CHAPTER The balance sheet identity or equation: Assets ϭ Liabilities [2.1] ϩ Shareholders’ equity The income statement equation: Revenues Ϫ Expenses ϭ Income [2.2] The cash flow identity: Cash flow from assets ϭ Cash flow to creditors ϩ [2.3] Cash flow to stockholders where a Cash flow from assets ϭ Operating cash flow (OCF) Ϫ Net capital spending Ϫ Change in net working capital (NWC) (1) Operating cash flow ϭ Earnings before interest and taxes (EBIT) ϩ Depreciation Ϫ Taxes (2) Net capital spending ϭ Ending net fixed assets Ϫ Beginning net fixed assets ϩ Depreciation 10 (3) Change in net working capital ϭ Ending NWC Ϫ Beginning NWC b Cash flow to creditors ϭ Interest paid Ϫ Net new borrowing c Cash flow to stockholders ϭ Dividends paid Ϫ Net new equity raised CHAPTER 11 12 The current ratio: Current assets Current ratio ϭ _ [3.1] Current liabilities The quick or acid-test ratio: 13 Current assets Ϫ Inventory Quick ratio ϭ [3.2] Current liabilities The cash ratio: Cash [3.3] 14 Cash ratio ϭ _ Current liabilities The ratio of net working capital to total assets: Net working capital to total assets Net working capital ϭ _ [3.4] Total assets The interval measure: Interval measure Current assets ϭ Average daily operating costs The total debt ratio: Total debt ratio Total assets Ϫ Total equity ϭ Total assets The debt-equity ratio: Debt-equity ratio ϭ Total debt͞Total equity The equity multiplier: Equity multiplier ϭ Total assets͞Total equity The long-term debt ratio: Long-term debt ratio Long-term debt ϭ _ Long-term debt ϩ Total equity The times interest earned (TIE) ratio: EBIT Times interest earned ratio ϭ _ Interest The cash coverage ratio: Cash coverage ratio EBIT ϩ Depreciation ϭ Interest The inventory turnover ratio: Inventory turnover Cost of goods sold ϭ Inventory The average days’ sales in inventory: Days’ sales in inventory 365 days ϭ Inventory turnover The receivables turnover ratio: Receivables turnover Sales ϭ _ Accounts receivable APPENDIX B KEY EQUATIONS [3.5] [3.6] [3.7] [3.8] [3.9] [3.10] [3.11] [3.12] [3.13] [3.14] B-1 ros3062x_App_B_Standard.indd 2/9/07 3:53:12 PM B-2 APPENDIX B Key Equations 15 The days’ sales in receivables: Days’ sales in receivables 365 days ϭ Receivables turnover 16 The net working capital (NWC) turnover ratio: Sales NWC turnover ϭ _ NWC 17 The fixed asset turnover ratio: Sales Fixed asset turnover ϭ _ Net fixed assets 18 The total asset turnover ratio: Sales Total asset turnover ϭ Total assets 19 Profit margin: Net income Profit margin ϭ Sales 20 Return on assets (ROA): Net income Return on assets ϭ Total assets 21 Return on equity (ROE): Net income Return on equity ϭ Total equity 22 The price-earnings (PE) ratio: Price per share PE ratio ϭ Earnings per share 23 The market-to-book ratio: Market-to-book ratio Market value per share ϭ _ Book value per share 24 The Du Pont identity: Sales ϫ Net income ϫ Assets ROE ϭ Assets Sales Equity The capital intensity ratio: [3.15] [3.16] [3.17] [3.18] [3.19] [3.20] [3.21] [3.22] [3.23] [3.24] ¥ ­ ¦ Return on assets ROE ϭ Profit margin ϫ Total asset turnover ϫ Equity multiplier CHAPTER The dividend payout ratio: Dividend payout ratio ϭ Cash dividends͞Net income The internal growth rate: ROA ϫ b Internal growth rate ϭ Ϫ ROA ϫ b The sustainable growth rate: ROE ϫ b Sustainable growth rate ϭ Ϫ ROE ϫ b ros3062x_App_B_Standard.indd Total assets Capital intensity ratio ϭ Sales ϭ Total asset turnover CHAPTER The future value of $1 invested for t periods at rate of r per period: Future value ϭ $1 ϫ (1 ϩ r)t The present value of $1 to be received t periods in the future at a discount rate of r: PV ϭ $1 ϫ [1͞(1 ϩ r)t] ϭ $1͞(1 ϩ r)t The relationship between future value and present value (the basic present value equation): PV ϫ (1 ϩ r)t ϭ FVt PV ϭ FVt ͞(1 ϩ r)t ϭ FVt ϫ [1͞(1 ϩ r)t] [5.1] [5.2] [5.3] CHAPTER The present value of an annuity of C dollars per period for t periods when the rate of return or interest rate is r: Annuity present value Ϫ Present value factor ϭ C ϫ r Ϫ [1͞(1 ϩ r)t] [6.1] ϭ C ϫ r ( { ) } The future value factor for an annuity: Annuity FV factor ϭ (Future value factor Ϫ 1)͞r ϭ [(1 ϩ r)t Ϫ 1]͞r Annuity due value ϭ Ordinary annuity value ϫ (1 ϩ r) [6.2] [6.3] Present value for a perpetuity: PV for a perpetuity ϭ C͞r ϭ C ϫ (1͞r) [6.4] Effective annual rate (EAR), where m is the number of times the interest is compounded during the year: EAR ϭ [1 ϩ (Quoted rate͞m)]m Ϫ [6.5] Effective annual rate (EAR), where q stands for the continuously compounded quoted rate: EAR ϭ e q Ϫ [6.6] CHAPTER [4.1] [4.2] [4.3] Bond value if bond has (1) a face value of F paid at maturity, (2) a coupon of C paid per period, (3) t periods to maturity, and (4) a yield of r per period: Bond value ϭ C ϫ [1 Ϫ 1͞(1 ϩ r)t]͞r ϩ F͞(1 ϩ r)t [7.1] Bond value Present value Present value ϭ ϩ of the coupons of the face amount 2/9/07 3:53:13 PM APPENDIX B The Fisher effect: ϩ R ϭ (1 ϩ r) ϫ (1 ϩ h) Rϭrϩhϩrϫh RϷrϩh [7.2] [7.3] [7.4] CHAPTER The dividend growth model: D0 ϫ (1 ϩ g) D1 P0 ϭ _ ϭ RϪg RϪg Required return: R ϭ D1͞P0 ϩ g [8.3] [8.5] Net present value (NPV): NPV ϭ Present value of future cash flows Ϫ Investment cost Payback period: Payback period ϭ Number of years that pass before the sum of an investment’s cash flows equals the cost of the investment Discounted payback period: Discounted payback period ϭ Number of years that pass before the sum of an investment’s discounted cash flows equals the cost of the investment The average accounting return (AAR): Average net income AAR ϭ _ Average book value Internal rate of return (IRR): IRR ϭ Discount rate of required return such that the net present value of an investment is zero Profitability index: PV of cash flows Profitability index ϭ Cost of investment CHAPTER 10 Bottom-up approach to operating cash flow (OCF): OCF ϭ Net income ϩ Depreciation [10.1] Top-down approach to operating cash flow (OCF): OCF ϭ Sales Ϫ Costs Ϫ Taxes [10.2] Tax shield approach to operating cash flow (OCF): OCF ϭ (Sales Ϫ Costs) ϫ (1 Ϫ T) ϩ Depreciation ϫ T [10.3] CHAPTER 11 ros3062x_App_B_Standard.indd 3 Cash break-even level: Q ϭ FC͞(P Ϫ v) Financial break-even level: Q ϭ (FC ϩ OCF*)͞(P Ϫ v) where OCF* ϭ Zero NPV cash flow Degree of operating leverage (DOL): DOL ϭ ϩ FC͞OCF B-3 [11.4] CHAPTER 12 Variance of returns, Var(R) or ␴ 2: CHAPTER Accounting break-even level: Q ϭ (FC ϩ D)͞(P Ϫ v) Relationship between operating cash flow (OCF) and sales volume: Q ϭ (FC ϩ OCF)͞(P Ϫ v) Key Equations [11.1] [(R Ϫ R¯ )2 ϩ · · · Var(R) ϭ _ TϪ1 ϩ (RT Ϫ R¯ )2] Standard deviation of returns, SD(R) or ␴: [12.3] SD(R) ϭ ͙ Var(R) CHAPTER 13 Risk premium: Risk premium ϭ Expected return – Risk-free rate Expected return on a portfolio: E(RP) ϭ x1 ϫ E(R1) ϩ x ϫ E(R2) ϩ · · · ϩ xn ϫ E(R n) The reward-to-risk ratio: E[Ri] Ϫ Rf Reward-to-risk ratio ϭ _ βi The capital asset pricing model (CAPM): E(Ri ) ϭ Rf ϩ [E(RM) Ϫ Rf ] ϫ βi [13.1] [13.2] [13.7] CHAPTER 14 Value of a call option at maturity: a C1 ϭ if (S1 Ϫ E) Յ b C1 ϭ S1 Ϫ E if (S1 Ϫ E) Ͼ Bounds on the value of a call option: a Upper bound: C0 Յ S0 b Lower bound: C0 Ն if S0 Ϫ E Ͻ C0 Ն S0 Ϫ E if S0 Ϫ E Ն S0 ϭ C0 ϩ E͞(1 ϩ R f ) C0 ϭ S0 Ϫ E͞(1 ϩ R f ) [14.1] [14.2] [14.3] [14.4] [14.5] [11.3] 2/9/07 3:53:14 PM B-4 APPENDIX B Key Equations Value of a call that is certain to finish in-the-money: Call option value ϭ Stock value Ϫ Present value of the exercise price C0 ϭ S0 Ϫ E͞(1 ϩ Rf )t [14.6] CHAPTER 15 Required return on equity, RE (dividend growth model): RE ϭ D1͞P0 ϩ g [15.1] Required return on equity, RE (CAPM): RE ϭ R f ϩ βE ϫ (RM Ϫ R f ) [15.2] Required return on preferred stock, RP: RP ϭ D͞P0 [15.3] The weighted average cost of capital (WACC): WACC ϭ (E͞V) ϫ RE ϩ (D͞V) ϫ RD ϫ (1 Ϫ TC) [15.6] Weighted average flotation cost, fA: E ϫ f ϩ Dϫf fA ϭ [15.8] E D V V CHAPTER 16 Rights offerings: a Number of new shares: Number of new shares Funds to be raised ϭ _ [16.1] Subscription price b Number of rights needed: Number of rights needed to buy a share of stock Old shares ϭ [16.2] New shares c Value of a right: Value of a right ϭ Rights-on price Ϫ Ex-rights price CHAPTER 19 The operating cycle: Operating cycle ϭ Inventory period ϩ Accounts receivable period The cash cycle: Cash cycle ϭ Operating cycle Ϫ Accounts payable period Float measurement: a Average daily float: Total float Average daily float ϭ _ Total days b Average daily float: Average daily float ϭ Average daily receipts ϫ Weighted average delay The Baumol-Allais-Tobin (BAT) model: a Opportunity costs: Opportunity costs ϭ (C͞2) ϫ R b Trading costs: Trading costs ϭ (T͞C) ϫ F c Total cost: Total cost ϭ Opportunity costs ϩ Trading costs d The optimal initial cash balance: C* ϭ ͙ළළළළළළළළළළ (2T ϫ F)͞R The Miller-Orr model: a The optimal cash balance: C* ϭ L ϩ (3͞4 ϫ F ϫ σ2͞R)1͞3 b The upper limit: U* ϭ ϫ C* Ϫ ϫ L CHAPTER 21 Modigliani-Miller Propositions (no taxes): The size of receivables: Accounts receivable ϭ Average daily sales ϫ ACP NPV of switching credit terms: a Present value of switching: PV ϭ [(P Ϫ v)(QЈ Ϫ Q)]͞R b Cost of switching: Cost of switching ϭ PQ ϩ v(QЈ Ϫ Q) c NPV of switching: NPV of switching ϭ Ϫ[PQ ϩ v(QЈ Ϫ Q)] ϩ (P Ϫ v) ϫ (QЈ Ϫ Q)͞R ros3062x_App_B_Standard.indd [17.1] [17.2] [17.3] [19.5] CHAPTER 20 CHAPTER 17 a Proposition I: VL ϭ VU b Proposition II: RE ϭ RA ϩ (RA Ϫ RD) ϫ (D͞E ) Modigliani-Miller propositions (with taxes): a Value of the interest tax shield: Value of the interest tax shield ϭ (TC ϫ RD ϫ D)͞RD ϭ TC ϫ D b Proposition I: VL ϭ VU ϩ TC ϫ D c Proposition II: RE ϭ RU ϩ (RU Ϫ RD) ϫ (D͞E ) ϫ (1 Ϫ TC) [19.4] [20.1] [20.2] [20A.1] [20A.2] [20A.3] [20A.4] [20A.5] [20A.6] [21.1] [21.4] [21.5] [21.6] [17.4] 2/9/07 3:53:15 PM APPENDIX B NPV of granting credit: a With no repeat business: NPV ϭ Ϫv ϩ (1 Ϫ ␲)P͞(1 ϩ R) b With repeat business: NPV ϭ Ϫv ϩ (1 Ϫ ␲)(P Ϫ v)͞R The economic order quantity (EOQ) model: a Total carrying costs: Total carrying costs ϭ Average inventory ϫ Carrying costs per unit ϭ (Q͞2) ϫ CC b Total restocking costs: Total restocking costs ϭ Fixed cost per order ϫ Number of orders ϭ F ϫ (T͞Q) c Total costs: Total costs ϭ Carrying costs ϩ Restocking costs ϭ (Q͞2) ϫ CC ϩ F ϫ (T͞Q) d The optimal order size Q*: Q* ϭ ros3062x_App_B_Standard.indd ළළළළළළළ 2T ϫ F CC ͙ _ Key Equations B-5 CHAPTER 22 [21.8] [21.9] [21.10] Purchasing power parity (PPP): E(St) ϭ S0 ϫ [1 ϩ (hFC Ϫ hUS)]t Interest rate parity (IRP): a Exact, single period: F1͞S0 ϭ (1 ϩ RFC)͞(1 ϩ RUS) b Approximate, multiperiod: Ft ϭ S0 ϫ [1 ϩ (RFC Ϫ RUS)]t Uncovered interest parity (UIP): E(St) ϭ S0 ϫ [1 ϩ (RFC Ϫ RUS)]t International Fisher effect (IFE): RUS Ϫ hUS ϭ RFC Ϫ hFC [22.3] [22.4] [22.7] [22.9] [22.10] [21.11] [21.12] [21.16] 2/9/07 3:53:16 PM ... Number of new shares: Number of new shares Funds to be raised ϭ _ [16.1] Subscription price b Number of rights needed: Number of rights needed to buy a share of stock Old shares ϭ ... [14.5] [11.3] 2/9/07 3:53:14 PM B- 4 APPENDIX B Key Equations Value of a call that is certain to finish in-the-money: Call option value ϭ Stock value Ϫ Present value of the exercise price C0 ϭ S0.. .B- 2 APPENDIX B Key Equations 15 The days’ sales in receivables: Days’ sales in receivables 365 days ϭ Receivables turnover 16 The net working capital