theory of application

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)