Achtergrond informatie
- Relative returns (without returns)
Δ Pt +1 Pt + Δ t−P t P t+1 −Pt
Rt +1= = =
Pt Pt Pt
- Log-returns (with returns)
r t +1=ln ( )
P t+ Δ t
Pt
These are close to relative returns for small time intervals.
- Volatility, the standard deviation of returns
σ =StDev (R)
Volatility is a measure of our uncertainty about the returns provided by the stock or any other financial
asset. It is the standard deviation of the return provided by the stock in one year (when the return is
expressed using continuous compounding).
Time aggregation: when Δ t is small, then σ √( Δt) is approximately equal to the
standard deviation of the percentage change (i.e. standard deviation of the relative return) in the
stock price in time Δ t )
If Rt −1 , R t are returns over two periods of time, the variance over two-period horizon
(assuming returns are independent) is:
2
Var R t ,2 =Var Rt +Var R t−1=2 σ
Taking the square root, we see that volatility scales as
σ Tperiods =σ 1 period × √ T
σ annual=σ daily × √ 251
The uncertainty about a future stock price return is measured by its standard deviation and it
increases with the square root of the time of the time period. This called the square root rule.
- Calculating historical volatility and average returns
1
Ŕ=
T ∑ Rt
R
¿
¿t
¿
¿
1
Var ( R )=σ 2R =
T −1
∑¿
Volatility=σ R=√ σ 2R
- Co-movements
Covariance, measures the degree of co-movement. If it is positive the two returns tend to move
together. When the covariance in negative the two returns tend to move in opposite directions.
, R
¿
(¿ i¿−Ŕ i)(R j− Ŕ j)
¿
1
( Ri −E ( Ri ) )( R j−E ( R j ) ] = T −1 ∑ ¿
Cov ( R i , R j )=E ¿
Correlation, a measure of the common risk
shared by stocks that does not depend on
their volatility. The correlation between the
stocks is always between -1 and 1. It measures
linear relationship between Ri and
RJ . If Ri changes by β then
R j changes (on average) by ρi , j × β .
Cov ( R i , R j )
Corr ( R i , R j )=ρi , j=
SD (Ri)× SD (R j)
- Portfolio
Returns
The weight of an investment is the fraction of the individual investment values to the total
portfolio value
value of investment i
weight=x i=
total value of portfolio
With this weight you are able to calculate the returns
R p=∑ x i Ri
The expected returns are given by
E ( R p ) =E ( ∑ x i Ri ) =∑ x i E ( Ri )
Volatility
The variance of a two-stock portfolio
x 21 Var ( R1 ) + x 22 Var ( R2 ) +2 x1 x 2 Cov (R1 , R2 )
The variance of an N-asset portfolio
N N
∑ x 2i Var ( R i)+2 ∑ ∑ xi x j Cov( Ri , R j )
i=1 j=i +1
The square root of the variance eventually gives you the volatility
If risks are independent, then the returns of two stocks are uncorrelated. This would imply that the
covariance is zero, because that is the measure of the risk they have in common. If the portfolio in
this case also consists of n assets all with equal weights, then the following formula gives you the
variance:
1
n ∑ 2
σi
Does the portfolio have risks that are not independent but equal correlations and volatilities, then
the variance is given by
σ 2 n−1 2
+ ρ σ which converges ¿ ρ σ 2 =Cov(i , j)
n n
, Hoorcollege slides 1
Return characteristics
Stock market indices show how a specified portfolio of share prices changes over time. If an index rises with
1%, the value of securities making up the index have risen with 1%.
A price weighted index weighs stocks based on their prices
stock value
total portfolio value
A value-weighted weighs stocks according to the total market value of their outstanding shares
total market value set of stocks
total market value of portfolio
The most widely used index is the Standard & Poor’s 500, which includes the largest 500 traded companies in
the US. It is a value-weighted index.
Prices and returns
We denote asset prices by Pt . We are more interested, however, in returns. This is the relative change in
the price of a financial asset over a given time interval (often expressed as a percentage). The dividend
component of returns is ignored for simplicity. There are two types of returns
simple (net) returns is the percentage change in prices.
For a stock
Pt −Pt −1
Rt =
Pt −1
For a portfolio
R p=∑ x i Ri
Continuously compounded returns
Y t =log ( 1+ Rt ) =log ( simpe gross returns )=log ( )Pt
Pt −1
=log( P t )−log ( P t−1)
In strict sense, only simple returns are correct. But the continuously compounded returns do have some
advantages. They are distributed symmetric, while simple returns are not. The difference between
- Relative returns (without returns)
Δ Pt +1 Pt + Δ t−P t P t+1 −Pt
Rt +1= = =
Pt Pt Pt
- Log-returns (with returns)
r t +1=ln ( )
P t+ Δ t
Pt
These are close to relative returns for small time intervals.
- Volatility, the standard deviation of returns
σ =StDev (R)
Volatility is a measure of our uncertainty about the returns provided by the stock or any other financial
asset. It is the standard deviation of the return provided by the stock in one year (when the return is
expressed using continuous compounding).
Time aggregation: when Δ t is small, then σ √( Δt) is approximately equal to the
standard deviation of the percentage change (i.e. standard deviation of the relative return) in the
stock price in time Δ t )
If Rt −1 , R t are returns over two periods of time, the variance over two-period horizon
(assuming returns are independent) is:
2
Var R t ,2 =Var Rt +Var R t−1=2 σ
Taking the square root, we see that volatility scales as
σ Tperiods =σ 1 period × √ T
σ annual=σ daily × √ 251
The uncertainty about a future stock price return is measured by its standard deviation and it
increases with the square root of the time of the time period. This called the square root rule.
- Calculating historical volatility and average returns
1
Ŕ=
T ∑ Rt
R
¿
¿t
¿
¿
1
Var ( R )=σ 2R =
T −1
∑¿
Volatility=σ R=√ σ 2R
- Co-movements
Covariance, measures the degree of co-movement. If it is positive the two returns tend to move
together. When the covariance in negative the two returns tend to move in opposite directions.
, R
¿
(¿ i¿−Ŕ i)(R j− Ŕ j)
¿
1
( Ri −E ( Ri ) )( R j−E ( R j ) ] = T −1 ∑ ¿
Cov ( R i , R j )=E ¿
Correlation, a measure of the common risk
shared by stocks that does not depend on
their volatility. The correlation between the
stocks is always between -1 and 1. It measures
linear relationship between Ri and
RJ . If Ri changes by β then
R j changes (on average) by ρi , j × β .
Cov ( R i , R j )
Corr ( R i , R j )=ρi , j=
SD (Ri)× SD (R j)
- Portfolio
Returns
The weight of an investment is the fraction of the individual investment values to the total
portfolio value
value of investment i
weight=x i=
total value of portfolio
With this weight you are able to calculate the returns
R p=∑ x i Ri
The expected returns are given by
E ( R p ) =E ( ∑ x i Ri ) =∑ x i E ( Ri )
Volatility
The variance of a two-stock portfolio
x 21 Var ( R1 ) + x 22 Var ( R2 ) +2 x1 x 2 Cov (R1 , R2 )
The variance of an N-asset portfolio
N N
∑ x 2i Var ( R i)+2 ∑ ∑ xi x j Cov( Ri , R j )
i=1 j=i +1
The square root of the variance eventually gives you the volatility
If risks are independent, then the returns of two stocks are uncorrelated. This would imply that the
covariance is zero, because that is the measure of the risk they have in common. If the portfolio in
this case also consists of n assets all with equal weights, then the following formula gives you the
variance:
1
n ∑ 2
σi
Does the portfolio have risks that are not independent but equal correlations and volatilities, then
the variance is given by
σ 2 n−1 2
+ ρ σ which converges ¿ ρ σ 2 =Cov(i , j)
n n
, Hoorcollege slides 1
Return characteristics
Stock market indices show how a specified portfolio of share prices changes over time. If an index rises with
1%, the value of securities making up the index have risen with 1%.
A price weighted index weighs stocks based on their prices
stock value
total portfolio value
A value-weighted weighs stocks according to the total market value of their outstanding shares
total market value set of stocks
total market value of portfolio
The most widely used index is the Standard & Poor’s 500, which includes the largest 500 traded companies in
the US. It is a value-weighted index.
Prices and returns
We denote asset prices by Pt . We are more interested, however, in returns. This is the relative change in
the price of a financial asset over a given time interval (often expressed as a percentage). The dividend
component of returns is ignored for simplicity. There are two types of returns
simple (net) returns is the percentage change in prices.
For a stock
Pt −Pt −1
Rt =
Pt −1
For a portfolio
R p=∑ x i Ri
Continuously compounded returns
Y t =log ( 1+ Rt ) =log ( simpe gross returns )=log ( )Pt
Pt −1
=log( P t )−log ( P t−1)
In strict sense, only simple returns are correct. But the continuously compounded returns do have some
advantages. They are distributed symmetric, while simple returns are not. The difference between
