CFA Level I Quantitative Methods

$1,000 is invested at 5% a year, compounded annually. What is it worth after 2 years?

$1,000 × 1.05² = $1,102.50.

What is the present value of $1,100 received in one year at a 10% discount rate?

$1,100 / 1.10 = $1,000.

What is the effective annual rate of 12% a year, compounded monthly?

(1 + 0.12/12)¹² − 1 ≈ 12.68%. 12.36% is semi-annual compounding.

Standard deviation is the square root of the:

Variance is the average squared deviation from the mean.

In a normal distribution, about 95% of observations lie within how many standard deviations of the mean?

68% lie within ±1, 90% within ±1.65, 95% within ±1.96 and 99% within ±2.58.

A share is bought at $50, pays a $2 dividend and is sold at $55. What is the holding period return?

(55 − 50 + 2) / 50 = 14%.

An asset returns +10% in year 1 and −10% in year 2. What is its geometric mean return?

√(1.10 × 0.90) − 1 = √0.99 − 1 ≈ −0.50%. The arithmetic mean is 0%.

Which return measure is not affected by the timing of client deposits and withdrawals?

That is why managers are judged on time-weighted returns.

What is the present value of $5 a year forever at a 4% discount rate?

A perpetuity is worth payment / rate = 5 / 0.04.

What is the present value of $100 at the end of each year for 3 years at 10%?

100 × [1 − 1.1⁻³] / 0.1 = 248.69. $273.55 is the annuity due (payments at the start of each year).

$100 is invested at 5% compounded continuously. What is it worth after 2 years?

100 × e^(0.05 × 2) = 100 × e^0.1 ≈ 110.52.

P(A) = 0.4, P(B) = 0.3 and P(A and B) = 0.1. What is P(A or B)?

P(A or B) = P(A) + P(B) − P(A and B).

1% of firms will default. A model flags 90% of firms that will default and 5% of those that won't. A firm is flagged. What is the probability it defaults?

Bayes: 0.009 / (0.009 + 0.0495) ≈ 0.154. False flags dominate when the event is rare.

An investment returns +10% with probability 0.5, 0% with 0.3 and −20% with 0.2. What is its expected return?

0.5 × 10% + 0.3 × 0% + 0.2 × (−20%) = 5% − 4% = 1%.

Two assets have a covariance of 0.006 and standard deviations of 0.10 and 0.20. What is their correlation?

Correlation = 0.006 / (0.10 × 0.20) = 0.30.

A return distribution is positively skewed. Which is usually true?

The long right tail pulls the mean up.

A leptokurtic return distribution has, compared with a normal distribution:

Excess kurtosis above zero means more extreme outcomes than the normal model predicts.

What is a Type I error?

Its probability is the significance level, α.

A test gives a p-value of 0.03. At a 5% significance level you:

Reject when the p-value is below α.

A population has a standard deviation of 20. What is the standard error of the mean for a sample of 100?

σ / √n = 20 / 10 = 2.

What does the central limit theorem say?

It holds for samples of about 30 or more with a finite variance.

A regression of Y on X has an R² of 0.64. What does it mean?

In a simple regression, R² is the square of the correlation, here 0.8.

A regression gives Y = 2 + 0.5X. What is the predicted Y when X = 10?

2 + 0.5 × 10 = 7.

Grouping customers by behaviour without any labels is an example of:

Clustering finds structure in unlabelled data.

A model fits its training data almost perfectly but predicts new data poorly. This is:

Cross-validation and simpler models help.

A sample mean is 8% with a standard error of 1%. What is the 95% confidence interval?

8% ± 1.96 × 1%. 5.42% to 10.58% would be the 99% interval.

Why are asset prices often modelled as lognormal?

The lognormal is bounded at zero and skewed to the right.

What does Monte Carlo simulation do?

Historical simulation is the method that replays past data.