KEY EQUATIONS
Ross-Westerfield-Jordan: Fundamentals of Corporate Finance, Ninth Edition, Alternate
Back Matter
Appendix B: Key Equations
? The McGraw-Hill Companies, 2010
APPENDIX
KEY EQUATIONS
B
CHAPTER 2
1. The balance sheet identity or equation:
Assets Liabilities
Shareholders' equity
[2.1]
2. The income statement equation:
Revenues Expenses Income
[2.2]
3. The cash flow identity:
Cash flow from assets
Cash flow to creditors
[2.3]
Cash flow to stockholders
where
a. Cash flow from assets Operating cash flow (OCF) Net capital spending
Change in net working capital (NWC) (1) Operating cash flow Earnings
before interest and taxes (EBIT) Depreciation Taxes (2) Net capital spending Ending net fixed assets Beginning net fixed assets Depreciation (3) Change in net working capital Ending NWC Beginning NWC
b. Cash flow to creditors Interest paid
Net new borrowing
c. Cash flow to stockholders Dividends paid Net new equity raised
CHAPTER 3
1. The current ratio:
Current
ratio
__C_u__rr_e_n_t_a_s_s_e_ts__ Current liabilities
[3.1]
2. The quick or acid-test ratio: Quick ratio _C__u_rCr_eu_nr_tr_ean_st_se_lit_as_b_il_itI_ine_vs_e_n_to_r_y[3.2]
3. The cash ratio:
Cash
ratio
______C_a_s_h______ Current liabilities
[3.3]
4. The ratio of net working capital to total assets:
Net working capital to total assets
_N_e_t_w__o_rk__in_g__c_a_p_it_a_l Total assets
[3.4]
5. The interval measure:
Interval measure
_______C_u_r_r_en_t_a_s_s_e_ts_______ Average daily operating costs
[3.5]
6. The total debt ratio:
Total debt ratio
_T_o_ta_l_a_s_s_e_ts____T_o_ta_l_e_q_u_i_ty_ Total assets
[3.6]
7. The debt-equity ratio:
Debt-equity ratio
Total debtTotal equity
[3.7]
8. The equity multiplier:
Equity multiplier
Total assetsTotal equity
[3.8]
9. The long-term debt ratio:
Long-term debt ratio
_______L_o_n_g_-_te_r_m__d_e_b_t ______ Long-term debt Total equity
[3.9]
10. The times interest earned (TIE) ratio:
Times
interest
earned
ratio
_E__B_I_T__ Interest
[3.10]
11. The cash coverage ratio:
Cash coverage ratio
_E_B_I_T____D_e_p_r_e_c_ia_t_io_n_ Interest
12. The inventory turnover ratio:
[3.11]
Inventory turnover
_C_o_s_t_o_f_g_o_o_d__s_s_o_ld_ Inventory
13. The average days' sales in inventory:
[3.12]
Days' sales in inventory
_____3_6_5_d_a_y_s_____ Inventory turnover
[3.13]
B-1
Ross-Westerfield-Jordan: Fundamentals of Corporate Finance, Ninth Edition, Alternate
Back Matter
Appendix B: Key Equations
? The McGraw-Hill Companies, 2010
B-2
A P P E N D I X B Key Equations
14. The receivables turnover ratio:
Receivables turnover
_______S_a_le_s_______ Accounts receivable
15. The days' sales in receivables:
Days' sales in receivables
______3_6_5_d_a_y_s______ Receivables turnover
16. The net working capital (NWC) turnover ratio:
NWC
turnover
_S_a_le_s_ NWC
17. The fixed asset turnover ratio:
Fixed
asset
turnover
_____S_a_le_s_____ Net fixed assets
18. The total asset turnover ratio:
Total
asset
turnover
___S_a_l_e_s___ Total assets
19. Profit margin:
Profit
margin
_N_e_t_i_n_c_o_m_e_ Sales
20. Return on assets (ROA):
Return
on
assets
_N_e_t_i_n_c_o_m_e_ Total assets
21. Return on equity (ROE):
Return
on
equity
_N_e_t_i_n_c_o_m_e_ Total equity
22. The price-earnings (PE) ratio:
PE
ratio
__P_r_i_c_e_p_e_r_s_h_a_r_e__ Earnings per share
23. The market-to-book ratio:
Market-to-book ratio
_M_a_r_k_e_t_v_a_l_u_e_p_e_r_s_h_a_r_e Book value per share
24. The Du Pont identity:
ROE
_N_e_t_i_n_c_o_m_e_ Sales
_S_a_l_e_s_ Assets
_A_s_s_e_ts_ Equity
?
?
Return on assets
ROE Profit margin Total asset turnover Equity multiplier
CHAPTER 4
1. The dividend payout ratio:
Dividend payout ratio Cash dividendsNet income
[3.14] [3.15] [3.16] [3.17] [3.18] [3.19] [3.20] [3.21] [3.22] [3.23] [3.24]
[4.1]
2. The internal growth rate:
Internal
growth
rate
__R__O_A____b___ 1 ROA b
[4.2]
3. The sustainable growth rate:
Sustainable
growth
rate
__R__O_E____b___ 1 ROE b
[4.3]
4. The capital intensity ratio:
Capital
intensity
ratio
_T_o_ta_l_a_s_s_e_t_s Sales
________1________ Total asset turnover
CHAPTER 5
1. The future value of $1 invested for t periods at rate of r per period:
Future value $1 (1 r)t
[5.1]
2. The present value of $1 to be received t periods in the future at a discount rate of r:
PV $1 [1(1 r)t] $1(1 r)t
[5.2]
3. The relationship between future value and present value (the basic present value equation):
PV (1 r)t FV
t
PV FV (1 r)t FV [1(1 r)t]
t
t
[5.3]
CHAPTER 6
1. 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
( ) C
_1___P_r_e_s_e_n_t _v_a_lu_e__fa_c_t_o_r r
{ } C
_1___[_1__(_1____r_)t_] r
[6.1]
2. The future value factor for an annuity:
Annuity FV factor
(Future value factor 1)r
[6.2]
[(1 r)t 1]r
3. Annuity due value Ordinary annuity value
(1 r)
[6.3]
4. Present value for a perpetuity:
PV for a perpetuity Cr C (1r)
[6.4]
5. Growing annuity present value
( ) C _1___r__11____g___g_r__t
[6.5]
6. Growing perpetuity present value
_r_C__g_
[6.6]
Ross-Westerfield-Jordan: Fundamentals of Corporate Finance, Ninth Edition, Alternate
Back Matter
Appendix B: Key Equations
? The McGraw-Hill Companies, 2010
A P P E N D I X B Key Equations
B-3
7. Effective annual rate (EAR), where m is the number of times the interest is compounded during the year:
EAR [1 (Quoted ratem)]m 1
8. Effective annual rate (EAR), where q stands for the continuously compounded quoted rate:
EAR eq 1
CHAPTER 7
1. 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
C [1 1(1 r)t]r F(1 r)t
[7.1]
Bond value
Present value of the coupons
of
Present value the face amount
2. The Fisher effect:
1 R (1 r) (1 h)
[7.2]
Rrhrh
[7.3]
Rrh
[7.4]
CHAPTER 8
1. The dividend growth model:
P0
_D_0___(_1____g_) Rg
__D__1__ Rg
[8.3]
2. Required return:
R D1P0 g
[8.7]
CHAPTER 9
1. Net present value (NPV):
NPV Present value of future cash flows Investment cost
2. Payback period:
Payback period Number of years that pass before the sum of an investment's cash flows equals the cost of the investment
3. Discounted payback period:
Discounted payback period Number of years that pass before the sum of an investment's discounted cash flows equals the cost of the investment
4. The average accounting return (AAR):
AAR
_A_v_e_r_a_g_e_n_e_t_i_n_c_o_m_e_ Average book value
5. Internal rate of return (IRR):
IRR Discount rate of required return such that the net present value of an investment is zero
6. Profitability index:
Profitability
index
_P__V__o_f _c_a_s_h_fl_o_w__s_ Cost of investment
CHAPTER 10
1. Bottom-up approach to operating cash flow (OCF):
OCF Net income Depreciation
[10.1]
2. Top-down approach to operating cash flow (OCF):
OCF Sales Costs Taxes
[10.2]
3. Tax shield approach to operating cash flow (OCF):
OCF (Sales Costs) (1 T) Depreciation T
[10.3]
CHAPTER 11
1. Accounting break-even level: Q (FC D)(P v)
2. Relationship between operating cash flow (OCF) and sales volume: Q (FC OCF)(P v)
3. Cash break-even level: Q FC(P v)
4. Financial break-even level: Q (FC OCF*)(P v)
where OCF* Zero NPV cash flow
5. Degree of operating leverage (DOL): DOL 1 FCOCF
[11.1] [11.3]
[11.4]
CHAPTER 12
1. Variance of returns, Var(R) or 2:
Var(R)
_T_1__1_[(R_1_
__
R
)2
?
?
?
(R R )2] T
2. Standard deviation of returns, SD(R) or :
SD(R) Var(R)
CHAPTER 13
1. Risk premium:
Risk premium Expected return ? Risk-free rate
2. Expected return on a portfolio:
E(RP) x1 E(R1) x2 E(R2) ? ? ? xn E(Rn)
[12.3]
[13.1] [13.2]
Ross-Westerfield-Jordan: Fundamentals of Corporate Finance, Ninth Edition, Alternate
Back Matter
Appendix B: Key Equations
? The McGraw-Hill Companies, 2010
B-4
A P P E N D I X B Key Equations
3. The reward-to-risk ratio:
E[R ] R
Reward-to-risk
ratio
____i _____f
i
4. The capital asset pricing model (CAPM):
E(Ri) Rf [E(RM) Rf] i
[13.7]
CHAPTER 14
1.
Required
return
on
equity,
R E
(dividend
growth
model):
RE D1P0 g
[14.1]
2. Required return on equity, RE (CAPM):
R R (R R )
E
f
E
M
f
[14.2]
3. Required return on preferred stock, R : P
R DP
P
0
[14.3]
4. The weighted average cost of capital (WACC):
WACC (EV) RE (DV) RD (1 TC)
5. Weighted average flotation cost, f : A
f
A
_E_ V
f
E
_D_ V
f
D
[14.6] [14.8]
CHAPTER 15
1. Rights offerings:
a. Number of new shares:
Number of new shares
_F_u_n_d_s_t_o_b_e__ra_i_s_ed_ Subscription price
b. Number of rights needed:
[15.1]
Number of rights needed to buy a share of stock
_O__ld__s_h_a_re_s_ New shares
[15.2]
c. Value of a right: Value of a right Rights-on price Ex-rights
price
CHAPTER 16
1. Modigliani-Miller propositions (no taxes):
a. Proposition I:
VL VU b. Proposition II:
R R (R R ) (DE )
E
A
A
D
2. Modigliani-Miller propositions (with taxes):
a. Value of the interest tax shield:
Present value of the interest tax shield
(T D R )R
C
DD
T D
C
[16.1] [16.2]
b. Proposition I:
V V T D
L
U
C
c. Proposition II:
R R (R R ) (DE )
E
U
U
D
(1 T )
C
CHAPTER 18
1. The operating cycle:
Operating cycle Inventory period Accounts receivable period
2. The cash cycle:
Cash cycle Operating cycle Accounts payable period
CHAPTER 19
1. Float measurement:
a. Average daily float:
Average
daily
float
_T_o_t_al__fl_o_a_t Total days
b. Average daily float:
Average daily float Average daily receipts Weighted average delay
2. The Baumol-Allais-Tobin (BAT) model:
a. Opportunity costs:
Opportunity costs (C2) R
b. Trading costs:
Trading costs (TC) F
c. Total cost:
Total cost Opportunity costs Trading costs
d. The optimal initial cash balance:
C* (2TF)R
3. The Miller-Orr model:
a. The optimal cash balance:
C* L (34 F 2R)13
b. The upper limit:
U* 3 C* 2 L
CHAPTER 20
1. The size of receivables:
Accounts receivable Average daily sales ACP
[16.3] [16.4]
[18.4] [18.5]
[19.1] [19.2] [19A.1] [19A.2] [19A.3] [19A.4] [19A.5] [19A.6]
[20.1]
Ross-Westerfield-Jordan: Fundamentals of Corporate Finance, Ninth Edition, Alternate
Back Matter
Appendix B: Key Equations
? The McGraw-Hill Companies, 2010
A P P E N D I X B Key Equations
B-5
2. NPV of switching credit terms: a. Present value of switching: PV [(P v)(Q Q)]R
b. Cost of switching: Cost of switching PQ v(Q Q)
c. NPV of switching: NPV of switching [PQ v(Q Q)] [(P v) (Q Q)]R
[20.4] [20.5] [20.6]
3. NPV of granting credit:
a. With no repeat business:
NPV v (1 )P(1 R)
b. With repeat business:
NPV v (1 )(P v)R
4. The economic order quantity (EOQ) model:
a. Total carrying costs:
Total carrying costs Average inventory
Carrying costs per unit (Q2) CC
b. Total restocking costs:
Total restocking costs Fixed cost per order
Number of orders F (TQ)
c. Total costs:
Total costs Carrying costs Restocking costs
(Q2) CC F (TQ)
d. The optimal order size Q*:
[20.8] [20.9]
[20.10] [20.11] [20.12]
Q* _2_T____F_ CC
[20.15]
CHAPTER 21
1. Purchasing power parity (PPP):
E(St) S0 [1 (hFC hUS)]t 2. Interest rate parity (IRP):
a. Exact, single period:
F1S0 (1 RFC)(1 RUS) b. Approximate, multiperiod:
F t
S0
[1
(R FC
R )]t US
[21.3] [21.4] [21.7]
3. Uncovered interest parity (UIP):
E(S ) S [1 (R R )]t
t
0
FC
US
4. International Fisher effect (IFE):
R h R h
US
US
FC
FC
[21.9] [21.10]
CHAPTER 24
1. Value of a call option at maturity:
a. C1 0 if (S1 E) 0
b.
C 1
S 1
E
if
(S1
E)
0
2. Bounds on the value of a call option:
[24.1] [24.2]
a. Upper bound:
C S
0
0
[24.3]
b. Lower bound:
C0 0 if S0 E 0 C0 S0 E if S0 E 0
[24.4]
3. S C E(1 R )
0
0
f
C0 S0 E(1 Rf)
[24.5]
4. Value of a call that is certain to finish in-the-money:
Call option value
Stock value
Present value of the exercise price
C0
S0
E(1
R f
)t
[24.6]
CHAPTER 25
1. Put-call parity condition:
S P PV(E) C
2. The Black-Scholes call option formula:
C S N(d1) E eRt N(d2) where
d 1
[ln(SE)
(R
22)
t](
t
)
d 2
d 1
t
3. Value of a risk-free bond:
Value of risky bond Put option
[25.2] [25.5] [25.6]
[25.7]
CHAPTER 26
4. The NPV of a merger:
NPV
V
* B
Cost
to
Firm
A
of
the acquisition
[26.1]
................
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