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help me finish this finance task , time is 10-12am  23th Nov GMT+8 time

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-

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)

The assignment help me finish this finance task , time is 10-12am  23th Nov GMT+8 time

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