MAF 203 Business Finance APPENDIX 1 â FORMULA SHEET ...
MAF 203 Business Finance APPENDIX 1 – FORMULA SHEET DESCRIPTION Present value of an ordinary annuity (PVA)
Future value of an ordinary annuity (FVA)
PMT (or CF) from FVA Present value of a perpetuity Value of an annuity due Present value of a growing annuity Present value of a growing perpetuity Effective annual interest rate Total holding period return Expected return on an asset Variance of return on an asset Standard deviation of return on an asset Expected return for a portfolio
FORMULA
𝐶𝐶𝐶𝐶 1 × �1 − � (1 + 𝑖𝑖)𝑛𝑛 𝑖𝑖 1 1− (1 + 𝑖𝑖)𝑛𝑛 = 𝐶𝐶𝐶𝐶 × � � 𝑖𝑖 𝑃𝑃𝑃𝑃𝑃𝑃𝑛𝑛 =
= 𝐶𝐶𝐶𝐶 × 𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝐶𝐶𝐶𝐶 𝐹𝐹𝐹𝐹𝐹𝐹𝑛𝑛 = × [(1 + 𝑖𝑖)𝑛𝑛 − 1] 𝑖𝑖 (1 + 𝑖𝑖)𝑛𝑛 − 1 = 𝐶𝐶𝐶𝐶 × � � 𝑖𝑖 = 𝐶𝐶𝐶𝐶 × 𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 (1 + 𝑖𝑖)𝑛𝑛 − 1 𝑃𝑃𝑃𝑃𝑃𝑃 = 𝐹𝐹𝐹𝐹𝐹𝐹/ � � i 𝐶𝐶𝐶𝐶 𝑃𝑃𝑃𝑃𝑃𝑃∞ = 𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑑𝑑𝑑𝑑𝑑𝑑 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 × (1 + 𝑖𝑖) 𝐶𝐶𝐶𝐶1 1 + 𝑔𝑔 𝑛𝑛 𝑃𝑃𝑃𝑃𝑃𝑃𝑛𝑛 = × �1 − � � � 𝑖𝑖 − 𝑔𝑔 1 + 𝑖𝑖 𝑃𝑃𝑃𝑃𝑃𝑃∞ =
𝐶𝐶𝐶𝐶1 𝑖𝑖 − 𝑔𝑔
𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑚𝑚 𝐸𝐸𝐸𝐸𝐸𝐸 = �1 + � −1 𝑚𝑚 𝑃𝑃1 − 𝑃𝑃0 𝐶𝐶𝐶𝐶1 𝑅𝑅𝑇𝑇 = 𝑅𝑅𝐶𝐶𝐶𝐶 + 𝑅𝑅𝐼𝐼 = + 𝑃𝑃0 𝑃𝑃0 𝑛𝑛
𝐸𝐸(𝑅𝑅𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 ) = �(𝑝𝑝𝑖𝑖 × 𝑅𝑅𝑖𝑖 ) 𝑛𝑛
𝑖𝑖=1
𝑉𝑉𝑉𝑉𝑉𝑉(𝑅𝑅) = �{(𝑝𝑝𝑖𝑖 × [𝑅𝑅𝑖𝑖 − 𝐸𝐸(𝑅𝑅𝑖𝑖 )]2 } 𝑖𝑖=1
𝑛𝑛
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆(𝑅𝑅) = ��{(𝑝𝑝𝑖𝑖 × [𝑅𝑅𝑖𝑖 − 𝐸𝐸(𝑅𝑅𝑖𝑖 )]2 } 𝑖𝑖=1
𝑛𝑛
𝐸𝐸(𝑅𝑅𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ) = �[𝑥𝑥𝑖𝑖 × 𝐸𝐸(𝑅𝑅𝑖𝑖 )] 𝑖𝑖=1
Expected return and systematic risk (CAPM) Portfolio beta Price of a bond Price of a bond making multiple payments per year Price of a zero coupon bond The general dividend valuation model
𝐸𝐸(𝑅𝑅𝑖𝑖 ) = 𝑅𝑅𝑓𝑓 + 𝛽𝛽𝑖𝑖 [𝐸𝐸(𝑅𝑅𝑚𝑚 ) − 𝑅𝑅𝑓𝑓 ] 𝑛𝑛
𝛽𝛽𝑛𝑛 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = �(𝑥𝑥𝑖𝑖 × 𝛽𝛽𝑖𝑖 ) 𝑖𝑖=1
𝐶𝐶 1 𝐹𝐹𝑛𝑛 𝑃𝑃𝐵𝐵 = × �1 − �+ 𝑛𝑛 (1 + 𝑖𝑖) 𝑖𝑖 (1 + 𝑖𝑖)𝑛𝑛 𝑃𝑃𝐵𝐵 =
𝐶𝐶/𝑚𝑚 1 𝐹𝐹𝑚𝑚𝑚𝑚 × �1 − �+ 𝑚𝑚𝑚𝑚 (1 + 𝑖𝑖/𝑚𝑚) 𝑖𝑖/𝑚𝑚 (1 + 𝑖𝑖/𝑚𝑚)𝑚𝑚𝑚𝑚
𝐹𝐹𝑚𝑚𝑚𝑚 (1 + 𝑖𝑖/𝑚𝑚)𝑚𝑚𝑚𝑚 𝐷𝐷1 𝐷𝐷2 𝐷𝐷3 𝐷𝐷4 𝑃𝑃0 = + + + +⋯ 2 3 (1 + 𝑅𝑅) (1 + 𝑅𝑅) (1 + 𝑅𝑅) (1 + 𝑅𝑅)4 𝐷𝐷∞ + (1 + 𝑅𝑅)∞ 𝑃𝑃𝐵𝐵 =
∞
𝑃𝑃0 = �
Zero growth dividend model Value of a dividend at time t in a constant growth scenario Constant growth dividend model Value of a share at time t when dividends grow at a constant rate Supernormal growth share valuation model Value of a preference share with a fixed maturity Value of a preference share in perpetuity Net present value
𝑡𝑡=1
𝐷𝐷𝑡𝑡 (1 + 𝑅𝑅)𝑡𝑡
𝑃𝑃0 =
𝐷𝐷1 𝑅𝑅
𝑃𝑃0 =
𝐷𝐷1 𝑅𝑅 − 𝑔𝑔
𝐷𝐷𝑡𝑡 = 𝐷𝐷0 × (1 + 𝑔𝑔)𝑡𝑡
𝑃𝑃𝑡𝑡 = 𝑃𝑃0 =
𝐷𝐷𝑡𝑡+1 𝑅𝑅 − 𝑔𝑔
𝐷𝐷1 𝐷𝐷2 𝐷𝐷𝑡𝑡 𝑃𝑃𝑡𝑡 + + ⋯ + + (1 + 𝑅𝑅) (1 + 𝑅𝑅)2 (1 + 𝑅𝑅)𝑡𝑡 (1 + 𝑅𝑅)𝑡𝑡
𝑃𝑃𝑃𝑃0 = 𝑃𝑃𝑃𝑃0 =
𝐷𝐷/𝑚𝑚 1 𝑃𝑃𝑚𝑚𝑚𝑚 × �1 − �+ 𝑚𝑚𝑚𝑚 (1 + 𝑖𝑖/𝑚𝑚) 𝑖𝑖/𝑚𝑚 (1 + 𝑖𝑖/𝑚𝑚)𝑚𝑚𝑚𝑚 𝐷𝐷 𝑖𝑖
𝑁𝑁𝑁𝑁𝑁𝑁 = 𝑁𝑁𝑁𝑁𝑁𝑁0 +
𝑁𝑁𝑁𝑁𝑁𝑁1 𝑁𝑁𝑁𝑁𝑁𝑁2 𝑁𝑁𝑁𝑁𝑁𝑁𝑛𝑛 + +⋯+ 2 1 + 𝑘𝑘 (1 + 𝑘𝑘) (1 + 𝑘𝑘)𝑛𝑛
∞
𝑁𝑁𝑁𝑁𝑁𝑁 = � 𝑡𝑡=0
𝑁𝑁𝑁𝑁𝑁𝑁𝑡𝑡 (1 + 𝑘𝑘)𝑡𝑡
𝑃𝑃𝑃𝑃 = 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝐶𝐶𝐶𝐶𝐶𝐶ℎ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑡𝑡ℎ𝑒𝑒 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 Accounting rate of return 𝐴𝐴𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑛𝑛𝑛𝑛𝑛𝑛 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 Incremental free cash flow 𝐹𝐹𝐹𝐹𝐹𝐹 = [(𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 − 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 − 𝐷𝐷&𝐴𝐴) × (1 − 𝑡𝑡𝑐𝑐 )] + 𝐷𝐷&𝐴𝐴 − 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 − 𝐴𝐴𝐴𝐴𝐴𝐴 𝑊𝑊𝑊𝑊 calculation
Payback period
Inflation and real components of cost of capital (Fisher Effect) Incremental additions to working capital Net present value in perpetuity Equivalent Annual cost Degree of pre-tax cash flow operating leverage Degree of accounting operating leverage Pre-tax operating cash flow break-even point Crossover level of unit sales for EBITDA Accounting operating profit break-even point Crossover level of unit sales for EBIT Profitability index
After tax cost of debt CAPM formula for cost of ordinary shares Constant growth dividend formula for the cost of ordinary shares
1 + 𝑘𝑘 = (1 + ∆𝑃𝑃𝑒𝑒 ) × (1 + 𝑟𝑟)
Add WC = Change in cash & cash equivalents + Change in accounts receivable +Change in inventories – Change in accounts payable (1 + 𝑘𝑘)𝑛𝑛 𝑁𝑁𝑁𝑁𝑁𝑁∞ = 𝑁𝑁𝑁𝑁𝑁𝑁0 � � (1 + 𝑘𝑘)𝑛𝑛 − 1 (1 + 𝑘𝑘)𝑛𝑛 𝐸𝐸𝐸𝐸𝐸𝐸0 = 𝑘𝑘 × 𝑁𝑁𝑁𝑁𝑁𝑁0 � � (1 + 𝑘𝑘)𝑛𝑛 − 1 𝐹𝐹𝐹𝐹 𝐶𝐶𝐶𝐶𝐶𝐶ℎ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝐷𝐷𝐷𝐷𝐷𝐷 = 1 + 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐹𝐹𝐹𝐹 + 𝐷𝐷&𝐴𝐴 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐷𝐷𝐷𝐷𝐷𝐷 = 1 + 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐹𝐹𝐹𝐹 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑉𝑉𝑉𝑉 𝐹𝐹𝐹𝐹𝐴𝐴𝐴𝐴𝐴𝐴 1 − 𝐹𝐹𝐹𝐹𝐴𝐴𝐴𝐴𝐴𝐴2 = 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴 1 − 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴 1
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 𝐶𝐶𝐶𝐶𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
𝐹𝐹𝐹𝐹 + 𝐷𝐷&𝐴𝐴 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑉𝑉𝑉𝑉 (𝐹𝐹𝐹𝐹 + 𝐷𝐷&𝐴𝐴)𝐴𝐴𝐴𝐴𝐴𝐴 1 − (𝐹𝐹𝐹𝐹 + 𝐷𝐷&𝐴𝐴)𝐴𝐴𝐴𝐴𝐴𝐴 2 𝐶𝐶𝐶𝐶𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴 1 − 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴 1 𝑁𝑁𝑁𝑁𝑁𝑁 + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑃𝑃𝑃𝑃 = 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
𝑘𝑘𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎−𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑘𝑘𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝−𝑡𝑡𝑡𝑡𝑡𝑡 × (1 − 𝑡𝑡)
𝑘𝑘𝑜𝑜𝑜𝑜 = 𝑅𝑅𝑟𝑟𝑟𝑟 + (𝛽𝛽𝑜𝑜𝑜𝑜 × 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) 𝑘𝑘𝑜𝑜𝑜𝑜 =
𝐷𝐷1 + 𝑔𝑔 𝑃𝑃0
Perpetuity formula for the cost of preference shares Traditional WACC Days sales outstanding (DSO) Days sales inventory (DSI)
𝑘𝑘𝑝𝑝𝑝𝑝 =
𝐷𝐷𝑝𝑝𝑝𝑝 𝑃𝑃𝑝𝑝𝑝𝑝
𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 𝑥𝑥𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑘𝑘𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 (1 − 𝑡𝑡) + 𝑥𝑥𝑝𝑝𝑝𝑝 𝑘𝑘𝑝𝑝𝑝𝑝 + 𝑥𝑥𝑜𝑜𝑜𝑜 𝑘𝑘𝑜𝑜𝑜𝑜 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐷𝐷𝐷𝐷𝐷𝐷 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠/365 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐷𝐷𝐷𝐷𝐷𝐷 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑓𝑓 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠/365 Days payables outstanding 𝐷𝐷𝐷𝐷𝐷𝐷 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠/365 (DPO)
Operating cycle Cash conversion cycle Economic Order Quantity (EOQ) WACC for company with no tax and no preference shares Cost of ordinary shares – M&M proposition 2
Operating cycle =DSO + DSI Cash conversion cycle = DSO + DSI - DPO
2 × 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 × 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝐸𝐸𝐸𝐸𝐸𝐸 = � 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 𝑥𝑥𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑘𝑘𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 + 𝑥𝑥𝑜𝑜𝑜𝑜 𝑘𝑘𝑜𝑜𝑜𝑜
𝑉𝑉𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑘𝑘𝑜𝑜𝑜𝑜 = 𝑘𝑘𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠 + � � × (𝑘𝑘𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 − 𝑘𝑘𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 ) 𝑉𝑉𝑜𝑜𝑜𝑜
Future value of an ordinary annuity (FVA)
PMT (or CF) from FVA Present value of a perpetuity Value of an annuity due Present value of a growing annuity Present value of a growing perpetuity Effective annual interest rate Total holding period return Expected return on an asset Variance of return on an asset Standard deviation of return on an asset Expected return for a portfolio
FORMULA
𝐶𝐶𝐶𝐶 1 × �1 − � (1 + 𝑖𝑖)𝑛𝑛 𝑖𝑖 1 1− (1 + 𝑖𝑖)𝑛𝑛 = 𝐶𝐶𝐶𝐶 × � � 𝑖𝑖 𝑃𝑃𝑃𝑃𝑃𝑃𝑛𝑛 =
= 𝐶𝐶𝐶𝐶 × 𝑃𝑃𝑃𝑃 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝐶𝐶𝐶𝐶 𝐹𝐹𝐹𝐹𝐹𝐹𝑛𝑛 = × [(1 + 𝑖𝑖)𝑛𝑛 − 1] 𝑖𝑖 (1 + 𝑖𝑖)𝑛𝑛 − 1 = 𝐶𝐶𝐶𝐶 × � � 𝑖𝑖 = 𝐶𝐶𝐶𝐶 × 𝐹𝐹𝐹𝐹 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 (1 + 𝑖𝑖)𝑛𝑛 − 1 𝑃𝑃𝑃𝑃𝑃𝑃 = 𝐹𝐹𝐹𝐹𝐹𝐹/ � � i 𝐶𝐶𝐶𝐶 𝑃𝑃𝑃𝑃𝑃𝑃∞ = 𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑑𝑑𝑑𝑑𝑑𝑑 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 × (1 + 𝑖𝑖) 𝐶𝐶𝐶𝐶1 1 + 𝑔𝑔 𝑛𝑛 𝑃𝑃𝑃𝑃𝑃𝑃𝑛𝑛 = × �1 − � � � 𝑖𝑖 − 𝑔𝑔 1 + 𝑖𝑖 𝑃𝑃𝑃𝑃𝑃𝑃∞ =
𝐶𝐶𝐶𝐶1 𝑖𝑖 − 𝑔𝑔
𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑚𝑚 𝐸𝐸𝐸𝐸𝐸𝐸 = �1 + � −1 𝑚𝑚 𝑃𝑃1 − 𝑃𝑃0 𝐶𝐶𝐶𝐶1 𝑅𝑅𝑇𝑇 = 𝑅𝑅𝐶𝐶𝐶𝐶 + 𝑅𝑅𝐼𝐼 = + 𝑃𝑃0 𝑃𝑃0 𝑛𝑛
𝐸𝐸(𝑅𝑅𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 ) = �(𝑝𝑝𝑖𝑖 × 𝑅𝑅𝑖𝑖 ) 𝑛𝑛
𝑖𝑖=1
𝑉𝑉𝑉𝑉𝑉𝑉(𝑅𝑅) = �{(𝑝𝑝𝑖𝑖 × [𝑅𝑅𝑖𝑖 − 𝐸𝐸(𝑅𝑅𝑖𝑖 )]2 } 𝑖𝑖=1
𝑛𝑛
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆(𝑅𝑅) = ��{(𝑝𝑝𝑖𝑖 × [𝑅𝑅𝑖𝑖 − 𝐸𝐸(𝑅𝑅𝑖𝑖 )]2 } 𝑖𝑖=1
𝑛𝑛
𝐸𝐸(𝑅𝑅𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ) = �[𝑥𝑥𝑖𝑖 × 𝐸𝐸(𝑅𝑅𝑖𝑖 )] 𝑖𝑖=1
Expected return and systematic risk (CAPM) Portfolio beta Price of a bond Price of a bond making multiple payments per year Price of a zero coupon bond The general dividend valuation model
𝐸𝐸(𝑅𝑅𝑖𝑖 ) = 𝑅𝑅𝑓𝑓 + 𝛽𝛽𝑖𝑖 [𝐸𝐸(𝑅𝑅𝑚𝑚 ) − 𝑅𝑅𝑓𝑓 ] 𝑛𝑛
𝛽𝛽𝑛𝑛 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = �(𝑥𝑥𝑖𝑖 × 𝛽𝛽𝑖𝑖 ) 𝑖𝑖=1
𝐶𝐶 1 𝐹𝐹𝑛𝑛 𝑃𝑃𝐵𝐵 = × �1 − �+ 𝑛𝑛 (1 + 𝑖𝑖) 𝑖𝑖 (1 + 𝑖𝑖)𝑛𝑛 𝑃𝑃𝐵𝐵 =
𝐶𝐶/𝑚𝑚 1 𝐹𝐹𝑚𝑚𝑚𝑚 × �1 − �+ 𝑚𝑚𝑚𝑚 (1 + 𝑖𝑖/𝑚𝑚) 𝑖𝑖/𝑚𝑚 (1 + 𝑖𝑖/𝑚𝑚)𝑚𝑚𝑚𝑚
𝐹𝐹𝑚𝑚𝑚𝑚 (1 + 𝑖𝑖/𝑚𝑚)𝑚𝑚𝑚𝑚 𝐷𝐷1 𝐷𝐷2 𝐷𝐷3 𝐷𝐷4 𝑃𝑃0 = + + + +⋯ 2 3 (1 + 𝑅𝑅) (1 + 𝑅𝑅) (1 + 𝑅𝑅) (1 + 𝑅𝑅)4 𝐷𝐷∞ + (1 + 𝑅𝑅)∞ 𝑃𝑃𝐵𝐵 =
∞
𝑃𝑃0 = �
Zero growth dividend model Value of a dividend at time t in a constant growth scenario Constant growth dividend model Value of a share at time t when dividends grow at a constant rate Supernormal growth share valuation model Value of a preference share with a fixed maturity Value of a preference share in perpetuity Net present value
𝑡𝑡=1
𝐷𝐷𝑡𝑡 (1 + 𝑅𝑅)𝑡𝑡
𝑃𝑃0 =
𝐷𝐷1 𝑅𝑅
𝑃𝑃0 =
𝐷𝐷1 𝑅𝑅 − 𝑔𝑔
𝐷𝐷𝑡𝑡 = 𝐷𝐷0 × (1 + 𝑔𝑔)𝑡𝑡
𝑃𝑃𝑡𝑡 = 𝑃𝑃0 =
𝐷𝐷𝑡𝑡+1 𝑅𝑅 − 𝑔𝑔
𝐷𝐷1 𝐷𝐷2 𝐷𝐷𝑡𝑡 𝑃𝑃𝑡𝑡 + + ⋯ + + (1 + 𝑅𝑅) (1 + 𝑅𝑅)2 (1 + 𝑅𝑅)𝑡𝑡 (1 + 𝑅𝑅)𝑡𝑡
𝑃𝑃𝑃𝑃0 = 𝑃𝑃𝑃𝑃0 =
𝐷𝐷/𝑚𝑚 1 𝑃𝑃𝑚𝑚𝑚𝑚 × �1 − �+ 𝑚𝑚𝑚𝑚 (1 + 𝑖𝑖/𝑚𝑚) 𝑖𝑖/𝑚𝑚 (1 + 𝑖𝑖/𝑚𝑚)𝑚𝑚𝑚𝑚 𝐷𝐷 𝑖𝑖
𝑁𝑁𝑁𝑁𝑁𝑁 = 𝑁𝑁𝑁𝑁𝑁𝑁0 +
𝑁𝑁𝑁𝑁𝑁𝑁1 𝑁𝑁𝑁𝑁𝑁𝑁2 𝑁𝑁𝑁𝑁𝑁𝑁𝑛𝑛 + +⋯+ 2 1 + 𝑘𝑘 (1 + 𝑘𝑘) (1 + 𝑘𝑘)𝑛𝑛
∞
𝑁𝑁𝑁𝑁𝑁𝑁 = � 𝑡𝑡=0
𝑁𝑁𝑁𝑁𝑁𝑁𝑡𝑡 (1 + 𝑘𝑘)𝑡𝑡
𝑃𝑃𝑃𝑃 = 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝐶𝐶𝐶𝐶𝐶𝐶ℎ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑡𝑡ℎ𝑒𝑒 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 Accounting rate of return 𝐴𝐴𝐴𝐴𝐴𝐴 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑛𝑛𝑛𝑛𝑛𝑛 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 Incremental free cash flow 𝐹𝐹𝐹𝐹𝐹𝐹 = [(𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 − 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 − 𝐷𝐷&𝐴𝐴) × (1 − 𝑡𝑡𝑐𝑐 )] + 𝐷𝐷&𝐴𝐴 − 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 − 𝐴𝐴𝐴𝐴𝐴𝐴 𝑊𝑊𝑊𝑊 calculation
Payback period
Inflation and real components of cost of capital (Fisher Effect) Incremental additions to working capital Net present value in perpetuity Equivalent Annual cost Degree of pre-tax cash flow operating leverage Degree of accounting operating leverage Pre-tax operating cash flow break-even point Crossover level of unit sales for EBITDA Accounting operating profit break-even point Crossover level of unit sales for EBIT Profitability index
After tax cost of debt CAPM formula for cost of ordinary shares Constant growth dividend formula for the cost of ordinary shares
1 + 𝑘𝑘 = (1 + ∆𝑃𝑃𝑒𝑒 ) × (1 + 𝑟𝑟)
Add WC = Change in cash & cash equivalents + Change in accounts receivable +Change in inventories – Change in accounts payable (1 + 𝑘𝑘)𝑛𝑛 𝑁𝑁𝑁𝑁𝑁𝑁∞ = 𝑁𝑁𝑁𝑁𝑁𝑁0 � � (1 + 𝑘𝑘)𝑛𝑛 − 1 (1 + 𝑘𝑘)𝑛𝑛 𝐸𝐸𝐸𝐸𝐸𝐸0 = 𝑘𝑘 × 𝑁𝑁𝑁𝑁𝑁𝑁0 � � (1 + 𝑘𝑘)𝑛𝑛 − 1 𝐹𝐹𝐹𝐹 𝐶𝐶𝐶𝐶𝐶𝐶ℎ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝐷𝐷𝐷𝐷𝐷𝐷 = 1 + 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐹𝐹𝐹𝐹 + 𝐷𝐷&𝐴𝐴 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐷𝐷𝐷𝐷𝐷𝐷 = 1 + 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐹𝐹𝐹𝐹 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑉𝑉𝑉𝑉 𝐹𝐹𝐹𝐹𝐴𝐴𝐴𝐴𝐴𝐴 1 − 𝐹𝐹𝐹𝐹𝐴𝐴𝐴𝐴𝐴𝐴2 = 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴 1 − 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴 1
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 𝐶𝐶𝐶𝐶𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
𝐹𝐹𝐹𝐹 + 𝐷𝐷&𝐴𝐴 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑉𝑉𝑉𝑉 (𝐹𝐹𝐹𝐹 + 𝐷𝐷&𝐴𝐴)𝐴𝐴𝐴𝐴𝐴𝐴 1 − (𝐹𝐹𝐹𝐹 + 𝐷𝐷&𝐴𝐴)𝐴𝐴𝐴𝐴𝐴𝐴 2 𝐶𝐶𝐶𝐶𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴 1 − 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴 1 𝑁𝑁𝑁𝑁𝑁𝑁 + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑃𝑃𝑃𝑃 = 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
𝑘𝑘𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎−𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑘𝑘𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝−𝑡𝑡𝑡𝑡𝑡𝑡 × (1 − 𝑡𝑡)
𝑘𝑘𝑜𝑜𝑜𝑜 = 𝑅𝑅𝑟𝑟𝑟𝑟 + (𝛽𝛽𝑜𝑜𝑜𝑜 × 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) 𝑘𝑘𝑜𝑜𝑜𝑜 =
𝐷𝐷1 + 𝑔𝑔 𝑃𝑃0
Perpetuity formula for the cost of preference shares Traditional WACC Days sales outstanding (DSO) Days sales inventory (DSI)
𝑘𝑘𝑝𝑝𝑝𝑝 =
𝐷𝐷𝑝𝑝𝑝𝑝 𝑃𝑃𝑝𝑝𝑝𝑝
𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 𝑥𝑥𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑘𝑘𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 (1 − 𝑡𝑡) + 𝑥𝑥𝑝𝑝𝑝𝑝 𝑘𝑘𝑝𝑝𝑝𝑝 + 𝑥𝑥𝑜𝑜𝑜𝑜 𝑘𝑘𝑜𝑜𝑜𝑜 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐷𝐷𝐷𝐷𝐷𝐷 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠/365 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐷𝐷𝐷𝐷𝐷𝐷 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑓𝑓 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠/365 Days payables outstanding 𝐷𝐷𝐷𝐷𝐷𝐷 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠/365 (DPO)
Operating cycle Cash conversion cycle Economic Order Quantity (EOQ) WACC for company with no tax and no preference shares Cost of ordinary shares – M&M proposition 2
Operating cycle =DSO + DSI Cash conversion cycle = DSO + DSI - DPO
2 × 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 × 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝐸𝐸𝐸𝐸𝐸𝐸 = � 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 𝑥𝑥𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑘𝑘𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 + 𝑥𝑥𝑜𝑜𝑜𝑜 𝑘𝑘𝑜𝑜𝑜𝑜
𝑉𝑉𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑘𝑘𝑜𝑜𝑜𝑜 = 𝑘𝑘𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑠𝑠 + � � × (𝑘𝑘𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 − 𝑘𝑘𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 ) 𝑉𝑉𝑜𝑜𝑜𝑜
