FINS2624: Final Exam FORMULA SHEET
•
The price of a bond, P is given by:
𝑃=
𝑐
1
𝐹𝑉
+
[1 −
]
𝑦
(1 + 𝑦)𝑇
(1 + 𝑦)𝑇
Where c is the dollar coupon, y is the yield-to-maturity, T is the time to maturity and FV is the face
value.
•
If X and Y are two random variables, 𝑎 and 𝑏 are constants
o
o
o
o
o
o
o
o
o
o
•
𝐸(𝑎𝑋 + 𝑏) = 𝑎𝐸(𝑋) + 𝑏
𝐸(𝑋 + 𝑌) = 𝐸(𝑋) + 𝐸(𝑌)
𝑉𝑎𝑟(𝑋) = 𝐸[𝑋 − 𝐸(𝑋)]2
𝑉𝑎𝑟(𝑎𝑋 + 𝑏) = 𝑎2 𝑉𝑎𝑟(𝑋)
𝑉𝑎𝑟(𝑋 + 𝑌) = 𝑉𝑎𝑟(𝑋) + 𝑉𝑎𝑟(𝑌) + 2𝐶𝑜𝑣(𝑋, 𝑌)
𝐶𝑜𝑣(𝑋, 𝑌) = 𝐸(𝑋𝑌) − 𝐸(𝑋)𝐸(𝑌)
𝐶𝑜𝑣(𝑋, 𝑋) = 𝑉𝑎𝑟(𝑋)
𝐶𝑜𝑣(𝑎𝑋 + 𝑏, 𝑐𝑌 + 𝑑) = 𝑎𝑐𝐶𝑜𝑣(𝑋, 𝑌)
𝐶𝑜𝑣(𝑋 + 𝑌, 𝑃 + 𝑄) = 𝐶𝑜𝑣(𝑋, 𝑃) + 𝐶𝑜𝑣(𝑋, 𝑄) + 𝐶𝑜𝑣(𝑌, 𝑃) + 𝐶𝑜𝑣(𝑌, 𝑄)
𝐶𝑜𝑣(𝑋, 𝑌) = 𝜌𝑋,𝑌 𝜎𝑋 𝜎𝑌
Variance of portfolio returns:
o 𝜎𝑃2 = 𝑉𝑎𝑟(𝑟𝑃 ) = 𝐶𝑜𝑣(𝑤𝐴 𝑟𝐴 , 𝑤𝐴 𝑟𝐴 ) + 𝐶𝑜𝑣(𝑤𝐵 𝑟𝐵 , 𝑤𝐴 𝑟𝐴 ) + 𝐶𝑜𝑣(𝑤𝐴 𝑟𝐴 , 𝑤𝐵 𝑟𝐵 ) +
𝐶𝑜𝑣(𝑤𝐵 𝑟𝐵 , 𝑤𝐵 𝑟𝐵 )
o
𝜎𝑃2 = 𝑤𝐴2 𝜎𝐴2 + 𝑤𝐵2 𝜎𝐵2 + 2𝑤𝐴 𝑤𝐵 𝜌𝜎𝐴 𝜎𝐵
•
Optimal weight on risky portfolio:
•
𝐸(𝑟𝑃∗ ) − 𝑟𝑓 1 𝐸(𝑟𝑃∗ ) − 𝑟𝑓
= ⋅
2
2
𝐴
𝐴𝜎𝑃∗
𝜎𝑃∗
Asset weights in optimal risky portfolio - two risky asset case:
𝑦∗ =
𝐸(𝑅1 )𝜎22 −𝐸(𝑅2 )𝐶𝑜𝑣(𝑅1 ,𝑅2 )
2
2
1 )𝜎2 +𝐸(𝑅2 )𝜎1 −(𝐸(𝑅1 )+𝐸(𝑅2 ))𝐶𝑜𝑣(𝑅1 ,𝑅2 )
𝑊1 = 𝐸(𝑅
•
The Optimal weight, 𝑤𝐴∗ of the active portfolio A is:
𝑤0
where, 𝑤𝐴0 =
𝐴
𝑤𝐴∗ = 1+𝑤0 (1−𝛽
𝐴
•
𝑤ℎ𝑒𝑟𝑒 𝐸(𝑅𝑖 ) = 𝐸(𝑟𝑖 ) − 𝑟𝑓
𝐴)
The optimal weight of an individual asset i in the active portfolio is:
𝑤𝐴,𝑖 =
2
𝛼𝑖 ⁄𝜎𝜖,𝑖
2
∑𝑁
𝑗=1 𝛼𝑗 ⁄𝜎𝜖,𝑗
2
𝛼𝐴 ⁄𝜎𝜖,𝐴
2
[𝐸(𝑟𝑀 )−𝑟𝑓 ]⁄𝜎𝑀
•
Black-Scholes pricing formula for European Call Options
𝑐𝑡 = 𝑆𝑡 𝑁(𝑑1 ) − 𝑋𝑒 −𝑟(𝑇−𝑡) 𝑁(𝑑2 )
where N() is the standard normal cumulative probability
𝑑1 =
𝑆
𝜎2
ln ( 𝑋𝑡 ) + (𝑟 + 2 )(𝑇 − 𝑡)
𝜎√(𝑇 − 𝑡)
𝑑2 = 𝑑1 − 𝜎√(𝑇 − 𝑡)
Where,
t refers to the current time point,
T refers to the expiration;
𝑆𝑡 is the current price of the underlying asset;
X is exercise price;
r is continuously compounded risk-free rate;
σ is return volatility of the underlying asset.
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