Finance II summary

Finance II

Lecture 1

Recap finance I

 Time value of money
CT
o PV 0 ( CT )=
( 1+ r )t
 Growing perpetuity (first payment next period)
C1
o V 0=
r −g
 Annuity (first payment next period, payments for t periods)
C1
o V 0= ¿
r
 Dividend discount model
¿1
o P=
r e −g
 Finance cares about market values, not book values
 Finance cares about cash, not book profits
 Investment decision rules (take positive NPV projects)

Lecture 2

Expected returns and volatility

 Higher risk means higher return
 Longer time means the risk can be evened out
 Asset returns:
o Realized returns = the return that actually occurs over a particular time period
¿t +1 + Pt +1 ¿t +1 P t+ 1−Pt
o Rt +1= = +
Pt Pt Pt
o Return = dividend yield + capital gain rate
 Asset returns random variables:
o Probability distributions
 When an investment is risky, it may earn different returns. Each possible
return has some likelihood of occurring. We can summarize this information
with a probability distribution, which assigns a profitability, Pr, that each
possible return, R, will occur
o Expected returns
 Calculated as a mean weighted average of the possible returns, where the
weights correspond to the probabilities
 Expected return = E[R] = ∑ Pr ¿ R
o Variance, volatility = stand dev. (risk)
 Variance = the expected squared deviation from the mean
Var(R) = E[(R – E[R])^2] = ∑ Pr ¿ ( R – E [ R ] )
2

 Standard deviation (volatility) = the square root of the variance
 SD(R) = √ Var (R)
 Volatility is mostly measured in % per annum

,  It is a measure of uncertainty about asset returns
 Scaling with different horizons
 σ T periods =σ 1 period∗ √T
 From daily to annually: σ annual=σ daily∗√ 252
 (T = 252, from 252 trading days a year, months = 12, weeks = 52)
o Higher moments (skewness, kurtosis)

Expected returns and volatility from historical data

 Average annual return
T
1 1
o R= ( R 1+ R 2+ …+ Rt )=
T T
∑ Rt
t=1
o Where Rt is the realized return of a security in year t, for the years 1 through T
 Variance estimate using realized returns
1
o Var ( R )=
T −1
∑ (Rt−R¿)2 ¿
o SD(R) = √ Var (R)
 Estimation error: using past returns to predict the future
o We can use a security’s historical average return to estimate its actual expected
return. However the average return is just an estimate of the expected return.
o Standard error = a statistical measure of the degree of estimation error
SD (R)
o SE=
√T
o 95% confidence interval is approximately: historical average return +- 1.96 *
standard error

Portfolios

 The expected return of a portfolio
o Portfolio weights = the fraction of the total investment in the portfolio held in each
individual investment in the portfolio
 The portfolio weights must add up to 1 or 100%
value of investment i
 x i=
total value of portfolio
o The return on the portfolio, Rp, is the weighted average of the returns on the
investments in the portfolio, where the weights correspond to portfolio weight
 R p =∑ xi Ri
o The expected return on the portfolio is the weighted average of the expected returns
of the investments within it
 E [ R¿¿ p]=∑ x i E [R¿¿ i ]¿ ¿

Diversification

 Diversification lowers risk in both direction; smaller losses, but also smaller gains

Covariance and correlation

 The volatility of a two-stock portfolio
o Diversification

, The amount of risk that is eliminated in a portfolio depends on the degree to
which the stocks face common risks and their prices move together 
covariance and correlation
 Determining covariance and correlation
o To find the risk of a portfolio, one must know the degree to which the assets’ returns
move together
o Covariance
 The expected product of the deviations of two returns from their expected
value
 Cov ( Ri , Rj )=E[ ( Ri−E [ Ri ] )( Rj−E [ RJ ] ) ]
 Historical covariance (estimate) between returns Ri and Rj
1
 Cov (Ri , Rj )=
T −1
∑ ( Ri−Ri ) ( Rj−R j)
 If the covariance is positive, the two returns tend to move together. If the
covariance is negative, the two returns tend to move in opposite directions
 Magnitude is however not easy to interpret
 Correlation
o A measure of the common risk shared by stocks that does not depend on their
volatility
Cov(Ri , Rj)
 Corr ( Ri , Rj )= pi , j=
SD ( Ri )∗SD( Rj)
 The correlation between two stocks will always be between -1 and +1
 It measures a linear relationship between Ri and Rj:
 If Ri changes by p%, we expect Rj changes by pi , j∗p %





 Volatility of a two-stock portfolio
o R p =∑ xi Ri
o Var(Rp) = Cov(Rp,Rp) = E [ ( Rp− E [ Rp ] ) ( Rp−E [ Rp ] ) ]

Lecture 3

Diversification

 The volatility of a large portfolio
o The variance of a portfolio is equal to the weighted average covariance of each stock
with the portfolio
o Var ( Rp )=∑ xiCov ( Ri , Rp )=∑ i ∑ j x i x j Cov ( Ri , Rj )
 Diversification with an equally weighted portfolio
o A portfolio in which the same amount is invested in each stock
o Variance of an equally weighted portfolio of n stocks
1
o Var ( Rp )= ∑ i ∑ j 2
Cov ( Ri , Rj )
n

, o Since there are n variance terms and n^2 – n covariance terms
o Var ( Rp )= 1/n (average variance of the individual stocks) + (1 – 1/n) (average
covariance between stocks)
 Diversification with general portfolios
o For a portfolio with arbitrary weights, the standard deviation is calculated as follows:
o Var ( Rp )= ∑ x i Cov( Ri , Rp)
o Var ( Rp )=∑ x i iSD ( Ri ) SD ( Rp ) Corr ( Ri , Rp)
o SD ( Rp ) =∑ xi SD ( Ri ) Corr ( Ri , Rp)
Cov ( Ri , Rp)
o Corr ( Ri , Rp ) =
SD ( Ri ) SD (Rp)
o Unless all of the stocks in a portfolio have a perfect positive correlation of 1 with one
another, the risk of the portfolio will be lower than the weighted average volatility of
the individual stocks

Efficient portfolio

 Efficient portfolios with two stocks
o Inefficient portfolio: it is possible to find another portfolio that is better in terms of
both expected return and volatility
o Efficient portfolio: there is no way to reduce volatility of the portfolio without
lowering the expected return




 Short sales and leverage
o Short position
 A negative investment in security