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

|------------|------------|-----------

P0P1/D1Pt/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))ab

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) / AB

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...