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
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
