ICAP PRC-2 Quantitative Analysis for Business cheat sheet

Percentage change = (new − old) / old × 100; a rise then fall of the same percentage never returns to the original value (e.g. +20% then −20% leaves 96%).

Simple interest: I = P·r·t; compound interest: A = P(1 + r/m)^(m·t), where m is compounding periods per year and r is the nominal annual rate.

Effective annual rate = (1 + r/m)^m − 1; continuous compounding gives EAR = e^r − 1.

Break-even quantity = Fixed costs / (Unit price − Unit variable cost); contribution margin per unit is price minus variable cost.

Markup on cost = (Price − Cost)/Cost; margin on price = (Price − Cost)/Price; converting markup to margin divides by (1 + markup).

Rules of indices: a^m·a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^0 = 1 for a ≠ 0.

Logarithms: log(xy) = log x + log y; log(x/y) = log x − log y; log x^n = n·log x; solving a^x = b gives x = log b / log a.

Sum of an arithmetic series = n/2·(2a + (n−1)d); sum of a geometric series = a(1 − r^n)/(1 − r) for r ≠ 1.

Mean = Σx/n; for grouped data use class midpoints weighted by frequency: x̄ = Σfx/Σf.

Median position in ordered data = (n+1)/2; for grouped data use interpolation within the median class using cumulative frequency.

Mode is the most frequent value; for grouped data use the modal class and the formula with d1 = f_m − f_(m−1) and d2 = f_m − f_(m+1).

Quartiles: Q1 at (n+1)/4 and Q3 at 3(n+1)/4; interquartile range = Q3 − Q1; semi-interquartile range = (Q3 − Q1)/2.

Sample variance s² = Σ(x − x̄)²/(n − 1); population variance σ² = Σ(x − μ)²/N; standard deviation is the positive square root.

Coefficient of variation = (s/x̄) × 100; it compares relative dispersion across data sets with different units or means.

Pearson's coefficient of skewness = 3(mean − median)/standard deviation; positive value indicates right skew.

In a histogram, frequency density = frequency / class width; areas, not heights, represent frequencies when widths differ.

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the intersection term is zero.

Multiplication rule: P(A ∩ B) = P(A)·P(B|A); independence holds when P(A ∩ B) = P(A)·P(B).

Bayes' theorem: P(A|B) = P(B|A)·P(A) / P(B), with P(B) expanded by the law of total probability.

Binomial: P(X = r) = C(n,r)·p^r·(1−p)^(n−r); mean = np; variance = np(1−p); requires fixed n and constant p.

Poisson: P(X = r) = e^(−λ)·λ^r / r!; mean and variance both equal λ; used for counts in a fixed interval.

Normal distribution: standardize with z = (x − μ)/σ; approximately 68% of values lie within ±1σ, 95% within ±2σ, 99.7% within ±3σ.

Complement rule: P(A′) = 1 − P(A); "at least one" problems are usually fastest via the complement.

Permutations nPr = n!/(n−r)!; combinations nCr = n!/(r!(n−r)!); use combinations when order is irrelevant.

Laspeyres price index = Σ(p1·q0)/Σ(p0·q0) × 100; uses base-period quantities and tends to overstate price rises.

Paasche price index = Σ(p1·q1)/Σ(p0·q1) × 100; uses current-period quantities and tends to understate price rises.

Fisher's ideal index = √(Laspeyres × Paasche); it satisfies both the time-reversal and factor-reversal tests.

Simple aggregate index = Σp1/Σp0 × 100; simple average of relatives = mean of (p1/p0 × 100).

Chain (link) relatives multiply together to give the fixed-base index: base index = product of link relatives / 100^(k−1).

Real value = money value / price index × 100; deflating converts nominal figures to base-period purchasing power.

Base shifting: new index = old index / old index of new base × 100; splicing joins two series by rescaling the older one.

Weights in a weighted average of relatives are usually base-period value shares p0·q0.

Multiplicative model: Y = T × S × C × I; additive model: Y = T + S + C + I, where T is trend, S seasonal, C cyclical, I irregular.

Centred moving average for an even period (e.g. 4) is a two-stage average of consecutive moving averages; it removes seasonality to isolate trend.

Seasonal index = (actual / moving average) × 100 in the multiplicative model; average the ratios for each season and adjust so they sum to 400 for quarterly data.

Deseasonalized value = actual / seasonal index; forecast = trend estimate × seasonal index.

Linear trend by least squares: Y = a + bX with b = (nΣXY − ΣXΣY)/(nΣX² − (ΣX)²) and a = Ȳ − bX̄.

Exponential smoothing: F(t+1) = α·A(t) + (1 − α)·F(t); a higher α gives more weight to recent actual values.

Mean absolute deviation = Σ|actual − forecast|/n; mean squared error = Σ(actual − forecast)²/n; lower values indicate a better fit.

Use an odd number of periods in a moving average to avoid the need for centring.

Pearson correlation r = Σ(x − x̄)(y − ȳ) / √(Σ(x − x̄)²·Σ(y − ȳ)²); r lies between −1 and +1.

Regression line of y on x: y = a + bx, with b = Cov(x,y)/Var(x) = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² and a = ȳ − b·x̄.

The two regression lines intersect at (x̄, ȳ); r² = b_xy × b_yx, the product of the two slopes.

Coefficient of determination r² is the proportion of variation in y explained by x; the remainder is unexplained.

Spearman's rank correlation = 1 − 6Σd²/(n(n² − 1)), where d is the difference in ranks; apply for ordinal data.

Correlation measures association only; it does not establish causation, and r = 0 does not rule out a non-linear relationship.

Extrapolating a regression line far beyond the observed range of x is unreliable.

Covariance sign indicates the direction of linear association; its magnitude depends on units, unlike r.

Present value of a single sum: PV = FV/(1 + r)^n; future value: FV = PV(1 + r)^n.

Present value of an ordinary annuity: PV = A·[1 − (1 + r)^(−n)]/r; annuity due multiplies this by (1 + r).

Net present value = Σ cash flow_t/(1 + r)^t − initial outlay; accept a project when NPV > 0.

Internal rate of return is the discount rate at which NPV = 0; accept when IRR exceeds the cost of capital.

Perpetuity with constant cash flow: PV = A/r; growing perpetuity: PV = A/(r − g) for r > g.

Sinking fund factor = r/[(1 + r)^n − 1]; capital recovery factor = r/[1 − (1 + r)^(−n)].

Loan amortization: each payment covers interest on the outstanding balance first, with the remainder reducing principal.

Use the periodic rate and number of periods consistently: a nominal annual rate compounded monthly uses r/12 and 12n periods.

Expected monetary value = Σ(probability × payoff) across states of nature for each alternative; choose the highest EMV.

Maximax criterion selects the alternative with the highest possible payoff (optimistic); maximin selects the highest of the minimum payoffs (pessimistic).

Minimax regret uses an opportunity loss table: regret = best payoff in a state − actual payoff; choose the alternative with the smallest maximum regret.

Expected value of perfect information = expected payoff with perfect information − maximum EMV.

Probabilities of states of nature must sum to 1; EMV is a long-run average, not a guarantee for a single decision.

A decision tree is evaluated by rolling back from terminal nodes, weighting branches by their probabilities.

The value of sample information is the increase in EMV from imperfect information and never exceeds EVPI.

Hurwicz criterion uses α·best payoff + (1 − α)·worst payoff, with α as the index of optimism.

A linear program has a linear objective function and linear constraints; the feasible region is the set satisfying all constraints including non-negativity.

The optimal solution of a two-variable LP lies at a corner (vertex) of the feasible region; evaluate the objective at each vertex.

Graphical method: plot each constraint as an equation, shade the feasible side, identify vertices, then substitute into the objective.

A binding constraint is satisfied with equality at the optimum; slack is the unused amount of a ≤ constraint, surplus the excess over a ≥ constraint.

Shadow price (dual value) is the change in the objective per unit increase in a constraint's right-hand side, valid within its range.

Convert to standard form by adding slack variables to ≤ constraints and subtracting surplus variables from ≥ constraints.

Simplex method moves between basic feasible solutions, improving the objective until no negative value remains in the objective row (maximization).

Infeasibility means no point satisfies all constraints; unboundedness means the objective can improve without limit.

Standard error of the mean = σ/√n; when σ is unknown and n is large, use s/√n.

Confidence interval for a mean: x̄ ± z·(σ/√n); for a proportion: p̂ ± z·√(p̂(1 − p̂)/n).

Common critical values: 90% → 1.645, 95% → 1.96, 99% → 2.576 (two-tailed).

Hypothesis test steps: state H0 and H1, choose significance level, compute the test statistic, compare with the critical value, then decide.

Test statistic for a mean: z = (x̄ − μ0)/(σ/√n); use the t-distribution with n − 1 degrees of freedom when σ is unknown and n is small.

Type I error rejects a true null hypothesis (probability α); Type II error fails to reject a false null (probability β); power = 1 − β.

A p-value below α leads to rejecting H0; a confidence interval that excludes the hypothesized value gives the same conclusion.

Sampling methods: simple random, stratified (within homogeneous groups), systematic (every kth item), and cluster (whole groups selected).