Lecture 1, Recap finance 1
The net present value NPV (time value of money):
Ct
PV 0 ( C t )=
( 1+ r )t
Growing perpetuity (continues forever) (first payment next period):
C1
PV 0=
r−g
Annuity (fixed term) (first payment next period, payments for t periods):
C1 1
PV 0= [1− ]
r (1+r )t
Growing annuity:
( )
C1 1+ g t
PV 0= [1− ]
r−g (1+r )
Lecture 2, Asset returns and volatility portfolio’s
2.1 Expected returns and volatility
Realized returns are the returns that actually occur over a particular time period.
Return = dividend yield + capital gain rate. In formula:
¿t +1 + Pt +1 ¿ P −Pt
Rt +1= −1= t +1 + t +1
Pt Pt Pt
The expected (mean) return is calculated as a weighted average of the possible returns,
where the weights correspond to the probabilities.
Expected return=E [ R ] =∑ P R ∙ R
R
The variance is the expected squared deviation from the mean.
Var ( R )=E ¿
The standard deviation is the root of the variance.
SD ( R )=√ Var (R)
,In finance, the standard deviation (σ ) of a return is also referred to as its volatility. The
standard deviation is easier to interpret because it’s in the same unit as the return itself.
Volatility is mostly measured in % per annum. It’s a measure of uncertainty about asset
returns.
Scaling of the volatility with different periods can be done by:
σ T periods=σ 1 period ∙ √ T for example, σ T annual=σ monthly ∙ √ 12
σ annual
and σ Tmonthly =
√12
2.2 Historic returns and volatility
When estimating the expected return, we use historical data to predict the future by taking
the average return.
T
1 1
R= ( R1 + R2 +…+ R T ) = ∑ Rt
T T t =1
When estimating the variance or the standard deviation using realized historical returns, we
use:
T
1
Var ( R )= ∑
T −1 t=1
( R t−R )
2
SD ( R )=√ Var (R)
The standard error is a statistical measure of the degree of estimation error.
SD ( R )
SE= Where T is the number of observations.
√T
The 95% confidence interval of the expected returns is approximately:
Historical average return ±1.96 ∙ standard error
2.3 Portfolios
A portfolio is a collection of assets. For example, gold, bonds, shares, real estate, etc.
Portfolio weights are the fractions of the total investment in the portfolio held in each
individual investment in the portfolio. The portfolio weights add up to 1.00 or 100%.
value of investment i
x i=
Total value of portfolio
, The return of a portfolio, Rp, is the weighted average of the returns on the investments in the
portfolio, where the weights correspond to portfolio weights:
R p =x1 R1 + x1 R2 +…+ x n R n=∑ x i R i
i
The expected return of a portfolio is the weighted average of the expected returns on the
investments in the portfolio:
E [ R¿¿ p]=∑ x i E [Ri ]¿
i
2.4 Diversification
Diversification lowers the risk in both directions: downward and upward. So, smaller losses,
but also smaller gains. Diversification eliminates stock specific risk.
2.5 Covariance and correlation
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.
The covariance is the expected product of the deviations of two returns from their expected
value:
Cov ( Ri , R j ) =E[ ( R i−E [ Ri ] )( R j−E [ R j ] ) ]
Historical covariance (estimate) between returns R i and Rj:
T
1
Cov ( Ri , R j ) = ∑ ( R −Ri ) ( R j , t−R j )
T −1 t=1 i , t
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.
The correlation is a measure of the common risk shared by stocks that does not depend on
their volatility.
Cov ( R i , R j )
Corr ( Ri , R j )= ρi, j =
SD (Ri )∙ SD (R j )
The correlation between two stocks is always between -1 and +1. It measures a linear
relationship between Ri and Rj. If Ri changes by p%, we expect Rj changes by ρi , j ∙ p %
The net present value NPV (time value of money):
Ct
PV 0 ( C t )=
( 1+ r )t
Growing perpetuity (continues forever) (first payment next period):
C1
PV 0=
r−g
Annuity (fixed term) (first payment next period, payments for t periods):
C1 1
PV 0= [1− ]
r (1+r )t
Growing annuity:
( )
C1 1+ g t
PV 0= [1− ]
r−g (1+r )
Lecture 2, Asset returns and volatility portfolio’s
2.1 Expected returns and volatility
Realized returns are the returns that actually occur over a particular time period.
Return = dividend yield + capital gain rate. In formula:
¿t +1 + Pt +1 ¿ P −Pt
Rt +1= −1= t +1 + t +1
Pt Pt Pt
The expected (mean) return is calculated as a weighted average of the possible returns,
where the weights correspond to the probabilities.
Expected return=E [ R ] =∑ P R ∙ R
R
The variance is the expected squared deviation from the mean.
Var ( R )=E ¿
The standard deviation is the root of the variance.
SD ( R )=√ Var (R)
,In finance, the standard deviation (σ ) of a return is also referred to as its volatility. The
standard deviation is easier to interpret because it’s in the same unit as the return itself.
Volatility is mostly measured in % per annum. It’s a measure of uncertainty about asset
returns.
Scaling of the volatility with different periods can be done by:
σ T periods=σ 1 period ∙ √ T for example, σ T annual=σ monthly ∙ √ 12
σ annual
and σ Tmonthly =
√12
2.2 Historic returns and volatility
When estimating the expected return, we use historical data to predict the future by taking
the average return.
T
1 1
R= ( R1 + R2 +…+ R T ) = ∑ Rt
T T t =1
When estimating the variance or the standard deviation using realized historical returns, we
use:
T
1
Var ( R )= ∑
T −1 t=1
( R t−R )
2
SD ( R )=√ Var (R)
The standard error is a statistical measure of the degree of estimation error.
SD ( R )
SE= Where T is the number of observations.
√T
The 95% confidence interval of the expected returns is approximately:
Historical average return ±1.96 ∙ standard error
2.3 Portfolios
A portfolio is a collection of assets. For example, gold, bonds, shares, real estate, etc.
Portfolio weights are the fractions of the total investment in the portfolio held in each
individual investment in the portfolio. The portfolio weights add up to 1.00 or 100%.
value of investment i
x i=
Total value of portfolio
, The return of a portfolio, Rp, is the weighted average of the returns on the investments in the
portfolio, where the weights correspond to portfolio weights:
R p =x1 R1 + x1 R2 +…+ x n R n=∑ x i R i
i
The expected return of a portfolio is the weighted average of the expected returns on the
investments in the portfolio:
E [ R¿¿ p]=∑ x i E [Ri ]¿
i
2.4 Diversification
Diversification lowers the risk in both directions: downward and upward. So, smaller losses,
but also smaller gains. Diversification eliminates stock specific risk.
2.5 Covariance and correlation
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.
The covariance is the expected product of the deviations of two returns from their expected
value:
Cov ( Ri , R j ) =E[ ( R i−E [ Ri ] )( R j−E [ R j ] ) ]
Historical covariance (estimate) between returns R i and Rj:
T
1
Cov ( Ri , R j ) = ∑ ( R −Ri ) ( R j , t−R j )
T −1 t=1 i , t
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.
The correlation is a measure of the common risk shared by stocks that does not depend on
their volatility.
Cov ( R i , R j )
Corr ( Ri , R j )= ρi, j =
SD (Ri )∙ SD (R j )
The correlation between two stocks is always between -1 and +1. It measures a linear
relationship between Ri and Rj. If Ri changes by p%, we expect Rj changes by ρi , j ∙ p %
