Portfolio Management

1. Modern Portfolio Theory

The HM model is also call Mean-Variance Model due to the fact that it is based on expected returns(mean) and the standard deviation(variance) of the various protfolios.

1.1 Meansurements of Return and Risk

Return: Average return (Arithmetic return)
It is used to estimate the expected return of next single period.

R = ( R 1 + R 2 + ⋅ ⋅ ⋅ + R n ) / n R=(R_1+R_2+···+R_n)/n R=(R1​+R2​+⋅⋅⋅+Rn​)/n

Risk: Variance and standard deviation

• Population variance
  σ 2 = ∑ i = 1 N ( X i − μ ) 2 N \sigma^2=\frac{\sum_{i=1}^{N}(X_i-\mu)^2}{N} σ2=N∑i=1N​(Xi​−μ)2​
• Population standard deviation
  σ = ∑ i = 1 N ( X i − μ ) 2 N \sigma=\sqrt\frac{\sum_{i=1}^{N}(X_i-\mu)^2}{N} σ=N∑i=1N​(Xi​−μ)2​ ​
• Sample variance
  σ 2 = ∑ i = 1 n ( X i − X ‾ ) 2 n − 1 \sigma^2=\frac{\sum_{i=1}^{n}(X_i-\overline{X})^2}{n-1} σ2=n−1∑i=1n​(Xi​−X)2​
• Sample standard deviation
  σ = ∑ i = 1 n ( X i − X ‾ ) 2 n − 1 \sigma=\sqrt\frac{\sum_{i=1}^{n}(X_i-\overline{X})^2}{n-1} σ=n−1∑i=1n​(Xi​−X)2​ ​

There is a positive relationship between expected return and risk.

1.2 Assumption of MPT

Capital markets are perfect

• There are no taxes or transaction costs
• All trader have costless access to all available information
• Perfect competition exists among all market participants

Returns are noramlly distributed

1.3 Utility Theory

Behavior of investors under uncertainty

Risk averse: Investors want to minimize their risk for the same amount of return and maximize their return for the same amount.

Risk neutral: Investors want to maximize return irrespective of risk.

Risk seeking: Investors want to maximize both risk and return.

Markowitz demonstrated that a “rational investor”(i.e., an investor who is risk averse and seeks to maximize utility) should evaluate potential protfolio alloactions based on the assciated means and variance for the expected rate of return distributions.

With all else being equal, investors prefer a higher mean return and a lower variance.

Utility function is a measure to rank different portfolios in the order of their preference.

U = E ( R ) − 1 2 A σ 2 U=E(R)-\frac{1}{2}A{\sigma}^2 U=E(R)−21​Aσ2

A A A is a measure of risk aversion.

• When A > 0 A>0 A>0, investor is risk-averse,
• When A = 0 A=0 A=0, investor is risk-neutral,
• When A < 0 A<0 A<0, investor is risk-seeking.

Unility theory allows us to quantify the rankings of investment choices using risk and return.

An indifference curve plots the combination of risk-return pairs that an investor would accept to maintain a given level of utility.

The investor is indifferent about the combinations on any one curve beasue they would provide the same level of overall utility.

Indifference curves are thus defined in terms of a trade-off between expected return and risk.

For a risk-averse investor, the curves are upward sloping and conves(get steeper) because of diminishing marginal utility of return(or wealth).

As risk increases, an investor needs greater return to compenstate for higher risk at an increasing rate.

For various types of investors, the more risk-averse the investor, the steeper the curve.

1.4 Efficient frontier

1.4.1 Return and Risk of Portfolio with Two Risky Assets

Return of portfolio

R p = w 1 R 1 + w 2 R 2 , R p = w 1 R 1 + ( 1 − w 1 ) R 2 R_p=w_1R_1+w_2R_2,\;R_p=w_1R_1+(1-w_1)R_2 Rp​=w1​R1​+w2​R2​,Rp​=w1​R1​+(1−w1​)R2​
Risk of portfolio
σ p = w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2 w 1 w 2 ∗ C o v ( R 1 , R 2 ) \sigma_p=\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2*Cov(R_1, R_2)} σp​=w12​σ12​+w22​σ22​+2w1​w2​∗Cov(R1​,R2​) ​
σ p = w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2 w 1 w 2 ρ 1 , 2 σ 1 σ 2 \sigma_p=\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{1,2}\sigma_1\sigma_2} σp​=w12​σ12​+w22​σ22​+2w1​w2​ρ1,2​σ1​σ2​ ​
Covariance for population
C o v ( x , y ) = ∑ i = 1 N ( X i − X ‾ ) ( Y i − Y ‾ ) N Cov(x,y)=\frac{\sum_{i=1}^N(X_i-\overline{X})(Y_i-\overline{Y})}{N} Cov(x,y)=N∑i=1N​(Xi​−X)(Yi​−Y)​

Covariance for sample
C o v ( x , y ) = ∑ i = 1 n ( X i − X ‾ ) ( Y i − Y ‾ ) n − 1 Cov(x,y)=\frac{\sum_{i=1}^n(X_i-\overline{X})(Y_i-\overline{Y})}{n-1} Cov(x,y)=n−1∑i=1n​(Xi​−X)(Yi​−Y)​

Correlation coefficient for population
ρ x y = C o v ( x , y ) σ x σ y \rho_{xy}=\frac{Cov(x,y)}{\sigma_x\sigma_y} ρxy​=σx​σy​Cov(x,y)​
Correlation coefficient for sample
r x y = C o v ( x , y ) s x s y r_{xy}=\frac{Cov(x,y)}{s_xs_y} rxy​=sx​sy​Cov(x,y)​

If ρ = 1 \rho = 1 ρ=1 (perfectly correlated)
σ p = w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2 w 1 w 2 ∗ σ 1 σ 2 = w 1 σ 1 + w 2 σ 2 \sigma_p=\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2*\sigma_1\sigma_2} =w_1\sigma_1+w_2\sigma_2 σp​=w12​σ12​+w22​σ22​+2w1​w2​∗σ1​σ2​ ​=w1​σ1​+w2​σ2​

If ρ < 1 \rho < 1 ρ<1 (less perfectly correlated)
σ p = w 1 2 σ 1 2 + w 2 2 σ 2 2 + 2 w 1 w 2 ρ 1 , 2 σ 1 σ 2 < w 1 σ 1 + w 2 σ 2 \sigma_p=\sqrt{w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\rho_{1,2}\sigma_1\sigma_2} < w_1\sigma_1+w_2\sigma_2 σp​=w12​σ12​+w22​σ22​+2w1​w2​ρ1,2​σ1​σ2​ ​<w1​σ1​+w2​σ2​

Diversification means investment in a portfolio of assets whose values do not move in lock-step with one another(i.e., are uncorrelated).

It allows investors to offset specific risk exposures.

The level of investment in a particular financial asset should be based on that asset’s contribution to the distribution of the portfolio’s overall return but not be judged in isolation.

1.4.2 Efficient Frontier of Risky Assets

All attainable portfolios with risky assets

Minimum-variance frontier of risky assets: the investment portfolios of risky assets that provide lminimum variance (the lowest risk) given a certain level of return.

Global minimum-variance portfolio: the investment portfolio that has the lowest variance on minimum-variance frontier of risky assets.

Efficient frontier of risky assets: the investment portfolios that not only provide the lowest risk given a certain level of return (Minimum-variance frontier), but also offer the highest return given certain level of risk.

1.4.3 Optimal portfolio selection

Investor should choose portfolio “X”, the tangent point of indifference curve to efficient frontier, to invests as it supplies the most statisfaction(utility).

2. Capital Market Theory

2.1 Capital Allocation Line(CAL)

2.1.1 Functions of CAL

The portfolios available to an investor through combining the risk-free asset with one risky asset.

R p = w i R i + w r f R r f , σ p = w i σ i R_p=w_iR_i+w_{rf}R_{rf}, \; \sigma_p = w_i\sigma_i Rp​=wi​Ri​+wrf​Rrf​,σp​=wi​σi​

E ( R p ) = R f + E ( R i ) − R f σ i ∗ σ p E(R_p) = R_f+\frac{E(R_i)-R_f}{\sigma_i}*\sigma_p E(Rp​)=Rf​+σi​E(Ri​)−Rf​​∗σp​

2.1.2 Selection among CALs

The CAL with highest Sharpe ratio should be selected, it provides the highest utility among all CALs.

The optimal CAL is tangent to efficient frontier of risky assets.

2.1.3 Optimal portfolio along CAL

Investor should choose portfolio “m” to invest as it supplies the most stratification(utility).

More risk-averse investor (A=4) will select portfolio “j” (less in risky asset), and less risk-averse investor (A=2) will select portfolio “k” (more in risky asset).

CAL( C ) should be select because it provides the highest utility among these three CALs. The Sharpe ratio of CAL( C ) is the highest.
The optimal investor portfolio is combined with the optimal risky portfolio and the risk-free asset.

2.2 Capital Market Line (CML)

Assuming all investors have a homogeneous expectation.

All investors have identical efficient frontier of risky portfolio and identical optimal risky portfilio, which is the market portfolio.

Capital market line(CML) is a special CAL that includes all possible combinations of risk-free asset and market portfolio.

CML is tangent to the efficient frontier at a point representing market portfolio.

The equation of the CML is:
E ( R p ) = R f + E ( R M ) − R f σ M ∗ σ p E(R_p)=R_f+\frac{E(R_M)-R_f}{\sigma_M}*\sigma_p E(Rp​)=Rf​+σM​E(RM​)−Rf​​∗σp​

• The intercept is risk-free rate.
• The slope is Sharpe Ratio of market portfolio.
• S R market -portfolio = E ( R M ) − R f σ M SR_{\text{market -portfolio}}=\frac{E(R_M)-R_f}{\sigma_M} SRmarket -portfolio​=σM​E(RM​)−Rf​​

CML is essentially the efficient frontier for all assets under the assumption that all investors have a homogeneous expectation.

Market portfolio assumes that the market achieves equilibrium and accordingly includes all of the risky assets in the economy weighted by their relative market values.

In practice, broad stock market indices are used to represent the market portfolio.

• E.g., For the United Stated, S&P index or the wider-based Russell 2000.

• E.g., For the U.K and European markets, FTSE 100 and the Euro Stoxx 50.

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  Asset Allocation

  Measurement

  需求侧/投资者

  供给侧/有效组合

  Risk

  Return

  Risk Aversion

  Utility Theory

  Utility Function

  Indifference Curve

  Portfolio Risk and Return

  Efficient Frontier

  Risk Free Asset

  CAL

  Largest Sharpe Ratio

  Homogeneous Expectation

3. Capital Asset Pricing Model

3.1 Assumptions of CAPM

Investors are expected to make decisions solely in terms of expected values and standard deviations of the returns on their portfolios.

Investor plan for the same single holding period.

Allocations can be made in an investment of any partial amount(infinitely divisible).

Including no transaction cost and no tax, or other frictions.

Short sale is allowed unlimitedly.

All participants can borrow and lend at a common risk-free rate.

Investor have homogeneous expectations or beliefs.

Any individual investor’s allocation decision cannot change the market prices(price taker).

All assets, including human capital, are tradable.

3.2 Systematic Risk & Unsystematic Risk

3.2.1 Systematic risk

The risk affects the entire market or economy, which cannot be avoided and is inherent in the overall market.

It is caused by macro factors: interest rates, GDP growth, supply shocks.

It is also named non-diversifiable risk or market risk,

Investor would be only rewarded for bearing systematic risk.

3.2.2 Unsystematic Risk

The risk that can be reduced or eliminated by holding well-diversified portfolios.

It is also named firm-specific risk or idiosyncratic risk.

Investor would not be rewarded for bearing unsystematic risk as it could be eliminated through diversification.

As more diversification is made within the portfolio, systematic risk would not change while unsystematic risk would decrease.

3.2.3 Measurement of systematic risk

Systematic risk can be measured by Beta( β \beta β) of the asset, which represents how sensitive an asset’s return is to the market as a whole.

From an investor’s perspective, β \beta β represents the portion of an asset’s total risk that cannot be diversified away and for which investors will expect compensation.
β i = C o v ( R i , R m ) σ m k t 2 = ρ i , m σ i σ m σ m 2 = ρ i , m σ i σ m \beta_i=\frac{Cov(R_i,R_m)}{\sigma^2_{mkt}} = \frac{\rho_{i,m}\sigma_i\sigma_m}{\sigma^2_m}=\rho_{i,m}\frac{\sigma_i}{\sigma_m} βi​=σmkt2​Cov(Ri​,Rm​)​=σm2​ρi,m​σi​σm​​=ρi,m​σm​σi​​

β \beta β can take on positive or negative values, depending on how an asset’s returns relate to those of the market portfolio.

A risk-free asset will have a beta of zero, because its returns are completely uncorrelated to the returns on the market portfolio.

The market portfolio has a beta of 1, because it is perfectly correlated with itself and has the same variance.

β m k t = C o v ( R i , R m ) σ m k t 2 = σ m k t 2 σ m k t 2 = 1 \beta_{mkt}=\frac{Cov(R_i,R_m)}{\sigma^2_{mkt}} = \frac{\sigma_{mkt}^2}{\sigma_{mkt}^2}=1 βmkt​=σmkt2​Cov(Ri​,Rm​)​=σmkt2​σmkt2​​=1

3.3 Capital asset pricing model(CAPM)

E ( R i ) = R f + β i [ E ( R m ) − R f ) ] E(R_i)= R_f+\beta_i[E(R_m)-R_f)] E(Ri​)=Rf​+βi​[E(Rm​)−Rf​)]

• E ( R i ) E(R_i) E(Ri​): expected return on risky asset i i i,
• E ( R m ) − R f E(R_m)-R_f E(Rm​)−Rf​: market portfolio risk premium.
• β i \beta_i βi​: systematical risk of asset i i i.
• β i [ E ( R m ) − R f ) ] \beta_i[E(R_m)-R_f)] βi​[E(Rm​)−Rf​)]: beta-adjusted risk premium on risky-asset i i i. expected return premium above the risk-free rate(as required by investors).

Rewriting the CAPM in terms of σ i \sigma_i σi​, ρ i , m \rho_{i,m} ρi,m​, σ m \sigma_m σm​ gives:
E ( R i ) = R f + σ i ρ i , m [ E ( R m ) − R f σ m ] E(R_i)= R_f+\sigma_i\rho_{i,m}[\frac{E(R_m)-R_f}{\sigma_m}] E(Ri​)=Rf​+σi​ρi,m​[σm​E(Rm​)−Rf​​]

This equation shows that excess expected return is the product of the systematic component of risk (i.e., σ i \sigma_i σi​, ρ i , m \rho_{i,m} ρi,m​) and the unit price of risk (i.e., [ E ( R m ) − R f σ m ] [\frac{E(R_m)-R_f}{\sigma_m}] [σm​E(Rm​)−Rf​​]).

3.4 Security Market Line (SML)

In a world where the market is in equilibrium and is expected to remain in equilibrium, no investor can achieve an abnormal return.(i.e., an expected return greater that return predicted by the CAPM risk-return relationship.)

All securities will lie on the Security Market Line.

In the real world, stocks and portfolios may yield a return in excess of ,or below, the return with fair compensation for risk exposure.

A graphical representation of the CAPM with beta on the x-axis and expected return on the y-axis. Intercept is R f R_f Rf​, slope is the market risk premium ( R m − R f ) (R_m-R_f) (Rm​−Rf​).

• Any asset that are properly priced plots on SML( A ).
• Any asset that are overpriced plots below SML( B ).
• Any asset that are underpriced plots above SML( C ).
  

CAPM theory asserts that investors would increase their allocations to asset C, driving up its price and decreasing its expected return.

Similarly, investors would decrease their allocaion to asset B, driving down its price and increasing its expected return.

The reallocation by investors would continue until both assets fell on the capital market line. (meaning that the market has reached equilibrium).

3.5 CML vs. SML

4. Performance Measures

4.1 Sharpe Performance Index (SPI, Sharpe Ratio)

The ratio of the mean excess return on portfolio i i i to the standard deviation of the returns of portfolio i i i.
S P I = E ( R i ) − R f σ i SPI=\frac{E(R_i)-R_f}{\sigma_i} SPI=σi​E(Ri​)−Rf​​

A measure of excess return per unit of risk (total risk), the higher is better.

• SPI greater than the slope of the CML: superior performance to equilibrium return.
• SPI below the slope of the CML: inferior performance to equilibrium return.
  

4.2 Treynor Performance Index (TPI)

The ratio of the mean excess return on portfolio i i i to the Beta( β \beta β) of portfolio i i i.

T P I = E ( R i ) − R f β i TPI=\frac{E(Ri)-R_f}{\beta_i} TPI=βi​E(Ri)−Rf​​

A measure of excess return per unit of risk (systematic risk), the higher is better.

For a well-diversified portfolio, beta is widely accepted as an appropriate measure of risk.

In equilibrium, T P I TPI TPI will be constant across all risky assets and equals to:

T P I = E ( R i ) − R f β i = E ( R m ) − R f TPI = \frac{E(R_i)-R_f}{\beta_i}=E(R_m)-R_f TPI=βi​E(Ri​)−Rf​​=E(Rm​)−Rf​

• E ( R m ) − R f E(R_m)-R_f E(Rm​)−Rf​ also called the alpha measure.
• TPI is greater than the alpha measure is considered to have a positive alpha (indicting superior performance) and vice versa.
• T P I > E ( R m ) − R f TPI>E(R_m)-R_f TPI>E(Rm​)−Rf​, superior performance
• T P I < E ( R m ) − R f TPI<E(R_m)-R_f TPI<E(Rm​)−Rf​, inferior performance

4.3 Jensen’s Performance Index (JPI)

The difference between actual return and return required to compensate for systematic risk (CAPM), also called Jensen’s Alpha.

J P I JPI JPI is like T P I TPI TPI, as both measures assume investors hold well-diversified portfolios.
α i = R i − [ R f + β ( R m − R f ) ] \alpha_i =R_i-[R_f+\beta(R_m-R_f)] αi​=Ri​−[Rf​+β(Rm​−Rf​)]

4.4 Sortino Ratio

The Sortino ratio (SR) is modification of S P I SPI SPI. Both ratios measure the risk-adjusted return of an asset or portfolio.

Sortino ratio is more appropriate for a case where returns are not symmetric.

S R = R p − T 1 N ∑ t = 1 N m i n ( 0 , R p t − T ) 2 SR=\frac{R_p-T}{\sqrt{\frac{1}{N}\sum^N_{t=1}min(0,R_{pt}-T)^2}} SR=N1​∑t=1N​min(0,Rpt​−T)2 ​Rp​−T​

• The denominator is the downside deviation, as measured by the standard deviation of negative returns.
• T T T is the target or required rate of return fro the investment strategy, also known as MAR or minimum accepted rate of return.
• T T T may be set to the risk free rate or another hurdle rate.

4.5 Tracking Error

The standard deviation of return difference between the portfolio and the benchmark.
T E = σ R p − R b TE=\sigma_{R_p-R_b} TE=σRp​−Rb​​

4.6 Information Ratio

The residual return of the managed portfolio relative to its benchmark divided by the tracking error.

I R = E ( R p ) − E ( R B ) σ R p − R B IR=\frac{E(R_p)-E(R_B)}{\sigma_{R_p-R_B}} IR=σRp​−RB​​E(Rp​)−E(RB​)​

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