Investment management
Lecture 1 – chapter 5, 6 and 7:
Chapter 5:
Realized return can be calculated as this formula, were
HPR = holding period return
D i ,t = cash dividend at t for stock i
Pi ,t = share price at t for stock i (price end of period)
Pi ,t −1 = share price at t – 1 for stock i (price beginning of period)
D i , t + Pi ,t −Pi ,t −1 ending price s h are−beginning price+dividend
Ri , t (HPR)= =
Pi ,t −1 beginning price
This formula calculates the actual, observed return an investor earns on a
stock during a specific time period, t. It reflects both income (dividends)
and capital gains/losses (changes in stock price).
Expected return:
Which is the average return an investor expects to earn on an investment,
considering potential scenarios (states of the world).
E [R]=∑ ps Rs
s
S = states of the world, different possible scenarios
ps = likelihood of a specific state of the world occurring, must sum up to 1
total
R s= return an investor would achieve if a particular state of the world s
occurs
,Risk premium
Represents the extra return investors expect/demand to earn from a risky
investment compared to s risk-free investment.
risk premium=E [Ri ]−r f
The variance ( σ 2 ¿ :
It is a measure of volatiltiy. It measures the dispersion of possible
outcomes around the expected value (uncertainty).
σ 2=∑ pi ¿¿
The standard deviation (σ ¿
It measures riks and represents the average amount by which returns
deviate from the expected return. Just go back to units from variance.
σ =√ σ 2
Larger variance or standard deviation indicates greater dispersion
(spread) of returns around the expected value. it suggest higher
uncertainty and thus greater risk.
Example:
Step 1: Expected Return
- E[R]==∑ pi Ri = (0.4 ⋅ 0. 10) + (0.3 ⋅ 0.05) + (0.3⋅ −0.05) = 0.04 >
4%
Step 2: variance
- σ 2 =(0.4 ⋅(0.10−0.04) 2)+( 0.3⋅(0.05−0.04) 2)+( 0.3⋅(−0.05−0.04) 2)
= 0.0039
Step 3: standard deviation
- σ =√ 0.0039 = 0.06254 > 6.25%
, Coveriance:
It measures the degree to which two variables move together:
σ ij= E[(Ri −μ j )( Ri−μ j)]
If σ ij> 0, Ri R j tend to move in the same direction (positive relation)
σ ij< 0, Ri R j tend to move in opposite directions (negative relationship)
σ ij= 0, they are uncorrelated
Correlation:
It quantifies the strenght and direction of the relationship between the 2
returns on assets ( Ri R j ¿
E[( R i−μi)−( R j−μ j)]
ρij =corr (Ri R j )=
σiσj
The correlation outcome is between -1 and +1
ρij = +1 > perfect positive relationship
ρij = -1 > perfect negative relationship
ρij = 0 > no linear relationship
Example:
Step 1: calculate mean returns
- μ1 = (0.4⋅0.10) + (0.3⋅0.05) + (0.3⋅−0.05) = 0.04 > 4%
- μ2 = (0.4⋅0.08) + (0.3⋅0.02) + (0.3⋅−0.03) = 0.029 > 2.9%
step 2: calculate coveriance
- σ1,2=(0.4⋅(0.10−0.04) (0.08−0.029)) + (0.3⋅ (0.05−0.04)
(0.02−0.029)) + (0.3⋅(−0.05 −0.04) (−0.03−0.029)) =0.00279
step 3: calculate the standard deviations
- for R1 > σ 1=(0.4 )¿ = 0.0039
- σ 1=√ 0.0039=0.06245
- for R2 > σ 2=( 0.4)¿ = 0.002109
- σ 2=√ 0.002109=0.04593
Lecture 1 – chapter 5, 6 and 7:
Chapter 5:
Realized return can be calculated as this formula, were
HPR = holding period return
D i ,t = cash dividend at t for stock i
Pi ,t = share price at t for stock i (price end of period)
Pi ,t −1 = share price at t – 1 for stock i (price beginning of period)
D i , t + Pi ,t −Pi ,t −1 ending price s h are−beginning price+dividend
Ri , t (HPR)= =
Pi ,t −1 beginning price
This formula calculates the actual, observed return an investor earns on a
stock during a specific time period, t. It reflects both income (dividends)
and capital gains/losses (changes in stock price).
Expected return:
Which is the average return an investor expects to earn on an investment,
considering potential scenarios (states of the world).
E [R]=∑ ps Rs
s
S = states of the world, different possible scenarios
ps = likelihood of a specific state of the world occurring, must sum up to 1
total
R s= return an investor would achieve if a particular state of the world s
occurs
,Risk premium
Represents the extra return investors expect/demand to earn from a risky
investment compared to s risk-free investment.
risk premium=E [Ri ]−r f
The variance ( σ 2 ¿ :
It is a measure of volatiltiy. It measures the dispersion of possible
outcomes around the expected value (uncertainty).
σ 2=∑ pi ¿¿
The standard deviation (σ ¿
It measures riks and represents the average amount by which returns
deviate from the expected return. Just go back to units from variance.
σ =√ σ 2
Larger variance or standard deviation indicates greater dispersion
(spread) of returns around the expected value. it suggest higher
uncertainty and thus greater risk.
Example:
Step 1: Expected Return
- E[R]==∑ pi Ri = (0.4 ⋅ 0. 10) + (0.3 ⋅ 0.05) + (0.3⋅ −0.05) = 0.04 >
4%
Step 2: variance
- σ 2 =(0.4 ⋅(0.10−0.04) 2)+( 0.3⋅(0.05−0.04) 2)+( 0.3⋅(−0.05−0.04) 2)
= 0.0039
Step 3: standard deviation
- σ =√ 0.0039 = 0.06254 > 6.25%
, Coveriance:
It measures the degree to which two variables move together:
σ ij= E[(Ri −μ j )( Ri−μ j)]
If σ ij> 0, Ri R j tend to move in the same direction (positive relation)
σ ij< 0, Ri R j tend to move in opposite directions (negative relationship)
σ ij= 0, they are uncorrelated
Correlation:
It quantifies the strenght and direction of the relationship between the 2
returns on assets ( Ri R j ¿
E[( R i−μi)−( R j−μ j)]
ρij =corr (Ri R j )=
σiσj
The correlation outcome is between -1 and +1
ρij = +1 > perfect positive relationship
ρij = -1 > perfect negative relationship
ρij = 0 > no linear relationship
Example:
Step 1: calculate mean returns
- μ1 = (0.4⋅0.10) + (0.3⋅0.05) + (0.3⋅−0.05) = 0.04 > 4%
- μ2 = (0.4⋅0.08) + (0.3⋅0.02) + (0.3⋅−0.03) = 0.029 > 2.9%
step 2: calculate coveriance
- σ1,2=(0.4⋅(0.10−0.04) (0.08−0.029)) + (0.3⋅ (0.05−0.04)
(0.02−0.029)) + (0.3⋅(−0.05 −0.04) (−0.03−0.029)) =0.00279
step 3: calculate the standard deviations
- for R1 > σ 1=(0.4 )¿ = 0.0039
- σ 1=√ 0.0039=0.06245
- for R2 > σ 2=( 0.4)¿ = 0.002109
- σ 2=√ 0.002109=0.04593
