Overview of Relevant Formulas Corporate Finance (B40.2302) _________________________________________________________________________________________ 1. Present value of $1 to be received after t years at discount rate r: 2. Present value of annuity of $1 per year for t years at discount rate r: $1 (1 + r )t

⎡1 − (1 + r ) − t ⎤ ⎢ ⎥ × $1 r ⎣ ⎦ 1 ⎡ (1 + g )t ⎤ 3. Present value of growing annuity of $1 at rate g per year at discount rate r: ⎢1 − ⎥ × $1 r − g ⎣ (1 + r )t ⎦ $1 r

4. Present value of perpetuity of $1 per year at discount rate r:

5. Present value of perpetuity of $1 with constant growth rate g at discount rate r:

$1 r−g

6. Measures of risk for individual financial asset i:

Variance of returns:
Standard deviation of returns: Covariance of returns assets i and j:

Var (ri ) = σ i2 = Expected value of [ri − E (ri )]2

σ i = σ i2
Cov(ri , rj ) = σ ij = Exp. value of [ri − E (ri )][rj − E (rj )]

Correlation between returns i and j: Expected portfolio return (N assets):

ρij =

σ ij σ iσ j
N i =1 N

E (rp ) = ∑ wi ri (weights wi)

Portfolio variance (N assets): 7. Beta of financial asset i:

σ = ∑∑ wi w jσ ij (weights wi, wj)
2 p i =1 j =1

N

βi =

Cov(ri , rm ) σ im = 2 Var (rm ) σm

8. Capital Asset Pricing Model → Expected return financial asset i: r = rD ×

E ( ri ) = rf + [ E (rm ) − rf ] × β i

9. Weighted Average Cost of Capital (WACC):

D E × (1 − T ) + rE × V V

D⎤ D ⎡ 10. Cost of capital levered firm: r = rA ⎢1 − T × ⎥ (scenario 1) or r = rA − T × rD × (scenario 2) V V⎦ ⎣ D D 11. Cost of equity levered firm: rE = rA + [rA − rD ] × × (1 − T ) (scenario 1) or rE = rA + [rA − rD ] × (scenario 2) E E D D 12. Equity beta levered firm: β E = β A + [ β A − β D ] × × (1 − T ) (scenario 1) or β E = β A + [ β A − β D ] × (scenario 2) E E 13. Asset beta levered firm: β A = (1 − T ) D E D E × βD + × β E (scenario 1) or β A = × β D + × β E (scenario 2) V V E + (1 − T ) D E + (1 − T ) D

...1、Formulas：
(1) Sustainable growth rate=retention ratio*ROE=PRAT (P: Profit margin, R:
retention ratio=retained E/E; A: Asset turnover ratio=sales/asset; T:
Asset/beginning of the period equity)
(2) Compounding an investment m times a year for T years provides for future
value of wealth: FV=C0*(1+r/m)^(m*T)
(3) Future value of an investment compounded continuously over many
periods: FV=C0*e^(r*T) e=2.718
(4) Perpetuity: PV=C/r; Growing Perpetuity: PV=C/(r-g) g=growth rate;
Annuity: PV=C/r*[1-1/(1+r)^T];
Growing Annuity: PV=C/(r-g) *
{1-[(1+g)/(1+r)]^T}
(5) Bonds: Level-coupon bonds(定息债券): PV= C/r*[1-1/(1+r)^T]+F/(1+r)^T
C=Coupon payment per period, F=Face value, T=Time to maturity. Pure
Discount Bonds(零息债券): PV=F/(1+r)^T
(6) Stocks: Zero growth: P0=Div/r
Constant growth: : P0=Div/(r-g);
R=Dividend yield +capital gain yield = Div/P0 + g
(7) Price Earnings Ratio(P/E ratio)=Price per share/earnings per share
(8) NPV=Total PV of future CF’s + Initial investment, initial investment should
be negative number
(9)CFs=OCF+Net capital spending+change in NWC:
Operating cash flow=EBIT-Taxes+Depreciation; Net capital spending (don’t
forget salvage value); NWC(don’t forget NWC return)
(10) Average Return=(R1+R2+….RT)/T; Standard deviation={[(R1-average
R)^2+(R2-average R)^2+…. (RT-average R)^2]/(T-1)}^(1/2) 注意权值不同时
的算法
(11) Rate of return on portfolio: rp=WbRb +WsRs; variance of the rate of
return
on
two
risky
assets
portfolio:
σ2 = (WB σB...

...rates of earnings or g = (1-payout)(return on equity)
-Firm has target capital structure, defined as mix of debt, preferred stock, and common equity that minimizes WACC
-Project Stand-alone Risk: risk the project would have if it were the firm’s only asset
-Corporate or within-firm risk reflects effect of project on firm’s risk, measured by project’s effect on firm’s earnings variability
-Market or Beta risk reflects effects of project on stockholders’ risk, measured by project’s effect on firm’s beta coefficient
-Risk-adjusted cost of capital is cost of capital appropriate for given project, given its risk. Greater the risk, higher the cost of capital
Chapter 11
-Corporate assets: include operating, financial, and non-operating
-Operating Assets: assets in place and growth options
-Non-operating assets include financial assets such as investments in marketable securities and non-controlling interests in stock of other companies
-Value of operations is present value of all future free cash flows when discounted at WACC
-Horizon Value: value of operations at end of explicit forecast period (terminal value), equal to present value of all free cash flows beyond forecast period, discounted at WACC
-Corporate valuation model: value of company = value of operations + value of non operating assets
-Intrinsic value of equity: total value of company minus value of debt + preferred stock; intrinsic price per share is value of...

...Advanced CorporateFinance I SS 2012
Problem Set 1 Valuing Cash Flows
Problem Set 1
Valuing Cash Flows
Exercise 1 (Ex. 11.2 - 11.6 GT): Assume that Marriott’s restaurant division has the following joint distribution with the market return: Market Scenario Bad Good Great .25 .50 .25 Probability Market Return (%) -15 5 25 YR 1. Cash Flow Forecast $40 million $50 million $60 million
Assume also that the CAPM holds. 11.2 Compute the expected year 1 restaurant cash ﬂow for Marriott. 11.3 Find the covariance of the cash ﬂow with the market return and its cash ﬂow beta. 11.4 Assuming that historical data suggests that the market risk premium is 8.4 percent per year and the market standard deviation is 40 percent per year, ﬁnd the certainty equivalent of the year 1 cash ﬂow. What are the advantages and disadvantages of using such historical data for market inputs as opposed to inputs from a set of scenarios, like those given in the table above exercise 11.2? 11.5 Discount your answer in exercise 11.4 at a risk-free rate of 4 percent per year to obtain the present value. 11.6 Explain why the answer to exercise 11.5 diﬀers from the answer in Example 11.2.
1
Advanced CorporateFinance I SS 2012
Problem Set 1 Valuing Cash Flows
Exercise 2 (Ex. 13.1 - 13.7 GT):) Exercises 13.1 - 13.7 make use of the following data: In 1985, General Motors (GM) was evaluating the acquisition of Hughes Aircraft Corporation....

...of capital:
A. will decrease as the risk level of the firm increases.
B. for a specific project is primarily dependent upon the source of the funds used for the project.
C. is independent of the firm's capital structure.
D. should be applied as the discount rate for any project considered by the firm.
E. depends upon how the funds raised are going to be spent.
6. The weighted average cost of capital for a wholesaler:
A. is equivalent to the aftertax cost of the firm's liabilities.
B. should be used as the required return when analyzing a potential acquisition of a retail outlet.
C. is the return investors require on the total assets of the firm.
D. remains constant when the debt-equity ratio changes.
E. is unaffected by changes in corporate tax rates.
7. Which one of the following is the primary determinant of a firm's cost of capital?
A. debt-equity ratio
B. applicable tax rate
C. cost of equity
D. cost of debt
E. use of the funds
8. Scholastic Toys is considering developing and distributing a new board game for children. The project is similar in risk to the firm's current operations. The firm maintains a debt-equity ratio of 0.40 and retains all profits to fund the firm's rapid growth. How should the firm determine its cost of equity?
A. by adding the market risk premium to the aftertax cost of debt
B. by multiplying the market risk premium by (1 - 0.40)
C. by using the dividend growth model
D. by using the capital asset pricing model
E. by...

...of expected future cash flow (discount rate).Net Present Value- Value CFs using project discount rate based on risk Investment Decision-which real assets the firm should acquire.Choose positive and greatest NPV.value through CF Financing Decision- how to raise money needed for a firm’s investments in real assets. Choose capital structure to minimize cost of capital, maximize value of the firm. value through the cost of capital
Valuation adjustments- Time, Risk, Inflation, LiquidityTruncated cash flows: (Time) receive $CFt each period until time T. Constant discount rate 10%. Investment of $100 in time 0. CFs of $22 in t=1 and $121 in t=2
Annuity: receive $CF each period until time N Perpetuity: receive $CF each period forever
Gordon Formula- (perpetuity) for valuing a firm with growing dividends
π =risk premium.the risk premium is everything above the risk free rate, r+π = Risk Adjusted Discount Rate (RADR)
Nominal rate - Actual rate of return( using actual dollars) rnomial=real+I, Real rate - Rate adjusted for inflation(using constant dollars), i.e. the return in today’s dollars.
Balance sheet (a given point of time)and income statement: Firm’s two main financial constructs Income statement-operating performance of the firm over a given time period
Cash flow statement- derived from income statement and changes in balance sheet FCF-Free Cash Flow- is cash available to distribute to stockholders and bondholders. Profits already have payments to...

...(investment’s sensitivity to market). (0 = riskless, 1 = as risky as entire market)
* βA = D/(D+E) βD + E/(D+E) βE, where D & E are Market Values of Debt & Equity
* E = Price/Share * # Shares
* D = Market Value of outstanding Debt
* βE = βA + D/E (βA – βD)
* If firm is unlevered (D=0), βE = βA. If little chance of default then βD = 0 and βA = βE E/(D+E)
* Use the risk measure associated with the project you are investing in – not necessarily the risk measure for your firm
* VL = VU + PV (Tax Benefits) + Corporate Benefits (Debt) – Costs of Financial Distress
* VL is the value of the firm with leverage, VU is the value of the all-equity firm
* 2 methods used to incorporate tax benefits to previous valuation techniques: APV & WACC
* APV (Adjusted Present Value – adjusts for tax by increasing the cash flows due to the tax benefit
* Increase each annual CF by Debt capacity * Debt Rate * Corporate Tax Rate = ITS (Interest Tax Shield)
* Step: Plug into CAPM to get Discount Rate Ra = Treasury + 6*Asset Beta if current
* WACC (Weighted Average Cost of Capital) – adjusts for taxes by decreasing the discount rate
* WACC = r* = rd (1-tc) [D/(D+E)] + rE [E/(D+E)] done on a Market Value basis (D can be book value but not E)
* Steps: (1) Relever Asset Beta to Equity Beta: Asset Beta / Proposed Equity Ratio, (2) Plug into CAPM to get Equity Discount Rate, (3) WACC =...

...e TCH321 – CORPORATEFINANCE MOCK EXAM Time: 1 hour 30 minutes The exam lasts 1 hour and 30 minutes and consists of 5 questions. Approved calculators are permitted. You are not allowed to use Excel. This is a closed book exam. You are NOT permitted to access any other material in either written or electronic form. All numerical answers should be reported to TWO decimal places. To ensure the accuracy of your answer, you should perform all intermediate calculations to at least THREE decimal places. Choose FIVE questions. DO show your working. Question 1. (20 marks) Suppose that you have the following information about a company Credit rating Beta Tax expense Pre-tax income Preferred dividend rate Preferred stock par value Preferred stock price Preferred stock outstanding Common stock price Common stock par value Common stock outstanding Expected next common stock dividend Long term bond yield-to-maturity Enterprise value Market risk premium 30 year Treasury bond yield-to-maturity AA 0.95 14,325,000 113,895,000 5.25% $100.00 $101.25 13,000,000 $53.29 $25.00 50,000,000 $1.95 7.55% 4,945,795,000 6.00% 4.75%
1
a. What is the estimated cost of common equity for the company? [4 marks]
b. What is the estimated after-tax cost of debt for the company? [4 marks]
c. What is the estimated cost of preferred equity for the company? [4 marks]
d. What is the estimated WACC of the company? [4 marks]
e. What is the implied...

...will be out of fashion and no longer in demand. The machines will be depreciated on a straightline basis over ﬁve years, and after ﬁve years will be sold at an estimated 20 million Euros. The company estimates that the EBITDA from the sale of purple trousers will be 12 million Euros per year for the coming 5 years. The company’s earnings are subject to a corporate tax rate of 40%. If the ﬁrm’s equity cost of capital is 9.6% what is the NPV of this project? (a) 0.48 million Euros (b) 0.72million Euros (c) 0.26 million Euros (d) 0.92 million Euros Instead of selling the machines after ﬁve years, the company can use them to produce grey trousers starting in year 6. If they do so, using these machines the company will generate free cash ﬂows of 2 million Euros per year in perpetuity, since grey trousers are classics and never go out of fashion. What is the NPV of the project if the company chooses this option? (a) 5.89 million Euros (b) 5.72 million Euros (c) 6.36 million Euros (d) 6.07 million Euros Suppose that the company has decided that they will use the machines to produce grey trousers after ﬁve years. The company can ﬁnance the purchase of new sewing machines entirely by debt by issuing 5-year bonds with 6% coupon rate sold at par. Assuming this additional borrowing is project-speciﬁc and hence will not alter the company’s capital structure, what is the value of the project with the tax shield? (a) 11.56 (b) 11.94 (c) 12.25 (d) 11.12
3...