Summary Investment Management
2020-2021
Week 1 (chapter 5 & chapter 6 & chapter 7)
Chapter 5
Holding Period Return (HPR); how much prices change with respect to some initial price
- What the return will be
- You cannot be sure about your eventual HPR
! "# $#
HPR = !" # !" !"#$ = return
!"#$
HPR = dividend yield + rate of capital gains
- Dividend yield; the percentage return from dividends
Expected rate of return; what you think the return will be
- Uncertainty around HPR
Expected HPR = 𝐸[𝑅%& ] 𝑝(𝑠) ; probability of each scenario
𝐸(𝑟) = ∑𝑝(𝑠)𝑟(𝑠) 𝑟(𝑠) ; HPR in each scenario
Variance; measure of volatility
- Measures dispersion of possible outcomes around the expected value
- Natural measure of uncertainty
- Volatility is reflected in deviations of actual returns from the mean return
𝜎 ' = ∑𝑝(𝑠)[𝑟(𝑠) − 𝐸(𝑟)]' 𝑝(𝑠) ; probability of each scenario
𝑟(𝑠) ; HPR in each scenario
𝐸(𝑟) ; expected rate of return
Standard Deviation; measure of risk
𝜎 = √𝜎 ' (square root of variance)
Risk-free rate; rate you would earn in risk-free assets (such as; T-bills/money market funds/bank)
Risk Premium; the difference between the expected HPR and the risk-free rate
- How much of an expected reward is offered for the risk involved in investing in stocks
- It is the extra return on top of the risk-free rate you demand for taking the riskier alternative
risk premium = expected HPR − 𝑟( = 𝐸(𝑟% ) − 𝑟(
Excess return; the difference between the return of a risky asset vs. the return of a risk-free asset
- Extra reward for taking risk
excess return = actual rate of return of a risky asset − actual risk free rate
excess return = 𝑅% − 𝑟(
Sharpe Ratio; reward-to-volatility ratio
- Reward = risk premium
- Volatility = risk (measured by SD)
- Sharpe ratio is a measure of attraction of a portfolio (measure of portfolio performance)
- The higher the Sharpe ratio, the greater the expected return corresponding to any level of volatility
(we seek to maximize the Sharpe ratio)
- A negative Sharpe ratio indicates that a risk-less asset would perform better
- Diversified portfolio gives highest Sharpe ratio
)*+, .)/0*10
Sharpe ratio =
23 45 /67/++ )/81)9
:(<% )$<&
Sharpe ratio = >%
,Chapter 6
A portfolio is more attractive when its expected return is higher and its risk lower. But when risk increases along with
return, the most attractive portfolio is not obvious. How can investors quantify the rate at which they are willing to
trade off return against risk?
-----> Utility!
Higher utility values are assigned to portfolios with more attractive risk-return profiles
𝐸(𝑟) ; expected rate of return
Utility;
? 𝐴 ; index of the investor’s risk aversion
𝑈 = 𝐸(𝑟) − ' 𝐴𝜎 ' 𝜎' ; variance of return
- Portfolios receive higher utility scores for higher expected returns and lower utility scores for higher volatility
- Investors choosing among competing investment portfolios will select the one providing the higher utility level
- The utility of the portfolio should be higher than the risk-free alternative (otherwise the investor would go for the
risk-free alternative)
Certainty equivalent; utility score of risky portfolio
- It is the rate that a risk-free investment would need to offer to provide the same utility score as the risky portfolio
(it is a natural way to compare utility values of competing portfolios)
- Certainty equivalent < risk-free rate ----> reject portfolio (prefers to invest in T-bills)
- Certainty equivalent > risk-free rate ----> accept portfolio (invest in risky portfolio)
Risk averse; 𝐴>0
Risk neutral; 𝐴 = 0 (judge solely on expected rates of return)
Risk lover; 𝐴 < 0 (adjusts expected returns upwards due to the ‘fun’ of taking risks)
Mean-Variance (M-V) Criterion (analysis);
- Portfolio A dominates B if;
𝐸(𝑟@ ) ≥ 𝐸(𝑟A ) and 𝜎@ ≤ 𝜎A
Equally preferred portfolios will lie in the mean-SD plane on the indifference
curve, which connects all portfolio points with the same utility value
Various points showing different 𝐸(𝑟# ) and 𝜎# combinations providing equally
utility to the investor
- It forms an indifference curve between risk and reward
How does the indifference curve of a less/more risk averse investor looks like compared to this indifference curve?
- More risk-averse investors have steeper indifference curves than less-risk averse investors
- Steeper curves mean that investors require a greater increase in expected return to compensate for an increase
in portfolio risk (higher Sharpe ratio)
Higher indifference curves correspond to higher level of utility
- The investor thus attempts to find the complete portfolio on the highest
possible indifference curve
,Rate of return on complete portfolio (c);
𝑟B = 𝑦 ∙ 𝑟# + (1 − 𝑦)𝑟(
Expected rate of return on complete portfolio (c);
𝐸(𝑟B ) = 𝑦 ∙ 𝐸(𝑟# ) + (1 − 𝑦)𝑟(
𝐸(𝑟B ) = 𝑟( + 𝑦[𝐸(𝑟# ) − 𝑟( ]
:(<% )$<&
𝐸(𝑟B ) = 𝑟( + U >%
V ∙ 𝜎C (equation CAL line)
Standard deviation of complete portfolio (c);
𝜎B = 𝑦𝜎# (𝜎B' = 𝑦 ' 𝜎#' )
- The SD of a risk-free asset is zero!
Plot of portfolio characteristics (investment opportunity set);
The investment opportunity set with a risky asset and a risk-free asset
If the investor chooses to invest solely in risk assets (y = 1) the complete portfolio
is P
If the investor chooses the invest solely in risk-free assets (y = 0) the complete
portfolio is F
The portfolios where y lies between 0 and 1, will graph on the straight-line
connecting point P and F
- Slope of the line is the Sharpe ratio
Investment opportunity set; the set of feasible expected return and standard deviation pairs of all portfolios resulting
from different values of y
Capital Allocation Line (CAL); depicts all risk-return combinations available to investors
- Slope CAL = Sharpe ratio
- The investor confronting the CAL must choose one optimal complete portfolio, C, from the set of feasible choices.
This choice entails a trade-off between risk and return
- Equation of the CAL line;
Equation of the CAL line;
𝐸(𝑟B ) = 𝑟( + (Sharpe ratio) ∙ 𝜎C
:(<% )$<&
𝐸(𝑟B ) = 𝑟( + U >%
V ∙ 𝜎C
Individuals differences in risk aversion led to different capital allocation choices even when facing an identical
opportunity set
- More risk-averse investors will choose to hold less of risky assets and more of the risk-free asset
!
Investors chooses the allocation to the risk asset, y, that maximizes their utility function (𝑈 = 𝐸(𝑟) − " 𝐴𝜎 " )
Optimal position in the risky asset (optimal complete portfolio);
:(<% )$<&
𝑦∗ = @>%(
,Chapter 7
There are two sources of risk;
1) Market/systematic/nondiversifiable risk
- Risk that comes from conditions in the general economy
- SD falls as the number of securities increase, but cannot be reduced to zero
2) Unique/firm-specific/nonsystematic/diversifiable risk
- Firm specific influences
- Can be eliminated by diversification
- Portfolio volatility should fall
Rate of return portfolio;
𝑟# = 𝑤E 𝑟E + 𝑤F 𝑟F + ⋯ + 𝑤G 𝑟G
Expected rate of return portfolio;
𝐸[𝑟# ] = 𝑤E 𝐸[𝑟E ] + 𝑤F 𝐸[𝑟F ] + ⋯ + 𝑤G 𝐸[𝑟G ]
Variance of portfolio; (two assets)
𝜎#' = 𝑤E' 𝜎E' + 𝑤F' 𝜎F' + 2𝑤E 𝑤F 𝐶𝑜𝑣(𝑟E , 𝑟F )
Covariance;
𝐶𝑜𝑣(𝑟E , 𝑟F ) = 𝜌EF 𝜎E 𝜎F
- We will always prefer to add to our portfolio assets with low, or even better, negative correlation with our existing
position
- Portfolios of less than perfectly correlated assets always offer some degree of diversification benefit (portfolio
has lower risk)
- 𝜌=1 ---> no benefit from diversification
- 𝜌 = −1 ---> perfect hedging opportunity and the maximum advantage from diversification
Markowitz; the identification of the efficient set of portfolios (efficient frontier of risky assets)
- The principal idea behind the frontier set of risky portfolios is that, for
any level of risk, we are interested only in that portfolio with the
highest expected return
- Markowitz says that you should maximize your Sharpe ratio in order
to find the optimal security
- Maximizing Sharpe ratio leads to a bullet-shaped envelope
Global Minimum-Variance Portfolio;
Point where there is the least amount of risk
Steps to find the optimal risky portfolio; (portfolio with bonds/stocks)
"#(%# )'%$ ()%& ' "#(%% )'%$ (+,-(%# ,%% )
1) 𝑤! =
"#(%# )'%$ ()%& / "#(%% )'%$ ()#
& ' "#(% )'% /#(% ) ' % (+,-(% ,% )
# $ % $ # %
2) 𝑤0 = 1 − 𝑤!
3) 𝐸[𝑟# ] = 𝑤E 𝐸[𝑟E ] + 𝑤F 𝐸[𝑟F ]
4) 𝜎#' = 𝑤E' 𝜎E' + 𝑤F' 𝜎F' + 2𝑤E 𝑤F 𝐶𝑜𝑣(𝑟E , 𝑟F )
5) 𝜎# = _𝜎#'
:(<% )$<&
6) Sharpe ratio = >%
(slope best CAL)
, Steps to find the optimal complete portfolio; (portfolio with t-bills/bonds/stocks)
:(<% )$<&
1) 𝑦 ∗ = @>%( 𝑦%
- The investor will invest 𝑦% of his overall wealth in portfolio P and (1 − 𝑦)% in T-bills
2) 𝑦 ∙ 𝑤!
(𝑦 ∙ 𝑤" )%
- This is the share of the complete portfolio invested in asset 𝑤! (𝑦 ∙ 𝑤! )%
3) 𝑦 ∙ 𝑤0
- This is the share of the complete portfolio invested in asset 𝑤0 (1 − 𝑦)%
Steps to find the minimum-variance portfolio;
)%& '+,-(%# ,%% )
1) 𝑤123 (𝑋) = & /) & '4+,-(% ,% )
)# % # %
2) 𝑤1230 = 1 − 𝑤123!
'
3) 𝜎H%I = 𝑤E' 𝜎E' + 𝑤F' 𝜎F' + 2𝑤E 𝑤F 𝐶𝑜𝑣(𝑟E , 𝑟F )
'
4) 𝜎H%I = `𝜎H%I
2020-2021
Week 1 (chapter 5 & chapter 6 & chapter 7)
Chapter 5
Holding Period Return (HPR); how much prices change with respect to some initial price
- What the return will be
- You cannot be sure about your eventual HPR
! "# $#
HPR = !" # !" !"#$ = return
!"#$
HPR = dividend yield + rate of capital gains
- Dividend yield; the percentage return from dividends
Expected rate of return; what you think the return will be
- Uncertainty around HPR
Expected HPR = 𝐸[𝑅%& ] 𝑝(𝑠) ; probability of each scenario
𝐸(𝑟) = ∑𝑝(𝑠)𝑟(𝑠) 𝑟(𝑠) ; HPR in each scenario
Variance; measure of volatility
- Measures dispersion of possible outcomes around the expected value
- Natural measure of uncertainty
- Volatility is reflected in deviations of actual returns from the mean return
𝜎 ' = ∑𝑝(𝑠)[𝑟(𝑠) − 𝐸(𝑟)]' 𝑝(𝑠) ; probability of each scenario
𝑟(𝑠) ; HPR in each scenario
𝐸(𝑟) ; expected rate of return
Standard Deviation; measure of risk
𝜎 = √𝜎 ' (square root of variance)
Risk-free rate; rate you would earn in risk-free assets (such as; T-bills/money market funds/bank)
Risk Premium; the difference between the expected HPR and the risk-free rate
- How much of an expected reward is offered for the risk involved in investing in stocks
- It is the extra return on top of the risk-free rate you demand for taking the riskier alternative
risk premium = expected HPR − 𝑟( = 𝐸(𝑟% ) − 𝑟(
Excess return; the difference between the return of a risky asset vs. the return of a risk-free asset
- Extra reward for taking risk
excess return = actual rate of return of a risky asset − actual risk free rate
excess return = 𝑅% − 𝑟(
Sharpe Ratio; reward-to-volatility ratio
- Reward = risk premium
- Volatility = risk (measured by SD)
- Sharpe ratio is a measure of attraction of a portfolio (measure of portfolio performance)
- The higher the Sharpe ratio, the greater the expected return corresponding to any level of volatility
(we seek to maximize the Sharpe ratio)
- A negative Sharpe ratio indicates that a risk-less asset would perform better
- Diversified portfolio gives highest Sharpe ratio
)*+, .)/0*10
Sharpe ratio =
23 45 /67/++ )/81)9
:(<% )$<&
Sharpe ratio = >%
,Chapter 6
A portfolio is more attractive when its expected return is higher and its risk lower. But when risk increases along with
return, the most attractive portfolio is not obvious. How can investors quantify the rate at which they are willing to
trade off return against risk?
-----> Utility!
Higher utility values are assigned to portfolios with more attractive risk-return profiles
𝐸(𝑟) ; expected rate of return
Utility;
? 𝐴 ; index of the investor’s risk aversion
𝑈 = 𝐸(𝑟) − ' 𝐴𝜎 ' 𝜎' ; variance of return
- Portfolios receive higher utility scores for higher expected returns and lower utility scores for higher volatility
- Investors choosing among competing investment portfolios will select the one providing the higher utility level
- The utility of the portfolio should be higher than the risk-free alternative (otherwise the investor would go for the
risk-free alternative)
Certainty equivalent; utility score of risky portfolio
- It is the rate that a risk-free investment would need to offer to provide the same utility score as the risky portfolio
(it is a natural way to compare utility values of competing portfolios)
- Certainty equivalent < risk-free rate ----> reject portfolio (prefers to invest in T-bills)
- Certainty equivalent > risk-free rate ----> accept portfolio (invest in risky portfolio)
Risk averse; 𝐴>0
Risk neutral; 𝐴 = 0 (judge solely on expected rates of return)
Risk lover; 𝐴 < 0 (adjusts expected returns upwards due to the ‘fun’ of taking risks)
Mean-Variance (M-V) Criterion (analysis);
- Portfolio A dominates B if;
𝐸(𝑟@ ) ≥ 𝐸(𝑟A ) and 𝜎@ ≤ 𝜎A
Equally preferred portfolios will lie in the mean-SD plane on the indifference
curve, which connects all portfolio points with the same utility value
Various points showing different 𝐸(𝑟# ) and 𝜎# combinations providing equally
utility to the investor
- It forms an indifference curve between risk and reward
How does the indifference curve of a less/more risk averse investor looks like compared to this indifference curve?
- More risk-averse investors have steeper indifference curves than less-risk averse investors
- Steeper curves mean that investors require a greater increase in expected return to compensate for an increase
in portfolio risk (higher Sharpe ratio)
Higher indifference curves correspond to higher level of utility
- The investor thus attempts to find the complete portfolio on the highest
possible indifference curve
,Rate of return on complete portfolio (c);
𝑟B = 𝑦 ∙ 𝑟# + (1 − 𝑦)𝑟(
Expected rate of return on complete portfolio (c);
𝐸(𝑟B ) = 𝑦 ∙ 𝐸(𝑟# ) + (1 − 𝑦)𝑟(
𝐸(𝑟B ) = 𝑟( + 𝑦[𝐸(𝑟# ) − 𝑟( ]
:(<% )$<&
𝐸(𝑟B ) = 𝑟( + U >%
V ∙ 𝜎C (equation CAL line)
Standard deviation of complete portfolio (c);
𝜎B = 𝑦𝜎# (𝜎B' = 𝑦 ' 𝜎#' )
- The SD of a risk-free asset is zero!
Plot of portfolio characteristics (investment opportunity set);
The investment opportunity set with a risky asset and a risk-free asset
If the investor chooses to invest solely in risk assets (y = 1) the complete portfolio
is P
If the investor chooses the invest solely in risk-free assets (y = 0) the complete
portfolio is F
The portfolios where y lies between 0 and 1, will graph on the straight-line
connecting point P and F
- Slope of the line is the Sharpe ratio
Investment opportunity set; the set of feasible expected return and standard deviation pairs of all portfolios resulting
from different values of y
Capital Allocation Line (CAL); depicts all risk-return combinations available to investors
- Slope CAL = Sharpe ratio
- The investor confronting the CAL must choose one optimal complete portfolio, C, from the set of feasible choices.
This choice entails a trade-off between risk and return
- Equation of the CAL line;
Equation of the CAL line;
𝐸(𝑟B ) = 𝑟( + (Sharpe ratio) ∙ 𝜎C
:(<% )$<&
𝐸(𝑟B ) = 𝑟( + U >%
V ∙ 𝜎C
Individuals differences in risk aversion led to different capital allocation choices even when facing an identical
opportunity set
- More risk-averse investors will choose to hold less of risky assets and more of the risk-free asset
!
Investors chooses the allocation to the risk asset, y, that maximizes their utility function (𝑈 = 𝐸(𝑟) − " 𝐴𝜎 " )
Optimal position in the risky asset (optimal complete portfolio);
:(<% )$<&
𝑦∗ = @>%(
,Chapter 7
There are two sources of risk;
1) Market/systematic/nondiversifiable risk
- Risk that comes from conditions in the general economy
- SD falls as the number of securities increase, but cannot be reduced to zero
2) Unique/firm-specific/nonsystematic/diversifiable risk
- Firm specific influences
- Can be eliminated by diversification
- Portfolio volatility should fall
Rate of return portfolio;
𝑟# = 𝑤E 𝑟E + 𝑤F 𝑟F + ⋯ + 𝑤G 𝑟G
Expected rate of return portfolio;
𝐸[𝑟# ] = 𝑤E 𝐸[𝑟E ] + 𝑤F 𝐸[𝑟F ] + ⋯ + 𝑤G 𝐸[𝑟G ]
Variance of portfolio; (two assets)
𝜎#' = 𝑤E' 𝜎E' + 𝑤F' 𝜎F' + 2𝑤E 𝑤F 𝐶𝑜𝑣(𝑟E , 𝑟F )
Covariance;
𝐶𝑜𝑣(𝑟E , 𝑟F ) = 𝜌EF 𝜎E 𝜎F
- We will always prefer to add to our portfolio assets with low, or even better, negative correlation with our existing
position
- Portfolios of less than perfectly correlated assets always offer some degree of diversification benefit (portfolio
has lower risk)
- 𝜌=1 ---> no benefit from diversification
- 𝜌 = −1 ---> perfect hedging opportunity and the maximum advantage from diversification
Markowitz; the identification of the efficient set of portfolios (efficient frontier of risky assets)
- The principal idea behind the frontier set of risky portfolios is that, for
any level of risk, we are interested only in that portfolio with the
highest expected return
- Markowitz says that you should maximize your Sharpe ratio in order
to find the optimal security
- Maximizing Sharpe ratio leads to a bullet-shaped envelope
Global Minimum-Variance Portfolio;
Point where there is the least amount of risk
Steps to find the optimal risky portfolio; (portfolio with bonds/stocks)
"#(%# )'%$ ()%& ' "#(%% )'%$ (+,-(%# ,%% )
1) 𝑤! =
"#(%# )'%$ ()%& / "#(%% )'%$ ()#
& ' "#(% )'% /#(% ) ' % (+,-(% ,% )
# $ % $ # %
2) 𝑤0 = 1 − 𝑤!
3) 𝐸[𝑟# ] = 𝑤E 𝐸[𝑟E ] + 𝑤F 𝐸[𝑟F ]
4) 𝜎#' = 𝑤E' 𝜎E' + 𝑤F' 𝜎F' + 2𝑤E 𝑤F 𝐶𝑜𝑣(𝑟E , 𝑟F )
5) 𝜎# = _𝜎#'
:(<% )$<&
6) Sharpe ratio = >%
(slope best CAL)
, Steps to find the optimal complete portfolio; (portfolio with t-bills/bonds/stocks)
:(<% )$<&
1) 𝑦 ∗ = @>%( 𝑦%
- The investor will invest 𝑦% of his overall wealth in portfolio P and (1 − 𝑦)% in T-bills
2) 𝑦 ∙ 𝑤!
(𝑦 ∙ 𝑤" )%
- This is the share of the complete portfolio invested in asset 𝑤! (𝑦 ∙ 𝑤! )%
3) 𝑦 ∙ 𝑤0
- This is the share of the complete portfolio invested in asset 𝑤0 (1 − 𝑦)%
Steps to find the minimum-variance portfolio;
)%& '+,-(%# ,%% )
1) 𝑤123 (𝑋) = & /) & '4+,-(% ,% )
)# % # %
2) 𝑤1230 = 1 − 𝑤123!
'
3) 𝜎H%I = 𝑤E' 𝜎E' + 𝑤F' 𝜎F' + 2𝑤E 𝑤F 𝐶𝑜𝑣(𝑟E , 𝑟F )
'
4) 𝜎H%I = `𝜎H%I
