PIPFA Quantitative Methods Syllabus

State the future value of an ordinary annuity formula.

FV = A × [((1 + r)ⁿ − 1) / r], where A = periodic payment, r = interest rate per period, n = number of payments.

State the present value of an ordinary annuity formula.

PV = A × [(1 − (1 + r)^(−n)) / r], where A = periodic payment, r = interest rate per period, n = number of payments.

What is a perpetuity and how is its present value calculated?

A perpetuity is an annuity with infinite equal payments. Its present value PV = A / r, where A is the periodic payment and r is the interest rate per period.

What does present value (discounting) represent, and what is the basic formula?

Present value is the current worth of a future sum, reflecting the time value of money. PV = FV / (1 + r)ⁿ = FV × (1 + r)^(−n), where r is the discount rate and n is the number of periods.

What is a discount factor and how is it used?

A discount factor is the multiplier (1 + r)^(−n) used to convert a future cash flow into its present value. Multiply each future cash flow by its discount factor and sum them to get total PV.

What is the purpose of presenting data, and what are the main methods?

Data presentation organises raw data to make it understandable and reveal patterns. Main methods: tabular (frequency tables), and graphical/diagrammatic (bar charts, pie charts, histograms, frequency polygons, ogives).

What is the difference between a histogram and a bar chart?

A histogram displays continuous (grouped) data with bars touching, area proportional to frequency. A bar chart displays discrete/categorical data with separate (non-touching) bars of equal width, height showing frequency.

What is an ogive (cumulative frequency curve) used for?

An ogive plots cumulative frequency against upper class boundaries. It is used to estimate the median, quartiles, percentiles, and the number of items above or below a given value.

Define the three main measures of central tendency.

Mean: arithmetic average (sum ÷ number of values). Median: the middle value when data is ordered. Mode: the most frequently occurring value.

State the formula for the arithmetic mean of grouped data.

Mean = Σ(fx) / Σf, where f is the frequency of each class and x is the class midpoint.

How do you find the median position for ungrouped data?

Arrange data in order. For n values, the median is the value at position (n + 1)/2. If n is even, the median is the average of the two middle values.

When is the median preferred over the mean as a measure of central tendency?

The median is preferred when the data is skewed or contains extreme values (outliers), because unlike the mean it is not distorted by them.

List the main measures of dispersion.

Range, interquartile range (and quartile deviation), mean (average) deviation, variance, standard deviation, and coefficient of variation.

What is the range and what is the interquartile range?

Range = Highest value − Lowest value. Interquartile range (IQR) = Q3 − Q1, the spread of the middle 50% of the data. Quartile deviation = (Q3 − Q1)/2.

State the formula for the standard deviation of a population.

σ = √[Σ(x − x̄)² / N], where x̄ is the mean and N is the number of values. It is the square root of the variance and measures average spread around the mean.

What is the coefficient of variation and why is it useful?

Coefficient of variation (CV) = (Standard deviation / Mean) × 100. It is a relative measure of dispersion that allows comparison of variability between data sets with different units or means.

What is the difference between variance and standard deviation?

Variance is the mean of the squared deviations from the mean (expressed in squared units). Standard deviation is the square root of the variance, expressed in the same units as the original data.

State the classical definition of probability for equally likely outcomes.

P(event) = (Number of favourable outcomes) / (Total number of possible outcomes). Probability ranges from 0 (impossible) to 1 (certain).

State the addition rule of probability for two events A and B.

P(A or B) = P(A) + P(B) − P(A and B). If A and B are mutually exclusive, P(A and B) = 0, so P(A or B) = P(A) + P(B).

State the multiplication rule of probability for independent events.

For independent events, P(A and B) = P(A) × P(B). For dependent events, P(A and B) = P(A) × P(B | A), where P(B | A) is the conditional probability of B given A.

What does it mean for two events to be mutually exclusive versus independent?

Mutually exclusive events cannot occur together (P(A and B) = 0). Independent events do not affect each other's probability (the occurrence of one does not change the probability of the other).

What are the key properties of the normal distribution?

It is bell-shaped, symmetrical about the mean, with mean = median = mode. The total area under the curve = 1, and it is fully described by its mean (μ) and standard deviation (σ).

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

Approximately 68% of values lie within ±1σ of the mean, 95% within ±2σ (more precisely 1.96σ), and 99.7% within ±3σ of the mean.

How do you calculate a z-score and what does it represent?

z = (x − μ) / σ. It standardises a value by measuring how many standard deviations x lies above (positive) or below (negative) the mean, allowing use of the standard normal table.

What conditions must be met for a binomial distribution to apply?

A fixed number of trials (n); each trial has only two outcomes (success/failure); the probability of success p is constant; and trials are independent.

State the binomial probability formula.

P(X = r) = ⁿCᵣ × pʳ × q^(n−r), where ⁿCᵣ = n!/[r!(n−r)!], p = probability of success, q = 1 − p, n = number of trials, r = number of successes.

State the mean and standard deviation of a binomial distribution.

Mean = np; Variance = npq; Standard deviation = √(npq), where n = number of trials, p = probability of success, q = 1 − p.

What is a price index number and what does it measure?

A price index number measures the relative change in the price of an item or basket of items between a base period and a current period. Base period index = 100. Simple price relative = (Current price / Base price) × 100.

What is the difference between the Laspeyres and Paasche price index?

Laspeyres uses base-period quantities as weights: Σ(p₁q₀)/Σ(p₀q₀) × 100. Paasche uses current-period quantities as weights: Σ(p₁q₁)/Σ(p₀q₁) × 100. Laspeyres tends to overstate, Paasche to understate, price changes.

What are the four components of a time series?

Trend (T) - long-term direction; Seasonal variation (S) - regular short-term cycles within a year; Cyclical variation (C) - longer wave-like business cycles; Irregular/Random variation (I) - unpredictable fluctuations.

What is the additive versus multiplicative model of a time series?

Additive model: Y = T + S + C + I (components added; seasonal effect constant in absolute terms). Multiplicative model: Y = T × S × C × I (components multiplied; seasonal effect proportional to the trend).

What is the method of moving averages used for in trend analysis?

Moving averages smooth out short-term seasonal and irregular fluctuations to reveal the underlying trend by averaging successive groups of observations (e.g. a 4-quarter moving average for quarterly data).

How is forecasting carried out using the least-squares trend line and seasonal factors?

Fit a trend line Y = a + bx by least squares, project the trend for the future period, then multiply (multiplicative model) or add (additive model) the relevant seasonal factor to obtain the forecast.

What does the correlation coefficient (r) measure, and what is its range?

It measures the strength and direction of the linear relationship between two variables. r ranges from −1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear relationship.

State Pearson's product-moment correlation coefficient formula.

r = [nΣxy − ΣxΣy] / √{[nΣx² − (Σx)²][nΣy² − (Σy)²]}, where n is the number of paired observations.

What is the coefficient of determination (r²) and how is it interpreted?

r² is the square of the correlation coefficient. It gives the proportion (or percentage) of the total variation in the dependent variable explained by the independent variable through the regression line.

What is Spearman's rank correlation coefficient and its formula?

It measures correlation between the ranks of two variables. R = 1 − [6Σd²] / [n(n² − 1)], where d is the difference between paired ranks and n is the number of pairs.

What is the purpose of simple linear regression?

To model the relationship between a dependent variable (Y) and one independent variable (X) by fitting a straight line, allowing prediction of Y for given values of X.

State the simple linear regression equation and what its coefficients represent.

Y = a + bX, where a is the intercept (value of Y when X = 0) and b is the slope/regression coefficient (the change in Y for each one-unit increase in X).

State the least-squares formulas for the slope (b) and intercept (a) in simple linear regression.

b = [nΣxy − ΣxΣy] / [nΣx² − (Σx)²] and a = ȳ − b·x̄ (equivalently a = (Σy − bΣx)/n). The line passes through the point (x̄, ȳ).