Formula Sheet
Present value of a cash flow
𝐶𝐶
𝑛𝑛
𝑃𝑃𝑃𝑃 = (1+𝑟𝑟)
𝑛𝑛
Present value of a growing perpetuity
𝑃𝑃𝑃𝑃 =
𝐶𝐶
𝑟𝑟−𝑔𝑔
𝑃𝑃𝑃𝑃 =
𝐶𝐶
𝑟𝑟
Present value of a perpetuity
Price of bond
Present value of a stream of cash flows
𝐶𝐶
𝐶𝐶
𝐶𝐶
𝐶𝐶
1
2
3
𝑛𝑛
𝑃𝑃𝑃𝑃 = 𝑐𝑐0 + (1+𝑟𝑟)
1 + (1+𝑟𝑟)2 + (1+𝑟𝑟)3 + ⋯ + (1+𝑟𝑟)𝑛𝑛
Dividend discount model
𝑃𝑃0 =
𝐷𝐷𝐷𝐷𝐷𝐷
𝑟𝑟−𝑔𝑔
Present value of an annuity
𝑃𝑃𝑃𝑃 = 𝐶𝐶 �
1−(1+𝑟𝑟)−𝑛𝑛
�
𝑟𝑟
1 − (1 + 𝑟𝑟)−𝑛𝑛
𝑃𝑃𝑃𝑃 = 𝐶𝐶 �
� + 𝐹𝐹(1 + 𝑟𝑟)−𝑛𝑛
𝑟𝑟
Free cash flow
FCF = EBIT × (1 - Tax rate) + Depreciation – Capital expenditure – Change in net working capital
EBIT = Revenue – Costs – Operating expenses - Depreciation
Profitability index
𝑃𝑃𝑃𝑃 =
𝑁𝑁𝑁𝑁𝑁𝑁
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼
Net present value
𝑇𝑇
𝑁𝑁𝑁𝑁𝑁𝑁 = �
𝑡𝑡=0
1−(1+𝑟𝑟)−𝑛𝑛
�
𝑟𝑟
𝑁𝑁𝑁𝑁𝑁𝑁 = 𝐸𝐸𝐸𝐸𝐸𝐸 × �
𝐹𝐹𝐹𝐹𝐹𝐹𝑡𝑡
(1 + 𝑟𝑟)𝑡𝑡
Sample mean
𝑥𝑥̅ =
Equivalent annual cost
𝑥𝑥1 +𝑥𝑥2 +⋯+𝑥𝑥𝑇𝑇
𝑇𝑇
Expected value
Sample standard deviation
s2 =
(𝑥𝑥1 −𝑥𝑥̅ )2 +(𝑥𝑥2 −𝑥𝑥̅ )2 +⋯+(𝑥𝑥𝑇𝑇 −𝑥𝑥̅ )2
𝐸𝐸(𝑋𝑋) = Pr(𝑋𝑋 = 𝑥𝑥1 ) 𝑥𝑥1 + Pr(𝑋𝑋 = 𝑥𝑥2 ) 𝑥𝑥2 + ⋯ + Pr(𝑋𝑋 = 𝑥𝑥𝑛𝑛 ) 𝑥𝑥𝑛𝑛
Population variance
𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋) = 𝐸𝐸{[𝑋𝑋 − 𝐸𝐸(𝑋𝑋)]2 } = 𝐸𝐸(𝑋𝑋 2 ) − [𝐸𝐸(𝑋𝑋)]2
Expectation algebra
𝐸𝐸(𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏) = 𝑎𝑎𝑎𝑎(𝑋𝑋) + 𝑏𝑏𝑏𝑏(𝑌𝑌)
𝑇𝑇−1
𝑠𝑠 = √𝑠𝑠 2
Variance algebra
𝑉𝑉𝑉𝑉𝑉𝑉(𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏) = 𝑎𝑎2 𝑉𝑉𝑉𝑉𝑉𝑉(𝑋𝑋) + 𝑏𝑏 2 𝑉𝑉𝑉𝑉𝑉𝑉(𝑌𝑌) + 2𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎(𝑋𝑋, 𝑌𝑌)
Covariance
𝐶𝐶𝐶𝐶𝐶𝐶(𝑋𝑋, 𝑌𝑌) = 𝐸𝐸[(𝑋𝑋 − 𝐸𝐸(𝑋𝑋))(𝑌𝑌 − 𝐸𝐸(𝑌𝑌))] = 𝐸𝐸(𝑋𝑋𝑋𝑋) − 𝐸𝐸(𝑋𝑋)𝐸𝐸(𝑌𝑌)
Holding return
𝑅𝑅𝑡𝑡,𝑡𝑡+1 =
𝑃𝑃𝑡𝑡+1 + 𝐷𝐷𝑡𝑡+1
𝑃𝑃𝑡𝑡+1 − 𝑃𝑃𝑡𝑡 𝐷𝐷𝑡𝑡+1
−1=
+
𝑃𝑃𝑡𝑡
𝑃𝑃𝑡𝑡
𝑃𝑃𝑡𝑡
Correlation
𝜌𝜌(𝑅𝑅1 , 𝑅𝑅2 ) =
Beta
𝛽𝛽𝑖𝑖 =
𝐶𝐶𝐶𝐶𝐶𝐶(𝑅𝑅1 , 𝑅𝑅2 )
𝜎𝜎(𝑅𝑅1 )𝜎𝜎(𝑅𝑅2 )
𝐶𝐶𝐶𝐶𝐶𝐶(𝑅𝑅𝑖𝑖 ,𝑅𝑅𝑚𝑚 )
𝑉𝑉𝑉𝑉𝑉𝑉(𝑅𝑅𝑚𝑚 )
=
CAPM
Cov(Ri ,Rm )∗𝜎𝜎(𝑅𝑅𝑖𝑖 )
𝜎𝜎(𝑅𝑅𝑚𝑚 )
𝐸𝐸(𝑅𝑅𝑖𝑖 ) − 𝑅𝑅𝑓𝑓 = 𝛽𝛽𝑖𝑖 [𝐸𝐸(𝑅𝑅𝑚𝑚 ) − 𝑅𝑅𝑓𝑓 ]
Weighted average cost of capital
𝑟𝑟𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 = 𝑟𝑟𝐸𝐸 𝐸𝐸% + 𝑟𝑟𝑝𝑝𝑝𝑝𝑝𝑝 𝑃𝑃% + 𝑟𝑟𝐷𝐷 (1 − 𝑇𝑇𝐶𝐶 )𝐷𝐷%
Capital market without frictions (M&M theorem)
Prop I: 𝑉𝑉𝐿𝐿 = 𝑉𝑉𝑈𝑈
𝐷𝐷
Prop II: rE = 𝑟𝑟𝑈𝑈 + 𝐸𝐸 ∗ (𝑟𝑟𝑈𝑈 − 𝑟𝑟𝐷𝐷 )
Capital market with only corporate taxes (M&M theorem)
Prop I: 𝑉𝑉𝐿𝐿 = 𝑉𝑉𝑈𝑈 + 𝑃𝑃𝑃𝑃(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)
𝑃𝑃𝑃𝑃(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖) = 𝐷𝐷 ∗ 𝑇𝑇𝐶𝐶
𝐷𝐷
Prop II: rE = 𝑟𝑟𝑈𝑈 + 𝐸𝐸 ∗ (𝑟𝑟𝑈𝑈 − 𝑟𝑟𝐷𝐷 )(1 − 𝑇𝑇𝑐𝑐 )