measurement error from unclear wording
  2. Survivorship bias
  3. Anchoring bias
  4. Sample selection bias Correct answer.

Explanation

The sampling frame excludes many groups, so the sample is not representative of the target population. This is a coverage or selection bias.

Question 9Quantitative Methods and Statistics

The central limit theorem implies that, for large sample size, the sampling distribution of the sample mean:

  1. approaches normality as n grows Correct answer.
  2. has standard deviation equal to the population standard deviation
  3. is normal only if the population is normal
  4. has mean equal to the sample mean

Explanation

For large n the sampling distribution of the mean is approximately normal regardless of the population shape, provided the variance is finite.

Question 10Quantitative Methods and Statistics

A 95% confidence interval for a mean is best interpreted as:

  1. In repeated samples, 95% of intervals constructed this way contain the true mean Correct answer.
  2. 95% of the sample observations fall inside the interval
  3. There is a 95% probability that the true mean lies in this interval
  4. 95% of the population values fall inside the interval

Explanation

The frequentist interpretation concerns coverage: the procedure produces intervals that capture the fixed parameter 95% of the time across repeated samples.

Question 11Quantitative Methods and Statistics

The p-value in a hypothesis test is:

  1. the probability of data at least as extreme, given the null Correct answer.
  2. the probability of a Type II error
  3. the probability that the null hypothesis is true
  4. the probability of rejecting the null when it is false

Explanation

The p-value is computed under the assumption that the null hypothesis holds and measures how extreme the observed statistic is.

Question 12Quantitative Methods and Statistics

Fund A has mean return 10% and standard deviation 15%; Fund B has mean 20% and standard deviation 18%. Which has the lower coefficient of variation?

  1. Fund B, with CV = 0.90 Correct answer.
  2. It cannot be compared from these figures
  3. Fund A, with CV = 1.50
  4. Both are equal

Explanation

The coefficient of variation is standard deviation divided by mean: A is 0.15/0.10 = 1.50 and B is 0.18/0.20 = 0.90. Fund B therefore has less relative risk per unit of return.

Question 13Quantitative Methods and Statistics

A regression reports R-squared = 0.64. This means that:

  1. 64% of the variance of the dependent variable is explained by the model Correct answer.
  2. the correlation between variables is 0.64
  3. there is a 64% probability the model is correct
  4. 64% of observations are correctly predicted

Explanation

R-squared is the proportion of total variation in the dependent variable explained by the regressors. An R-squared of 0.64 indicates that 64% of that variance is explained.

Question 14Quantitative Methods and Statistics

Which statement about covariance and correlation is correct?

  1. Covariance magnitude indicates strength independent of scale
  2. Covariance is bounded between −1 and +1
  3. A correlation of zero proves independence
  4. Correlation is unit-free and measures linear association Correct answer.

Explanation

Correlation standardizes covariance by the product of the standard deviations, making it unit-free and bounded. It measures the strength and direction of linear association.

Question 15Quantitative Methods and Statistics

A disease has prevalence 1%. A test has sensitivity 90% and specificity 95%. Given a positive test, the probability of disease is closest to:

  1. 1%
  2. 15.4% Correct answer.
  3. 90%
  4. 50%

Explanation

By Bayes' theorem, P(disease | positive) = (0.01 × 0.90) / [(0.01 × 0.90) + (0.99 × 0.05)] = 0.009 / 0.0585, which is about 15.4%.

Question 16Quantitative Methods and Statistics

A discrete variable X takes values 0, 1, and 2 with probabilities 0.5, 0.3, and 0.2. Its expected value is:

  1. 0.5
  2. 1.0
  3. 1.5
  4. 0.7 Correct answer.

Explanation

The expected value is the probability-weighted sum: 0 × 0.5 + 1 × 0.3 + 2 × 0.2 = 0.7. The probabilities sum to one, as required.

Question 17Quantitative Methods and Statistics

For a fair coin flipped four times, the probability of exactly two heads is:

  1. 0.50
  2. 0.625
  3. 0.375 Correct answer.
  4. 0.25

Explanation

Using the binomial formula, C(4,2)(0.5)^2(0.5)^2 = 6/16 = 0.375. There are six favorable orderings out of sixteen equally likely sequences.

Question 18Quantitative Methods and Statistics

How many different ways can 3 items be selected from a set of 7, if order does not matter?

  1. 343
  2. 21
  3. 210
  4. 35 Correct answer.

Explanation

The number of combinations is C(7,3) = 7! / (3! × 4!) = 35. Order matters for permutations but not for combinations.

Question 19Quantitative Methods and Statistics

A 4-quarter moving average is computed for quarterly sales. Compared with the raw series, the moving average will:

  1. smooth short-term fluctuations but lag turning points Correct answer.
  2. eliminate seasonality completely
  3. have greater variance than the raw series
  4. respond faster to turning points

Explanation

Averaging over four periods reduces noise and short-term fluctuation, but it also delays detection of changes, creating lag.

Question 20Quantitative Methods and Statistics

A strategy has a monthly Sharpe ratio of 0.40. Assuming independent monthly returns, the annualized Sharpe ratio is closest to:

  1. 4.80
  2. 0.80
  3. 0.40
  4. 1.39 Correct answer.

Explanation

Annualized Sharpe equals the monthly Sharpe ratio multiplied by the square root of 12, since variance scales linearly with time. Thus 0.40 × 3.464 is about 1.39.

Question 21Quantitative Methods and Statistics

In a regression with heteroskedastic errors, which consequence is most accurate?

  1. Estimates stay unbiased but standard errors are invalid Correct answer.
  2. The coefficient estimates become biased and inconsistent
  3. The residuals are guaranteed to be autocorrelated
  4. R-squared becomes exactly zero

Explanation

Non-constant error variance violates the standard OLS standard-error formulas. Under exogeneity the coefficients remain unbiased, but conventional inference is unreliable.

Question 22Quantitative Methods and Statistics

Severe multicollinearity among regressors typically causes:

  1. Residual variance to drop to zero
  2. Standard errors to inflate and precision to fall Correct answer.
  3. The model's R-squared to become negative
  4. Omitted variable bias in every coefficient

Explanation

Highly correlated regressors make individual coefficient estimates imprecise, inflating their standard errors. Predictions may remain reliable even when individual coefficients are poorly estimated.

Question 23Quantitative Methods and Statistics

Which property makes logarithmic returns useful in multi-period analysis?

  1. They are always larger than simple returns
  2. They cannot be negative
  3. They are additive over time Correct answer.
  4. They equal simple returns exactly for all values

Explanation

The log return over multiple periods is the sum of the single-period log returns, which simplifies compounding. Log returns approximate simple returns closely only for small price changes.

Question 24Quantitative Methods and Statistics

Why does the sample variance divide by n − 1 rather than n?

  1. The population mean is assumed to be known
  2. One observation is always discarded from the data
  3. It makes the variance estimate larger in every sample
  4. It corrects bias from using the sample mean Correct answer.

Explanation

Deviations from the sample mean sum to zero, so only n − 1 of them are free to vary. Dividing by n − 1 yields an unbiased estimator of the population variance.

Question 25Quantitative Methods and Statistics

Monte Carlo simulation is best described as:

  1. removing all estimation error from a model
  2. proving asymptotic normality analytically
  3. solving linear systems exactly in one step
  4. estimating outcomes by repeated random sampling Correct answer.

Explanation

Monte Carlo methods draw many random scenarios to approximate distributions or expectations when analytic solutions are difficult. The approach is numerical and stochastic rather than exact.

Question 26Quantitative Methods and Statistics

A portfolio's annual return has mean 8% and standard deviation 12%. In a particular year the return is 20%. Its z-score is closest to:

  1. 1.00 Correct answer.
  2. 2.40
  3. 1.20
  4. 0.80

Explanation

The z-score is (observed value − mean) / standard deviation. Here (20% − 8%) / 12% = 12% / 12% = 1.00, so the return is one standard deviation above the mean.

Question 27Quantitative Methods and Statistics

A Type I error in hypothesis testing occurs when:

  1. a true null hypothesis is rejected Correct answer.
  2. a false null hypothesis is rejected
  3. a true null hypothesis is not rejected
  4. a false null hypothesis is not rejected

Explanation

A Type I error is rejecting a null hypothesis that is actually true. Its probability is the significance level, α, chosen before the test.

Question 28Quantitative Methods and Statistics

Which summary statistic is least affected by a few extreme outliers?

  1. The median Correct answer.
  2. The standard deviation
  3. The arithmetic mean
  4. The range

Explanation

The median depends on the middle position of ordered data, so a few extreme values do not move it much. The mean, standard deviation, and range all use extreme values directly.

Question 29Quantitative Methods and Statistics

Which sampling method divides the population into non-overlapping strata and randomly samples within each stratum?

  1. Simple random sampling
  2. Systematic sampling
  3. Stratified sampling Correct answer.
  4. Cluster sampling

Explanation

Stratified sampling divides the population into non-overlapping strata and draws random samples within each stratum. This helps ensure representation of each subgroup.

Question 30Quantitative Methods and Statistics

For a sample of 64 observations with sample standard deviation 20, the standard error of the sample mean is:

  1. 0.3125
  2. 5.0
  3. 2.5 Correct answer.
  4. 20.0

Explanation

The standard error of the sample mean is s / √n. With s = 20 and n = 64, it is 20 / 8 = 2.5.

Question 31Quantitative Methods and Statistics

Events A and B are mutually exclusive, with P(A) = 0.30 and P(B) = 0.45. What is P(A or B)?

  1. 0.15
  2. 0.135
  3. 0.75 Correct answer.
  4. 0.60

Explanation

For mutually exclusive events, P(A or B) = P(A) + P(B). Thus 0.30 + 0.45 = 0.75.

Question 32Quantitative Methods and Statistics

A binomial random variable has n = 20 trials and success probability p = 0.25. Its expected value and variance are:

  1. 5 and 3.75 Correct answer.
  2. 5 and 1.25
  3. 5 and 5
  4. 4 and 3.75

Explanation

For a binomial variable, E(X) = np and Var(X) = np(1 − p). Here 20 × 0.25 = 5 and 20 × 0.25 × 0.75 = 3.75.

Question 33Quantitative Methods and Statistics

When the population standard deviation is unknown and the sample is small, the appropriate distribution for a confidence interval for a mean is:

  1. t with n − 1 df Correct answer.
  2. F with n − 1 df
  3. Standard normal
  4. Chi-square with n df

Explanation

When the population standard deviation is unknown and the sample is small, use a t-distribution with n − 1 degrees of freedom. It has heavier tails than the standard normal to account for estimating the standard deviation.

Question 34Quantitative Methods and Statistics

Adjusted R-squared differs from R-squared because adjusted R-squared:

  1. It always increases when any regressor is added
  2. It penalizes regressors that do not improve fit Correct answer.
  3. It is always larger than R-squared
  4. It measures explained variance exactly

Explanation

Adjusted R-squared applies a penalty for adding regressors, so it can fall when a new variable does not improve fit enough to justify its cost in degrees of freedom. R-squared itself never falls when a regressor is added.

Question 35Quantitative Methods and Statistics

In a simple linear regression of y on x, the estimated slope is 2.5. This means:

  1. x explains 2.5% of the variation in y
  2. a one-unit rise in y predicts a 2.5-unit rise in x
  3. the correlation between x and y equals 2.5
  4. a one-unit rise in x predicts a 2.5-unit rise in y Correct answer.

Explanation

The slope estimates the change in predicted y for a one-unit increase in x, holding other regressors fixed in multiple regression. A slope of 2.5 therefore means predicted y rises by 2.5 units per one-unit rise in x.

Question 36Quantitative Methods and Statistics

The number of defects per item follows a Poisson distribution with mean 0.5. The probability of exactly 2 defects is closest to:

  1. 0.2500
  2. 0.1250
  3. 0.3033
  4. 0.0758 Correct answer.

Explanation

For a Poisson variable with mean λ, P(X = k) = e^(−λ) λ^k / k!. With λ = 0.5 and k = 2, this is e^(−0.5) × 0.25 / 2 ≈ 0.0758.

Question 37Quantitative Methods and Statistics

Two datasets have different units. To compare relative variability, the most appropriate measure is:

  1. Coefficient of variation Correct answer.
  2. Variance in squared units
  3. Standard deviation in original units
  4. Interquartile range

Explanation

The coefficient of variation, standard deviation divided by the mean, is unitless. That makes it suitable for comparing relative variability across data measured in different units.

Question 38Quantitative Methods and Statistics

X and Y are random variables with P(X = 1) = 0.5, P(Y = 1) = 0.4, and P(X = 1 and Y = 1) = 0.2. Are the events X = 1 and Y = 1 independent?

  1. No, because 0.5 and 0.4 are unequal
  2. No, because 0.2 is less than 0.5
  3. Yes, because 0.2 is greater than zero
  4. Yes, because 0.2 equals 0.5 × 0.4 Correct answer.

Explanation

Two events are independent if P(A and B) = P(A)P(B). Here 0.5 × 0.4 = 0.2, which equals the given joint probability, so the events are independent.

Question 39Quantitative Methods and Statistics

In a regression of y on x, omitting a variable z causes bias in the estimated coefficient on x when z is:

  1. correlated with both x and y Correct answer.
  2. independent of both x and y
  3. correlated with y but not with x
  4. correlated with x but not with y

Explanation

Omitted variable bias in the coefficient on x requires the omitted variable z to be correlated with x and to affect y, equivalently to be correlated with y after conditioning on x. If either link is missing, the bias is absent.

Question 40Quantitative Methods and Statistics

X and Y are independent random variables with Var(X) = 4 and Var(Y) = 9. What is Var(3X − 2Y)?

  1. 36
  2. 0
  3. 48
  4. 72 Correct answer.

Explanation

For independent variables, Var(aX + bY) = a²Var(X) + b²Var(Y), and the sign of b does not matter because it is squared. Thus Var(3X − 2Y) = 9(4) + 4(9) = 36 + 36 = 72.

Question 41Financial Markets and Instruments

Which instrument is a short-dated sovereign debt security sold at a discount to face value and redeemed at par?

  1. Corporate bond
  2. Preference share
  3. Treasury bill Correct answer.
  4. Treasury note

Explanation

Treasury bills are issued at a discount and repay face value at maturity, so the investor's return is the difference between purchase price and par. Notes and bonds instead carry periodic coupon payments and typically have longer maturities.

Question 42Financial Markets and Instruments

A bond's coupon rate is best described as:

  1. the annual coupon over face value Correct answer.
  2. the spread paid over the benchmark rate
  3. the fixed price paid at issuance
  4. the yield earned if held to maturity

Explanation

The coupon rate is the fixed annual interest stated on the bond, expressed as a percentage of the bond's face (par) value. It is set at issuance and does not change with the bond's market price.

Question 43Financial Markets and Instruments

Which right is normally attached to an ordinary (common) share?

  1. A guaranteed repurchase by the issuer
  2. Voting at general meetings Correct answer.
  3. A dividend fixed before any other distribution
  4. Priority over creditors on liquidation

Explanation

Ordinary shareholders are residual owners and typically carry voting rights at shareholder meetings. Their dividends are discretionary and they rank behind creditors and preference holders in a liquidation.

Question 44Financial Markets and Instruments

A market order instructs a broker to:

  1. execute only in the closing auction
  2. trade only at a stated price or better
  3. trade immediately at the best available price Correct answer.
  4. execute only if a price trigger is hit

Explanation

A market order prioritises immediate execution over price, so it fills against the best available opposing orders. The trade-off is that the execution price is not guaranteed.

Question 45Financial Markets and Instruments

In the foreign exchange quote EUR/USD = 1.10, the base currency is:

  1. the currency chosen by the trading venue
  2. the US dollar
  3. the euro Correct answer.
  4. the currency with the higher interest rate

Explanation

An FX quote shows how much of the quote (price) currency is needed for one unit of the base currency, and the base currency is the one written first. Here one euro buys 1.10 US dollars.

Question 46Financial Markets and Instruments

Compared with a forward contract, a futures contract is characteristically:

  1. standardised and exchange-traded Correct answer.
  2. completely free of counterparty risk
  3. settled only by physical delivery
  4. available only on equities

Explanation

Futures are standardised contracts traded on organised exchanges and cleared through a central counterparty, with daily margining. Forwards are private, customised agreements between two parties.

Question 47Financial Markets and Instruments

The primary market is the place where:

  1. existing securities change hands between investors
  2. new securities are issued and sold to investors Correct answer.
  3. central banks set their policy rates
  4. derivative trades are cleared

Explanation

In the primary market an issuer raises capital by selling newly created securities, for example through an IPO or a bond issue. Subsequent trading of those securities occurs in the secondary market.

Question 48Financial Markets and Instruments

An open-ended fund differs from a closed-ended fund because it:

  1. must be liquidated after one year
  2. trades only on an exchange
  3. issues a fixed number of shares at launch
  4. redeems units on demand at NAV Correct answer.

Explanation

Open-ended funds create and cancel units as investors subscribe or redeem, with transactions priced at net asset value. Closed-ended funds have a fixed share count and their shares trade on an exchange.

Question 49Financial Markets and Instruments

A bond with a face value of $1,000 pays an annual coupon of 6% and trades at $1,250. Its current yield is:

  1. 4.0%
  2. 6.0%
  3. 4.8% Correct answer.
  4. 7.5%

Explanation

Current yield equals the annual coupon divided by the current market price: $60 / $1,250 = 4.8%. It ignores any capital gain or loss to maturity, which is why it differs from yield to maturity.

Question 50Financial Markets and Instruments

Holding all else equal, which of the following bonds has the greatest sensitivity to a change in yields?

  1. A 30-year zero-coupon bond Correct answer.
  2. A 2-year bond with a 9% coupon
  3. A 5-year bond with a 7% coupon
  4. A 10-year bond with a 6% coupon

Explanation

Interest rate sensitivity rises with maturity and falls with coupon size, since high coupons return cash sooner. A long-dated zero-coupon bond has all its cash flow at the far end and therefore the longest duration.