Corporate Finance Equations Sheet

Assignment 1
NPV: = -PF + FV /(1+r)
PV = FV/(1+r) or
PV = C1/1-r + C2/(1-r)2 + .. + CT/(1-r)T
Rate of return: R=(Vf-Vi)/Vf
Rate r compounded m times a year:
FV = C(1+r/m)mt
10% semiannually = 10.25% annually, Hence 10.25 is said to be the Effective Annual Yield (EAY)
1+EAY = (1+r/m)mt
Assignment 2
Perpetuity
The value of D received each year, forever: PV = D/r
Annuity
The value of D received each year for T years:
PV = (D/r)*[1 – 1/(1+r)T]
Growing Perpetuity
PV = D/(R-g)
R: the cost of capital, interest rate
G:growth
Growing Annuity
PV = (D/(r-g))*[1 – (1+g)T/(1+r)T]
Dividend & Stock Price
|------------|------------|-----------
P0 P1/D1 Pt/Dt
Rate of Return of Stock r = (D1 + P1)/P0 - 1
Given annual growth rate g1, g2, then g3 remained constant forever:
Di = Di-1*(1+r)
P2 = D3 /(R-g)
P0 = D1 /(1+r)+ (D2+P2)/(1+r)2
Variation: quarterly basis
Di = Di-1*(1+r)4
P2 = D3 /(R-g/4)
Variation2: Change of Growth
Find the R before Change:
P0 = D1 /(R+g)
Use it to find future P
P1 = D2 /(R+gnew)
Bond
Repayment = total bond*(face value + last payment price)
Efficient Market
-Stock prices fully reflect available information
-Competition among investors eliminates abnormal profit
Foundation of ME
Rationality: adjust their estimate of stock prices in a rational way
Independent deviation from rationality: # optimist = perssimistic
Arbitrage: 0 investment, no risk, but + reward
Different Type of Efficiency
Weak: Prices reflects all information in past prices and vol.
Semi-Strong: Prices reflect all publicly available information: historical price, published acc statement, info on annuals report
Strong : reflect all information, public and private(e.g insiders), it implies that anything pertinent to the stock and known to at least one investor is already incorporated in the price.
What neglects?
1.Investors can throw darts.
2.Prices are random and uncaused.
Expected Return
E(r total) = Wb E(rb) + Wa E (ra)
Variance
Var(A, B) = Wa2(Var(A)) + Wb2(Var(B)) + 2(Wa)(Wb)(Cov(A, B))
Or
Var(A, B) = Wa2(Var(A)) + Wb2(Var(B)) + 2(Wa)(Wb)(Corr(A, B))ab

Covariance & Correlation Stock A | Return(%) | Prop | Good | RA Good | P Good | Medium | RA Medium | P Medium | Bad | RA Bad | P Bad | Stock B | Return(%) | Prop | Good | RB Good | P Good | Medium | RB Medium | P Medium | Bad | RB Bad | P Bad |

Expected Return of Stock A =
P Good* RA Good +
P Medium*RA Medium +
P Medium*RA Medium = E(A)
Standard Deviation of Stock A =
(P Good*( RA Good- E(A))2 +
P Medium *( RA Medium - E(A))2 +
P Bad *( RA Bad - E(A))2 )1/2 = A

Covariance
Cov( A, B) =
P1(RA – E(A)) (RB – E(B))Good +
P2(RA – E(A)) (RD – E(B))Medium +
P3(RA – E(A)) (RD – E(B))Bad

Correlation
A,B = Cov( A, B) / AB

Total Risk = Systematic risk + unsystematic risk
Systematic: non-divertible market risk, affect a large number of assets
Unsystematic: affect a single asset, can be eliminated by combining

PortfolioOptimal Decision Rule
1. Use estimation of return, var, and convar to determine point M and CML (without personal)
2. Investors to decided how he will combine pt M with the riskless asset.

Step 1: Tangency = M
Step 2: Rational investors will passively hold an index fund
Step 3: Al efficient portfolios have same price of risk: CML slope

Capital Market Line x = standard deviation y = Expected return

beta (measure of a stock’s systematic risk only =>
cov(Ri, Rm) / var (Rm)

which is unitless!
< 1 = the asset has less systematic risk than the overall market

E[Rportfolio] = wE[RA] + (1-w) E[RB]
portfolio = w A + (1-w)B

Capital Asset Pricing Model
Assume no monopoly trading power
No tax, transaction cost, regulations
Investors tradeoff against stand_dev
All investor analyze the securities the same way

E[Ri] = RF + i *(E[RB] – RF)

Security Market Line (SML) x =  y = Expected return

CML:
- traces the efficient set of portfolio formed with both risky assets and riskless assets
- Examine efficient portfolio risk premium against appropriate risk measure
- with well diversified portfolio, the relevant measure of risk is total risk, we examine stand_dev

SML:
-Examine all assets risk premiums against appropriate risk measure
-Only beta risk is priced, it is appropriate for measuring all assets’ risk
- beta and stand_dev are equivalent for efficient portfolio

Assignment 3
Discounted Cash Flow

Project involve replacement of Machine A with B

FCF1->T-1 = (increase in sales – Cost + Depreciation(B-A))(1- Corporate Tax) + DepreciationB
FCFT = FCF1->T-1 + Salvage value
FCF0 = (-CostB+ MarketValueA-Gain*(Corporate Tax)
Gain = MarketValue - BookValue

IRR
Rate that makes the NPV = 0
- IRR ignore the scale project
- May result multiple solutions
- Borrowing and lending
Incremental IRR
Subtract the most expensive project cash flow and find the IRR
Payback Rule
PBP = 1/r*(1 – 1/(1+r)T)
# year it takes to recover the initial investment
-ignore time value of money
-ignore cash flow beyond the cutoff
-arbitrary cutting pt
-good if rough idea is needed, or while liquidating project
Cost of Capital(WACC)
-Find the weight of the stocks, bonds
- Cost of equity = common stock
Use formula rf + marketPremium
-Cost of debt(bonds) is YTM of the bond, times it to 1-Tc
- Cost of preferred stock = percent outstanding/market price
DDM
R = Div/P + g
P price per share of a stock
Div dividend per share = Dt-1 x (1+g)
-DDM appear to contain more error
Assignment 4
M&M Proposition I
VL = VU
-When there is no taxes and capital markets functions well, it makes no difference whether firm borrows or individual shareholder borrow
-firm value depends on assets, and is preserved regardless of the nature of the claims against it
With tax, VL = VU + TcD
Increase with leverage
M&M Proposition II rE = rA + (D/E)(rA-rD)
Leverage increases the risk and return to stockholders
With tax, rE = rA + (D/E)(rA-rD)(1-Tc)
Some of the increase in equity risk and return is offset by interest tax shield
Firm Leverage/Repurchasing
Expected Return(Before announcement)
Net Earnings = Earning/year – tax
Expected Return = Net Earnings /Equity(total value of stocks) = rA
Expected Return(After announcement)
VL = VU + TcD
Vu = market Value
Tc = Tax
D = Debt
Required Return on Equity/Debt
VL = D+ E
Look for either D or E rE = rA + (D/E)(rA-rD)(1-Tc) rD = bond rate
Stock Values
Market Value/# stock outstanding
Dividend and stock price
Current value of the firm (Vo) =
Net income + Expected Value/ (1+tax)
P0 = Current value of the firm (Vo)/ # outstanding shares
P1 = Expected Value/ (1+tax)/ # outstanding shares
P0-P1 = price of dividends
Current Share Holder Wealth
New Div Value +
[(Expected value of the firm +
New Shares values) / 1+r]
New shares value = Old div – new dividend
-repurchase is obvious incentive as the capital gain is taxed at more favorable rate