Finance

Arbitrage
NPV = PV(Benefits) - PV(Costs)
No arbitrage should exist in competitive markets
Law of one price: The law of one price is the economic theory that the price of a given security, commodity or asset has the same price when exchange rates are taken into consideration.

Cash flows:
Future value of a cash flow: C * (1 + r) n
Present value of cash flow: C/ (1 + r) power n
Stream of cash flows: PV =  sum of the present value of individual cash flows: Cn/((1 + r) power n

Perpetuity: PV of perpetuity = C/r
Annuity: PV of annuity = C/r ( 1 - 1/(1+r) power n)
PV(Growing perpetuity) = C/ (r -g)
PV(Growing Annuity) = C/(r - g) * (1 -  (1 +g) power n)
                                                             _____________
       
                                                             (( 1 + r) power n)

IRR (internal rate of return) is the rate at which the NPV = 0

Session 5: Risk and Return

Expected return = E[R] = ΣR PR * R
PR is the probability of the return and R is the return

Variance(R) = ΣR PR * (R - E[R])²
 Risk = Standard Deviation = √Variance(R)

Since probability is not always known, calculated expected return and risk using historical data:
Return R_t = Dividend yield + Capital Gains Rate
Dividend yield = Dividend _t+1  / P _t
Capital Gains Rate = (P_t+1 - P_t )   /  P_t
Expected Return = E(R) =  arithmetic average return =1/T *  Σ R_t
Variance(R) = 1 / (T - 1) * Σ (Rt - E(R))²
 Risk = Standard Deviation = √Variance(R)

Individual stocks have two components of risk: firm specific idiosyncratic) and market wide(systematic)Firm specific risk cancel out when combined into an index or portfolio due to diversification
An efficient portfolio is one that contains only market wide risk
𝛃 measures the % change in an individual security for a 1% change in the return of an efficient portfolio.

Capital Asset Pricing Model(CAPM) equation

Expected Return E[R] = Risk Free Interest Rate + Risk Premium
E[R] = r_f + 𝛃 * (E(R_Mkt) - r_f)


Now we know how to calculate the expected return of a stock using CAPM, the following section covers the calculations for Expected return and risk of a Portfolio:

R_P = ΣX_iR_i;         X_i = Value of investment / total value of portfolio.
E[R_P] = ΣX_iE[R_i];   E[R_i] is calculated from CAPM equation above using risk free interest rate and market risk premium

Covariance and Correlation

Risk is eliminated through diversification  dependent on how much common risks securities face and how much their prices move together

Covariance(Ri,Rj) = E[(Ri - E[Rj]) * (Rj - E(Ri))]

Correlation standardizes covariance by dividing by product of respective volatilities

Correlation(Ri, Rj) = Covariance(Ri, Rj)/(SD(Ri) * SD(Rj))

Volatility Var(Rp)/Risk of a two security portfolio = x₁² Var(R₁) + x₂²*Var(R₂) + 2x₁x₂Cov(R₁R₂)

B_i^Mkt = SD(R_i) * Corr(R_i, R_mkt)  /  SD(R_mkt)
E[R_i] = r_f + B_i^Mkt * (E(R_Mkt) - r_f)

ɑ = Expected return - actual return
ɑs = E[Rs] - { r_f + Bs * (E(R_Mkt) - r_f) }