Formula Sheet

FORMULA SHEET
The price of a bond, 𝑃, is given by:

𝑃=

𝑐
1
𝐹𝑉
[1 −
]
+
(1 + 𝑦)𝑇
(1 + 𝑦)𝑇
𝑦

where 𝑐 is the 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 and α and 𝛽 are two constants then:
𝐸 (𝑋 + 𝑌) = 𝐸 (𝑋) + 𝐸(𝑌)
𝐸 (𝛼𝑋) = 𝛼𝐸 (𝑋)
𝑉𝑎𝑟(𝑋) = 𝐶𝑜𝑣(𝑋, 𝑋)
𝐶𝑜𝑣(𝑋, 𝑌) = 𝐶𝑜𝑣(𝑌, 𝑋)
𝐶𝑜𝑣(𝑋, 𝑌) = 𝜌𝑋𝑌 𝜎𝑋 𝜎𝑌
𝐶𝑜𝑣(𝛼𝑋, 𝛽𝑌) = 𝛼𝛽𝐶𝑜𝑣(𝑋, 𝑌)
The solution to the quadratic equation
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
is
𝑥1⁄2 =

−𝑏 ± √𝑏2 − 4𝑎𝑐
2𝑎

The Modified Duration of a bond, 𝐷 ∗, is given by
𝐷∗ =

𝐷
(1 + 𝑦)

where 𝐷 is the Macaulay Duration and 𝑦 is the yield-to maturity
The return on a portfolio, 𝑟𝑃 , consisting of assets 𝐴 and 𝐵 is given by
𝑟𝑃 = 𝑤𝐴 𝑟𝐴 + 𝑤𝐵 𝑟𝐵
where 𝑤𝐴 , 𝑤𝐵 are the weights allocated to assets A and B and 𝑟𝐴 , 𝑟𝐵 are the returns
respectively