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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