FINC6001 – Finance: Theory to Applications Final Exam Formula Sheet Margin Margin= equity in account value of stock Expected rate of return on a portfolio 𝐸(𝑟𝑝) = 𝑤𝐷𝐸(𝑟𝐷)+𝑤𝐸𝐸(𝑟𝐸) Variance of the return on a portfolio 𝜎𝑝 2 = (𝑤𝐷𝜎𝐷) 2 +(𝑤𝐸𝜎𝐸) 2 +2(𝑤𝐷𝜎𝐷)(𝑤𝐸𝜎𝐸)𝜌𝐷𝐸 Portfolio variance (n assets) when securities have the same standard (σ) and share a common correlation coefficient (ρ) 𝜎𝑝 2 = 1 𝑛 𝜎2 + 𝑛 −1 𝑛 𝜌𝜎2 Correlation between assets D and E 𝜌𝐷𝐸 = 𝐶𝑜𝑟𝑟(𝑟𝐷,𝑟𝐸) = 𝐶𝑜𝑣(𝑟𝐷,𝑟𝐸) 𝜎𝐷𝜎𝐸 Sharpe ratio of a portfolio 𝑆𝑝 = 𝐸(𝑟𝑝)−𝑟𝑓 𝜎𝑝 Sharpe ratio maximising portfolio weights with two risky assets (D and E) and a risk-free asset 𝑤𝐷 = [𝐸(𝑟𝐷)−𝑟𝑓]𝜎𝐸 2 −[𝐸(𝑟𝐸)−𝑟𝑓]𝜎𝐷𝜎𝐸𝜌𝐷𝐸 [𝐸(𝑟𝐷)−𝑟𝑓]𝜎𝐸 2 +[𝐸(𝑟𝐸)−𝑟𝑓]𝜎𝐷 2 −[𝐸(𝑟𝐷)−𝑟𝑓 +𝐸(𝑟𝐸)−𝑟𝑓]𝜎𝐷𝜎𝐸𝜌𝐷𝐸 𝑤𝐸 = 1−𝑤𝐷 Optimal capital allocation to the risky asset/portfolio 𝑦 = 𝐸(𝑟𝑝)−𝑟𝑓 𝐴𝜎𝑝 2 Single index model (SIM) in excess returns 𝑅𝑖 = 𝛼𝑖 +𝛽𝑖𝑅𝑀 +𝑒𝑖 Security risk in the SIM Total risk = Systematic risk + Firm-specific risk 𝜎2 = 𝛽2𝜎𝑀 2 +𝜎𝑒 2 𝐶𝑜𝑣(𝑟𝑖,𝑟𝑗) = Product of betas x Market-index risk = 𝛽𝑖𝛽𝑗𝜎𝑀 2 Treynor-Black optimisation procedure 𝑤𝑖 0 = 𝛼𝑖 𝜎2(𝑒𝑖) (1) ⇒ 𝑤𝑖 = 𝑤𝑖 0 ∑ 𝑤𝑖 0𝑛 𝑖 (2) ⇒ { 𝛼𝐴 = ∑𝑤𝑖𝛼𝑖 𝑛 𝑖=1 𝜎2(𝑒𝐴) = ∑𝑤𝑖 2 𝑛 𝑖=1 𝜎2(𝑒𝑖) 𝛽𝐴 = ∑𝑤𝑖𝛽𝑖 𝑛 𝑖=1 (3) ⇒ 𝑤𝐴 0 = [ 𝛼𝐴 𝜎2(𝑒𝐴) ⁄ 𝐸(𝑅𝑀) 𝜎𝑀 2⁄ ] (4) ⇒ 𝑤𝐴 ∗ = 𝑤𝐴 0 1+(1−𝛽𝐴)𝑤𝐴 0 (5) ⇒ { 𝑤𝑀 ∗ = 1−𝑤𝐴 ∗ 𝑤𝑖 ∗ = 𝑤𝐴 ∗𝑤𝑖 (6) ⇒ { 𝐸(𝑅𝑃) = (𝑤𝑀 ∗ +𝑤𝐴 ∗𝛽𝐴)𝐸(𝑅𝑀)+𝑤𝐴 ∗𝛼𝐴 𝜎𝑃 2 = (𝑤𝑀 ∗ +𝑤𝐴 ∗𝛽𝐴) 2𝜎𝑀 2 +[𝑤𝐴 ∗𝜎(𝑒𝐴)] 2 Multifactor model (2 factors): 𝑅𝑖 = 𝐸(𝑅𝑖)+𝛽𝑖1𝐹1 +𝛽𝑖2𝐹2 +𝑒𝑖 Multifactor SML (2 factors): 𝐸(𝑟𝑖) = 𝑟𝑓 +𝛽𝑖1[𝐸(𝑟1)−𝑟𝑓]+𝛽𝑖2[𝐸(𝑟2)−𝑟𝑓] Fama-French 3 factor model: 𝑅𝑖𝑡 = 𝛼𝑖 +𝛽𝑖𝑀𝑅𝑀𝑡 +𝛽𝑖𝑆𝑀𝐵𝑆𝑀𝐵𝑡 +𝛽𝑖𝐻𝑀𝐿𝐻𝑀𝐿𝑡 +𝑒𝑖𝑡 Fama-French 3 factor model (APT): 𝐸(𝑟𝑖)−𝑟𝑓 = 𝑎𝑖 +𝑏𝑖[𝐸(𝑟𝑀)−𝑟𝑓]+𝑠𝑖𝐸(𝑆𝑀𝐵)+ℎ𝑖𝐸(𝐻𝑀𝐿) M2 of portfolio P: 𝑀2 = 𝜎𝑀(𝑆𝑝 −𝑆𝑀) Treynor measure: 𝑇𝑝 = 𝑟𝑝 −𝑟𝑓 𝛽𝑝 Jensen’s alpha: 𝛼𝑝 = �̅�𝑝 −[�̅�𝑓 +𝛽𝑝(�̅�𝑀 −�̅�𝑓)] Information ratio: 𝛼𝑝 𝜎(𝑒𝑝) Morningstar risk- adjusted return: 𝑀𝑅𝐴𝑅(𝛾) = [ 1 𝑇 Σ𝑡=1 𝑇 ( 1+𝑟𝑡 1+𝑟𝑓𝑡 ) −𝛾 ] −12/𝛾 −1 Stock index futures Hedge ratio = Hedge value Total position value Optimal hedge ratio = ℎ∗ = 𝜌( 𝜎𝑠 𝜎𝑓 ) Number of contracts required to hedge the risk in a stock portfolio = 𝑉𝑝 𝑉𝐹 × 𝛽𝑝 𝛽𝐹 Interest rate futures Duration of interest rate futures contract = 𝐷𝐹 = 𝐷𝑈 +𝑀𝐹 Number of contracts required to hedge the risk in a bond portfolio = 𝐷𝑝 𝐷𝐹 × 𝑉𝑝 𝑉𝐹 Bargaining model (2 players with ’A’ making the initial offer; 3 dates) Player A: 1−𝛽(1−𝛼); Player B: 𝛽(1−𝛼) ...