Investment Management Certificate (IMC) Quantitative Methods and the Time Value of Money Syllabus

What is the range as a measure of dispersion?

The range is the difference between the highest and lowest values: $\text{Range} = x_{max} - x_{min}$. It is simple but sensitive to outliers.

What is the formula for population variance?

$\sigma^{2} = \frac{1}{N} \sum_{i=1}^{N} (x_{i} - \mu)^{2}$ — the average of squared deviations from the population mean $\mu$.

What is the formula for sample variance and why does it use $n-1$?

$s^{2} = \frac{1}{n-1} \sum_{i=1}^{n} (x_{i} - \bar{x})^{2}$. The $n-1$ divisor (Bessel's correction) gives an unbiased estimate of the population variance.

How is standard deviation related to variance?

Standard deviation is the square root of variance: $\sigma = \sqrt{\sigma^{2}}$. It expresses dispersion in the same units as the original data.

What is the coefficient of variation and what does it measure?

$CV = \frac{\sigma}{\bar{x}}$ — the ratio of standard deviation to mean. It measures relative dispersion (risk per unit of return), enabling comparison across data sets with different units or means.

Define a positively (right) skewed distribution.

A positively skewed distribution has a long right tail. Typically $\text{mode} < \text{median} < \text{mean}$, as large positive outliers pull the mean upward.

Define a negatively (left) skewed distribution.

A negatively skewed distribution has a long left tail. Typically $\text{mean} < \text{median} < \text{mode}$, as large negative outliers pull the mean downward.

What does kurtosis measure?

Kurtosis measures the 'peakedness' and tail thickness of a distribution. Leptokurtic ($>3$) distributions have fat tails; platykurtic ($<3$) have thin tails; the normal distribution has kurtosis of $3$ (excess kurtosis of $0$).

What is a histogram?

A histogram is a bar chart displaying the frequency distribution of continuous data grouped into intervals (bins). The bar area/height represents the frequency of observations in each interval.

What is a frequency distribution?

A frequency distribution tabulates data by grouping observations into classes/intervals and showing the number (frequency) of observations falling into each class.

What does a scatter plot display?

A scatter plot shows the relationship between two variables by plotting paired observations as points on $x$ and $y$ axes, revealing the direction, strength, and form of any correlation.

What is the difference between a probability and an odds expression?

Probability expresses likelihood as $P(E)$ between $0$ and $1$. Odds express the ratio of favourable to unfavourable outcomes; odds 'for' $= \frac{P(E)}{1 - P(E)}$.

State the addition rule of probability for two events.

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$. For mutually exclusive events, $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.

State the multiplication rule of probability for independent events.

For independent events, $P(A \cap B) = P(A) \times P(B)$. More generally $P(A \cap B) = P(A) \times P(B \mid A)$.

What is conditional probability?

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ — the probability that event $A$ occurs given that event $B$ has occurred.

What is expected value (mean) of a discrete random variable?

$E(X) = \sum_{i} p_{i} x_{i}$ — the probability-weighted sum of all possible outcomes.

What are the key properties of the normal distribution?

It is symmetric and bell-shaped, fully described by its mean $\mu$ and standard deviation $\sigma$, with mean = median = mode, and tails extending to infinity. It is the basis for much of modern finance theory.

State the empirical (68-95-99.7) rule for the normal distribution.

About $68\%$ of observations lie within $\mu \pm 1\sigma$, about $95\%$ within $\mu \pm 2\sigma$ (more precisely $1.96\sigma$), and about $99.7\%$ within $\mu \pm 3\sigma$.

What is the standard normal distribution and the z-score formula?

The standard normal distribution has $\mu = 0$ and $\sigma = 1$. Any value is standardised via $z = \frac{x - \mu}{\sigma}$, expressing how many standard deviations $x$ lies from the mean.

What is covariance and what does its sign indicate?

Covariance measures how two variables move together: $\text{Cov}(X,Y) = \frac{1}{n}\sum (x_{i}-\bar{x})(y_{i}-\bar{y})$. A positive value means they move in the same direction; negative means opposite directions.

What is the correlation coefficient and its formula?

$\rho_{X,Y} = \frac{\text{Cov}(X,Y)}{\sigma_{X}\,\sigma_{Y}}$ — a standardised measure of linear association ranging from $-1$ to $+1$.

What do correlation coefficients of $+1$, $0$, and $-1$ indicate?

$+1$ = perfect positive linear relationship; $-1$ = perfect negative linear relationship; $0$ = no linear relationship. Values between indicate partial linear association.

Why does correlation not imply causation?

A high correlation only shows that two variables move together; it does not prove one causes the other. The relationship may be coincidental (spurious) or driven by a third underlying variable.

What is the equation of a simple linear regression line?

$y = a + bx + \epsilon$, where $a$ is the intercept, $b$ the slope (sensitivity of $y$ to $x$), $x$ the independent variable, and $\epsilon$ the error term.

What does the coefficient of determination $R^{2}$ measure?

$R^{2}$ measures the proportion of the variation in the dependent variable explained by the independent variable(s). In simple regression it equals the square of the correlation coefficient and ranges from $0$ to $1$.

What is the holding period return (HPR) formula?

$HPR = \frac{P_{1} - P_{0} + D}{P_{0}}$, where $P_{0}$ is the purchase price, $P_{1}$ the ending price, and $D$ the income/dividend received over the period.

How do you annualise a holding period return?

$\text{Annualised return} = (1 + HPR)^{\frac{1}{t}} - 1$, where $t$ is the holding period in years. This converts a return into an equivalent compound annual rate.

What is the difference between real and nominal returns?

The nominal return includes inflation; the real return strips inflation out, measuring purchasing-power gain. The real return reflects the actual increase in goods/services you can buy.

State the Fisher equation linking real and nominal returns.

$1 + r_{nominal} = (1 + r_{real})(1 + i)$, where $i$ is inflation. The approximation $r_{real} \approx r_{nominal} - i$ holds for small rates.

What is the difference between a price-weighted and a market-capitalisation-weighted index?

A price-weighted index (e.g. Dow Jones) weights constituents by share price, so high-priced shares dominate. A market-cap-weighted index (e.g. FTSE 100) weights by total market value, so large companies dominate, reflecting their economic size.