IAI Actuarial Core Principles · Business Finance

Key principles of finance: formula sheet

Full chapter guide

Key formulas

Accumulation (future value): FV = PV × (1 + i)^n; i is the effective rate per period and n is the number of the same periods.
Present value (discounting): PV = FV × v^n, where v = 1 ÷ (1 + i); v is the discount factor for one period.
Nominal to effective rate: 1 + i = (1 + i(m) ÷ m)^m; i(m) is the nominal annual rate compounded m times a year. i is the effective annual rate.
Force of interest and continuous compounding: 1 + i = e^δ, so PV = FV × e^(−δn); δ is the force of interest, the nominal rate with continuous compounding.
Annuity-immediate present value: a(n) = (1 − v^n) ÷ i; n payments of 1, the first one period from now.
Annuity-due present value: ä(n) = (1 − v^n) ÷ d = (1 + i) × a(n), where d = i ÷ (1 + i); First payment is made immediately.
Annuity-immediate accumulated value: s(n) = ((1 + i)^n − 1) ÷ i; Value of the payments at the time of the last payment.
Perpetuity: PV = C ÷ i (immediate); PV = C ÷ d (due); Valid only for i > 0. C is the level payment.
Growing perpetuity: PV = C ÷ (i − g); First payment C is one period away. Requires i > g. g is the constant growth rate.
Net present value: NPV = Σ CF(t) × v^t; CF(t) is the net cash flow at time t. Include the initial outlay at t = 0.
Expected return of an asset: E(R) = Σ pᵢ × Rᵢ; pᵢ are the probabilities of each outcome and they must add to 1.
Variance and standard deviation: σ² = Σ pᵢ × (Rᵢ − E(R))²; σ = √σ²; Equivalent form: σ² = E(R²) − [E(R)]². Standard deviation is in the same units as return.
Risk premium: Risk premium = E(R) − Rf; Rf is the risk-free rate.
Portfolio expected return: E(Rp) = w₁E(R₁) + w₂E(R₂); Weights add to 1. It is a simple weighted average, so no diversification effect appears here.
Two-asset portfolio variance: σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂; Here ρ₁₂σ₁σ₂ = Cov(R₁, R₂). Portfolio standard deviation is the square root.
Correlation: ρ₁₂ = Cov(R₁, R₂) ÷ (σ₁σ₂); Lies between −1 and +1. Portfolio risk falls as ρ falls.
Total risk split: Total risk = systematic risk + unsystematic risk; In variance terms the two parts add. Diversification reduces only the unsystematic part.
CAPM: E(Ri) = Rf + βi × (E(Rm) − Rf); Rf is the risk-free rate. E(Rm) − Rf is the market risk premium. Result is the required return on asset i.
Beta: βi = Cov(Ri, Rm) ÷ Var(Rm) = ρim × σi ÷ σm; ρim is the correlation between the asset and the market. σ is standard deviation.
Cost of equity from dividend growth model: Ke = D1 ÷ P0 + g; Assumes dividends grow at a constant rate g forever. D1 is next year's dividend. P0 is the current ex-dividend price.
After-tax cost of debt (irredeemable): Kd = i × (1 − T); i is the interest rate on the debt at its market value. T is the corporate tax rate. Assumes interest is fully tax-deductible.
Cost of redeemable debt: Kd (pre-tax) is the IRR of: market price now, interest flows, redemption value; Find the rate that makes the present value of the flows equal the market price. Then multiply by (1 − T) for the after-tax cost.
WACC: WACC = [E × Ke + D × Kd(1 − T)] ÷ (E + D); Use market values of equity E and debt D. Add terms for preference shares if present.
Portfolio beta: βp = Σ wi × βi; wi are the proportions of the portfolio's value in each asset. Weights sum to 1.
Net present value: NPV = Σ [CFt ÷ (1 + r)^t], summed from t = 0 to n; CF0 is normally negative (the outlay). r is the cost of capital. Accept if NPV > 0.
Annuity present value: PV = C × [1 − (1 + r)^−n] ÷ r; Use for level cash flows at the end of each year for n years.
Perpetuity present value: PV = C ÷ r; Level cash flow C at the end of each year forever.
IRR condition: Σ [CFt ÷ (1 + IRR)^t] = 0; Accept if IRR > required return, for a conventional project.
IRR by linear interpolation: IRR ≈ r1 + [NPV1 ÷ (NPV1 − NPV2)] × (r2 − r1); NPV1 and NPV2 are computed at two trial rates r1 and r2, ideally one giving a positive and one a negative NPV, close together. The result is an estimate.
Simple payback: Payback = full years before recovery + (unrecovered amount ÷ cash flow in recovery year); Assumes cash flows arise evenly through the year. Use cumulative cash flows.
Law of one price: If two portfolios give identical cash flows in every state, then price A = price B; If prices differ, buy the cheaper and sell the dearer to lock in a riskless profit.
Arbitrage condition: Initial cost ≤ 0, payoff ≥ 0 in all states, payoff > 0 in at least one state with positive probability; A strategy must satisfy all of these. Mere high expected profit is not arbitrage.
Forward price (no income, continuous compounding): F = S × e^(δT); S is the spot price, δ the risk-free force of interest, T the time to delivery. Use F = S × (1 + i)^T with annual compounding.
Forward price with known income: F = (S − PV of income) × (1 + i)^T; Subtract the present value of income paid before delivery. Valid under no-arbitrage with no transaction costs.
Information sets in EMH: Weak ⊂ Semi-strong ⊂ Strong; Each stronger form includes all the information of the weaker one.
MM Proposition I (no tax): V_G = V_U; Value of a geared firm equals value of an identical ungeared firm. Holds only under the perfect market assumptions.
MM Proposition II (no tax, risk-free debt): ke = k0 + (k0 − kd) × D/E; Cost of equity rises linearly with the debt to equity ratio. D and E are market values.
WACC (no tax): WACC = ke × E/(D+E) + kd × D/(D+E) = k0; Unchanged by gearing in the no-tax case.
MM with corporate tax (permanent debt): V_G = V_U + t × D; t is the corporate tax rate and D the market value of permanent debt. Ignores distress costs.
Cost of equity with tax: ke = k0 + (k0 − kd) × (1 − t) × D/E; WACC falls as gearing rises because of the tax shield.
After-tax cost of debt: kd(1 − t); Used in WACC when interest is deductible.
Dividend irrelevance (no frictions): Value = PV of future free cash flows to investors, independent of payout; Ex-dividend share price falls by about the dividend paid in perfect markets.
Shareholder wealth (return to shareholders): Total shareholder return = (P₁ − P₀ + D) ÷ P₀; P₀ is the opening share price, P₁ the closing price and D the dividends in the period. Wealth is created through both price rise and dividends.
Agency cost: Agency cost = monitoring cost + bonding cost + residual loss; Residual loss is the value lost because the agent's decisions still differ from the principal's best choice.
Value of a share (dividend view): P₀ = Σ Dₜ ÷ (1 + ke)ᵗ, for t = 1 to ∞; Links wealth maximisation to future dividends, their timing and risk through the cost of equity ke.

Quick revision

Present value = cash flow ÷ (1 + i)^t, where i is the discount rate per period and t is the number of periods.
A higher discount rate lowers present value.
Investors demand higher expected return for taking more risk.
CAPM: E(Ri) = Rf + βi × (E(Rm) − Rf). Beta measures sensitivity to market risk.
Only systematic risk is rewarded in CAPM, because specific risk can be diversified away.
NPV = Σ discounted cash inflows − initial outlay. Accept a project if NPV > 0.
IRR is the discount rate that makes NPV equal to zero. Compare it with the required return.
NPV and IRR can rank mutually exclusive projects differently. NPV measures value added in rupees.
Arbitrage means making a risk-free profit with no net investment. In an efficient market it should not persist.
Weak, semi-strong and strong forms of efficiency differ by the information reflected in prices.
Debt is usually cheaper than equity because it carries lower risk and interest may be tax deductible, but more debt raises financial risk.
Agency problems arise when managers' interests differ from shareholders' interests.

Common mistakes

Using a nominal rate directly as the periodic rate. Fix: Convert first. A nominal 12% compounded monthly means 1% per month. The effective annual rate is 1.01^12 − 1, about 12.68%.
Mismatching the period of the rate and the time unit. Fix: Make n and i refer to the same period. Convert the annual effective rate to a quarterly rate using (1 + i)^(1/4) − 1.
Taking the square root of portfolio variance before adding the terms, or averaging standard deviations. Fix: Add the variance terms first. Only when ρ = 1 does the standard deviation equal the weighted average of the standard deviations.
Forgetting the factor 2 in the covariance term. Fix: Write the formula in full every time: w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂.
Using the market return in place of the market risk premium in CAPM. Fix: Always compute Rm − Rf first. Then multiply by beta and add Rf.
Forgetting the tax adjustment on debt, or applying it to equity. Fix: Apply (1 − T) only to the interest cost of debt. Dividends and equity returns are not adjusted.
Including depreciation or sunk costs in the cash flows. Fix: Use only future incremental cash flows. Add back depreciation if you started from profit, and ignore costs already incurred.
Forgetting working capital and its recovery at the end. Fix: Put the increase in working capital as an outflow when it arises and the release as an inflow in the final year.
Saying an efficient market means prices never fall or that no one makes money. Fix: State that efficiency means no consistent excess return after allowing for risk. Returns still occur and prices still fall.
Mixing up weak and semi-strong form. Fix: Link each form to its set: weak is past prices only, semi-strong is all public information, strong is all information.

Exam tips

Always state the rate basis in your answer, for example 'effective annual 8%'. Markers give credit for clear assumptions.
In MCQs, look first at the rate wording. Many wrong options come from using nominal as effective or the reverse.
Write the timeline even for short questions. Timing errors are the most common source of lost marks.
Show the formula and the substituted values before the number. This secures method marks if your arithmetic slips.
Keep at least five decimal places in discount factors until the final step, then round the answer.
Show every term of the portfolio variance formula on separate lines. Method marks are given even if arithmetic slips.
In written answers, name the type of risk (systematic or unsystematic) and say which one diversification affects.
State assumptions, such as risk-averse investors and that standard deviation is the risk measure.