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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(π‘Ÿπ‘) = 𝑀𝐷𝐸(π‘Ÿπ·)+𝑀𝐸𝐸(π‘ŸπΈ)

Variance of the return on
a portfolio

πœŽπ‘
2 = (π‘€π·πœŽπ·)

2 +(π‘€πΈπœŽπΈ)
2 +2(π‘€π·πœŽπ·)(π‘€πΈπœŽπΈ)𝜌𝐷𝐸

Portfolio variance (n
assets) when securities
have the same standard
(Οƒ) and share a common

correlation coefficient (ρ)

πœŽπ‘
2 =

1

𝑛
𝜎2 +

𝑛 βˆ’1

𝑛
𝜌𝜎2

Correlation between
assets D and E

𝜌𝐷𝐸 = πΆπ‘œπ‘Ÿπ‘Ÿ(π‘Ÿπ·,π‘ŸπΈ) =
πΆπ‘œπ‘£(π‘Ÿπ·,π‘ŸπΈ)

𝜎𝐷𝜎𝐸
Sharpe ratio of a portfolio 𝑆𝑝 =
𝐸(π‘Ÿπ‘)βˆ’π‘Ÿπ‘“

πœŽπ‘
Sharpe ratio maximising
portfolio weights with

two risky assets (D and E)
and a risk-free asset

𝑀𝐷 =
[𝐸(π‘Ÿπ·)βˆ’π‘Ÿπ‘“]𝜎𝐸

2 βˆ’[𝐸(π‘ŸπΈ)βˆ’π‘Ÿπ‘“]𝜎𝐷𝜎𝐸𝜌𝐷𝐸

[𝐸(π‘Ÿπ·)βˆ’π‘Ÿπ‘“]𝜎𝐸
2 +[𝐸(π‘ŸπΈ)βˆ’π‘Ÿπ‘“]𝜎𝐷

2 βˆ’[𝐸(π‘Ÿπ·)βˆ’π‘Ÿπ‘“ +𝐸(π‘ŸπΈ)βˆ’π‘Ÿπ‘“]𝜎𝐷𝜎𝐸𝜌𝐷𝐸
𝑀𝐸 = 1βˆ’π‘€π·

Optimal capital allocation
to the risky

asset/portfolio
𝑦 =

𝐸(π‘Ÿπ‘)βˆ’π‘Ÿπ‘“

π΄πœŽπ‘
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 = 𝛽2πœŽπ‘€
2 +πœŽπ‘’

2

πΆπ‘œπ‘£(π‘Ÿπ‘–,π‘Ÿπ‘—) = Product of betas x Market-index risk = π›½π‘–π›½π‘—πœŽπ‘€
2

Treynor-Black
optimisation procedure

𝑀𝑖
0 =

𝛼𝑖
𝜎2(𝑒𝑖)

(1)
β‡’ 𝑀𝑖 =

𝑀𝑖
0

βˆ‘ 𝑀𝑖
0𝑛

𝑖

(2)
β‡’

{
𝛼𝐴 = βˆ‘π‘€π‘–π›Όπ‘–

𝑛

𝑖=1

𝜎2(𝑒𝐴) = βˆ‘π‘€π‘–
2

𝑛

𝑖=1

𝜎2(𝑒𝑖)

𝛽𝐴 = βˆ‘π‘€π‘–π›½π‘–

𝑛

𝑖=1

(3)
β‡’ 𝑀𝐴

0 = [

𝛼𝐴
𝜎2(𝑒𝐴)
⁄

𝐸(𝑅𝑀)
πœŽπ‘€
2⁄
]

(4)
β‡’ 𝑀𝐴

βˆ— =
𝑀𝐴
0

1+(1βˆ’π›½π΄)𝑀𝐴
0

(5)
β‡’ {
𝑀𝑀
βˆ— = 1βˆ’π‘€π΄

βˆ—

𝑀𝑖
βˆ— = 𝑀𝐴

βˆ—π‘€π‘–

(6)
β‡’ {
𝐸(𝑅𝑃) = (𝑀𝑀

βˆ— +𝑀𝐴
βˆ—π›½π΄)𝐸(𝑅𝑀)+𝑀𝐴

βˆ—π›Όπ΄
πœŽπ‘ƒ
2 = (𝑀𝑀

βˆ— +𝑀𝐴
βˆ—π›½π΄)

2πœŽπ‘€
2 +[𝑀𝐴

βˆ—πœŽ(𝑒𝐴)]
2

Multifactor model (2
factors):

𝑅𝑖 = 𝐸(𝑅𝑖)+𝛽𝑖1𝐹1 +𝛽𝑖2𝐹2 +𝑒𝑖

Multifactor SML (2
factors):

𝐸(π‘Ÿπ‘–) = π‘Ÿπ‘“ +𝛽𝑖1[𝐸(π‘Ÿ1)βˆ’π‘Ÿπ‘“]+𝛽𝑖2[𝐸(π‘Ÿ2)βˆ’π‘Ÿπ‘“]

Fama-French 3 factor
model:

𝑅𝑖𝑑 = 𝛼𝑖 +𝛽𝑖𝑀𝑅𝑀𝑑 +𝛽𝑖𝑆𝑀𝐡𝑆𝑀𝐡𝑑 +𝛽𝑖𝐻𝑀𝐿𝐻𝑀𝐿𝑑 +𝑒𝑖𝑑

Fama-French 3 factor
model (APT):

𝐸(π‘Ÿπ‘–)βˆ’π‘Ÿπ‘“ = π‘Žπ‘– +𝑏𝑖[𝐸(π‘Ÿπ‘€)βˆ’π‘Ÿπ‘“]+𝑠𝑖𝐸(𝑆𝑀𝐡)+β„Žπ‘–πΈ(𝐻𝑀𝐿)

M2 of portfolio P: 𝑀2 = πœŽπ‘€(𝑆𝑝 βˆ’π‘†π‘€)

Treynor measure: 𝑇𝑝 =
π‘Ÿπ‘ βˆ’π‘Ÿπ‘“

𝛽𝑝
Jensen’s alpha: 𝛼𝑝 = �̅�𝑝 βˆ’[�̅�𝑓 +𝛽𝑝(�̅�𝑀 βˆ’οΏ½Μ…οΏ½π‘“)]

Information ratio:
𝛼𝑝

𝜎(𝑒𝑝)
Morningstar risk-
adjusted return: 𝑀𝑅𝐴𝑅(𝛾) = [

1

𝑇
Σ𝑑=1
𝑇 (

1+π‘Ÿπ‘‘
1+π‘Ÿπ‘“π‘‘

)

βˆ’π›Ύ

]

βˆ’12/𝛾

βˆ’1

Stock index futures

Hedge ratio =
Hedge value

Total position value
Optimal hedge ratio = β„Žβˆ— = 𝜌(
πœŽπ‘ 
πœŽπ‘“
)

Number of contracts required to hedge the risk in a stock portfolio =
𝑉𝑝

𝑉𝐹
Γ—
𝛽𝑝

𝛽𝐹
Interest rate futures

Duration of interest rate futures contract = 𝐷𝐹 = π·π‘ˆ +𝑀𝐹

Number of contracts required to hedge the risk in a bond portfolio =
𝐷𝑝

𝐷𝐹
Γ—
𝑉𝑝

𝑉𝐹
Bargaining model (2
players with ’A’ making

the initial offer; 3
dates)

Player A: 1βˆ’π›½(1βˆ’π›Ό); Player B: 𝛽(1βˆ’π›Ό)

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