1.Markowitz The basic idea of
Risk can be measured in a sense .
Various risks may inhibit each other , Or maybe “ hedging ”. therefore , Don't invest “ Put the eggs in a basket ”, And we need to decentralize .
In some way “ Optimal investment " In the sense of , A large return means a greater risk .
2.Markowitz Model overview
Markowitz at 1952 Proposed in “ Mean variance combination model ” Under the assumption that securities lending is prohibited and there is no risk-free lending , Find out the effectiveness of portfolio by the mean and variance of individual stock return in portfolio
boundary ( Efficient
Frontier), That is, the portfolio with the minimum variance under a certain level of return , And the conclusion is that investors only choose portfolio on the effective boundary . According to the concept of Markowitz portfolio , To invest

Minimum portfolio risk , Apart from diversifying into different stocks , Stocks with low correlation coefficient should also be selected . therefore , Markowitz's mean variance combination model ” It's not just about diversifying money into different kinds of stocks , It also implies that funds should be invested in stocks of different industries . At the same time, Markowitz mean - Variance model is also a normative mathematical model to provide a technical path to determine the effective boundary .
Implementation method ：
profit —— Expected return of portfolio
risk —— Variance of portfolio
first , Two related characteristics of a portfolio are : (1) Its expected rate of return (2) A measure of the degree to which a possible rate of return deviates from its expectation , Variance as a measure is the easiest to deal with in analysis .
secondly , Rational investors will choose and hold efficient portfolios , That is, the portfolio with the maximum expected return at a given risk level , Or portfolios that minimize risk at a given expected return level .

again , Through the expected rate of return on a security , The variance of return rate and the relationship between return rate of one security and other securities ( Measure with covariance ) Proper analysis of these three kinds of information , It is theoretically feasible to identify the effective portfolio .

last , By solving quadratic programming , The set of effective portfolios can be calculated , The calculation results indicate the proportion of various securities in the investors' funds , To achieve portfolio effectiveness —— That is, maximizing the expected return on a given risk , Or minimize risk for a given expected return .
3. hypothesis
① Single period investment

Single period investment refers to investors' investment at the beginning of the period , Return at the end of the period . Single period model is an approximate description of reality , Such as zero interest bond , Investment in European options, etc . Although many problems are not single period models , But as a simplification , Yes : The analysis of single period model becomes the basis of our analysis of multi period model .
② Investors know the probability distribution of investment return in advance , And the yield satisfies the condition of normal distribution .
③ The utility function of investors is quadratic , Namely u(W)=a+bW+CW2u (W)=a+bW+CW^2u(W)=a+bW+CW2.
④ Investors with expected rate of return ( Mean return ) To measure the overall level of future real yield , Variance of yield (
Or standard deviation ) To measure the uncertainty of yield ( risk ), Therefore, investors only care about the expected return and variance of investment in decision-making .
⑤ Investors are insatiable and risk averse , Follow the principle of dominance , Namely : At the same risk level , Select securities with high yield ; At the same yield level , Choose less risky securities .
4. Price and rate of return
For single investment , Assumed at time 0 At price S0S_0S0​ Purchase an asset , At time 1 Gain from selling the asset S1S_1S1​, Then return on investment r=(S1−S0)/S0
r=(S_1-S_0)/S_0r=(S1​−S0​)/S0​.
For a portfolio , Its rate of return can be calculated in the same way ：
rp=(W1−W0)/W0⇒W0(1+rp)=W1 r_p=(W_1-W_0)/W_0\Rightarrow W_0(1+r_p)=W_1 rp​=(W1​−
W0​)/W0​⇒W0​(1+rp​)=W1​

W0W_0W0​ remember t=0t=0t=0 The composite price of the securities included in the portfolio ,W1W_1W1​ remember t=1t=1t=1 The composite price of the securities included in the portfolio .
We note that , Investors must t=0t=0t=0 Always make decisions about what combination to buy . In doing so , For most of the various combinations considered , Investors don't know W1W_1W1​
Value of , Because they don't know what the return on these portfolios is . thus , According to Markowitz's Theory , Investors should consider the return rate of any of these combinations as a random variable in statistics ; Such variables can be described by their matrices , Two of them are expected ( Or mean ) And standard deviation .
5. Expected return on securities
5.1 Expectations for individual securities
E(r)=∑sPr(s)r(s) E(r)=\sum_sPr(s)r(s) E(r)=s∑​Pr(s)r(s)

E(r)—— Expected value of yield ;r(s)——s Rate of return in state ;Pr(s)——r(s) Probability of state occurrence
E(r)—— Expected value of yield ;r(s)——s Rate of return in state ;Pr(s)——r(s) Probability of state occurrence E(r)—— Expected value of yield ;r(s)——s Rate of return in state ;Pr(s
)——r(s) Probability of state occurrence
E(rp)=∑i=1NxiE(ri) E(r_p)=\sum_{i=1}^Nx_iE(r_i) E(rp​)=i=1∑N​xi​E(ri​)

5.2 Expected return of a portfolio

The expected return of a portfolio is the weighted average of the expected return of the securities it contains , Weight by proportion . The contribution of each security to the expected return of the portfolio depends on its expected return , And its share in the initial value of the portfolio , And nothing to do with anything else . that , One just wants to
Investors with the largest expected return will hold a security , This kind of security is the one with the largest expected return . Few investors do , Few investment advisers offer such an extreme
proposal . contrary , Investors will diversify their investment , That is, their portfolio will contain more than one security . This is because decentralization reduces the risk measured by standard deviation .
5.3.1 variance —— Variance of expected return of a security
The expected rate of return of a security describes the average rate of return weighted by probability . But it's not enough , We also need a useful risk measure , It should consider the possibilities in some way “ bad ” Probability of results and
“ bad ” Magnitude of the result . The probability of a large number of different possible results of substitution measures , The risk measure will estimate the possible deviation between the actual result and the expected result in some way , Variance is such a measure , Because it estimates the possible deviation between the actual rate of return and the expected rate of return .

In securities investment , One - It is generally believed that the distribution of investment income is symmetrical , That is to say, the possibility that the actual income is lower than the expected income is the same as the possibility that the actual income is higher than the expected income . The larger the deviation between the actual yield and the expected yield , The greater the risk of investing in the security , So the risk to a single security , Usually expressed as variance or standard deviation in Statistics .
Follow the above representation , The variance of a security in this period is the deviation of the possible future return from the expected return ( Usually called dispersion ) Weighted average of square of , Probability that the weight is the corresponding possible value . Remember the variance is σ2
\sigma^2σ2, Namely :
σ2=∑sPr(s)[r(s)−E(r)]2 \sigma^2=\sum_sPr(s)[r(s)-E(r)]^2 σ2=s∑​Pr(s)[r(s)−E(r)]
2

5.3.2 variance —— Variance of expected return of two securities portfolios
The variance is σ1,σ2\sigma_1,\sigma_2σ1​,σ2​ Two assets of w1,w2w_1,w_2w1​,w2​ 's weight makes up a portfolio , Variance is ：
σp2=w12σ12+w22σ22+2w1w2cor(r1,r2)
\sigma_p^2=w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2cor(r_1,r_2)σp2​=w12​σ12​+w22​
σ22​+2w1​w2​cor(r1​,r2​)

5.4.1 covariance
Covariance is a statistical measure of the relationship between two random variables , That is, it measures two random variables , Such as securities AAA and BBB The interaction between the rate of return of .
σAB=cov(rA,rB)=E(rA−E(rA))(rB−E(rB))
\sigma_{AB}=cov(r_A,r_B)=E(r_A-E(r_A))(r_B-E(r_B))σAB​=cov(rA​,rB​)=E(rA​−E(rA​)
)(rB​−E(rB​))

A positive covariance indicates that the return of a security tends to move in the same direction . for example , One
The situation of higher than expected return of one security is likely to be accompanied by the situation of higher than expected return of another security . A negative covariance indicates the tendency of a security to move against another security . for example , The situation of one security's higher than expected return is likely to be accompanied by the situation of another security's lower than expected return . A relatively small or 0 The covariance of the value indicates that there is only a small or no interaction between the two securities .
5.4.2 correlation coefficient
negotiable securities AAA and BBB Correlation coefficient of ：
ρAB=σABσAσB \rho_{AB}=\frac{\sigma_{AB}}{\sigma_A\sigma_B} ρAB​=σA​σB​σAB​​

Completely negative correlation will make the risk disappear
Full positive correlation does not reduce risk
stay -1.0 and +1.0 Correlation between can reduce risk but not all .
5.4.3 Variance covariance matrix of multiple portfolio
σp2=∑i=1n∑j=1nwiwjσij=w′Qw
\sigma_p^2=\sum_{i=1}^n\sum_{j=1}^nw_iw_j\sigma_{ij}=w'Qwσp2​=i=1∑n​j=1∑n​wi​wj​
σij​=w′Qw

Q=[σ11⋯σ1N⋮⋯⋮σN1⋯σNN]
Q=\begin{bmatrix}\sigma_{11}&\cdots&\sigma_{1N}\\\vdots&\cdots&\vdots\\\sigma_{N1}&\cdots&\sigma_{NN}\end{bmatrix}
Q=⎣⎢⎡​σ11​⋮σN1​​⋯⋯⋯​σ1N​⋮σNN​​⎦⎥⎤​

6. Variance of portfolio and risk decentralization
（ One ） The causes of portfolio risk dispersion
Assume there are securities on the market 1,2,3,⋯,N1,2,3,\cdots,N1,2,3,⋯,N, negotiable securities iii The expected rate of return is EiE_iEi​, Variance is σi\sigma_iσi​
, negotiable securities iii And securities jjj The covariance of is σij\sigma_{ij}σij​, The investor's portfolio is ( Investment in securities iii Proportion of )：wiw_iwi​,∑i=1nwi=1
\sum_{i=1}^nw_i=1∑i=1n​wi​=1.
Then the expected return and variance of the portfolio are ：
∑i=1nwiE(ri)=E(rp)σp2=∑i=1n∑j=1nwiwjσij \sum_{i=1}^nw_iE(r_i)=E(r_p)\\
\sigma^2_p=\sum_{i=1}^n\sum_{j=1}^nw_iw_j\sigma_{ij}i=1∑n​wi​E(ri​)=E(rp​)σp2​=i
=1∑n​j=1∑n​wi​wj​σij​

（ Two ） A case study of expected return and risk of portfolio
A The value of a company's stock is sensitive to the price of sugar . Over the years , When Caribbean Sugar production falls , The price of sugar skyrocketed , and A The company will suffer huge losses , See the table below .

Normal year of sugar production Normal year of sugar production Abnormal year
bull market bear market Sugar production crisis
probability 0.5 0.3 0.2
Yield % 25 10 -25
B Analysis of the company's stock situation ：

Normal year of sugar production Normal year of sugar production Abnormal year
bull market bear market Sugar production crisis
probability 0.5 0.3 0.2
Yield % 1 -5 35
Suppose an investor considers the following assets to choose from , One is holding A Shares in the company , One is to buy risk-free assets , Another is holding B Shares in the company . Known investors 50% Held by A Shares in the company , in addition
50% How to choose . The return rate of risk-free assets is 5%.

Options Average yield Yield variance
Total investment A Company shares 10.50% 18.90%
Total investment B Company shares 6.0% 14.7%
Half invested in treasury bills , half A shares 7.75% 9.45%
Half invested in A shares , half B shares 8.25% 4.83%
The influence of covariance on portfolio risk : Positive covariance improves the variance of portfolio , Negative covariance reduces the variance of portfolio , It stabilizes the return on the portfolio .
Methods of risk management : Hedging —— Purchase of assets negatively related to existing assets , This negative correlation makes the hedged asset have the nature of reducing risk .
Adding risk-free assets to portfolio is a simple risk management strategy , Hedging strategy is a powerful way to replace this strategy .

7.Markowitz The mathematical model of portfolio selection
Suppose there is nnn Securities , Their yields are random variables r1,r2,⋯ ,rnr_1,r_2,\cdots,r_nr1​,r2​,⋯,rn​. Portfolio means nnn
A combination of securities , It can use one in Mathematics nnn Dimension vector w=(w1,w2,⋯ ,wn)w=(w_1,w_2,\cdots,w_n)w=(w1​,w2​,⋯,wn​)
To represent , Where real number wiw_iwi​ For and on behalf of iii The proportion of the price of securities in the total value , This portfolio www The rate of return of will be a random variable ：
rp=∑i=1nwiri r_p=\sum_{i=1}^nw_ir_i rp​=i=1∑n​wi​ri​

Markowitz The question to consider is how to determine wiw_iwi​, Make portfolio www At expected rate of return E[rp]E[r_p]E[rp​] One
timing , risk ( Variance or standard deviation of yield ) minimum .
order ：μi=E(ri),Vij=cov(ri,rj)\mu_i=E(r_i),V_{ij}=cov(r_i,r_j)μi​=E(ri​),Vij​=cov(r
i​,rj​)
min σ2=∑i,j=1nVijwiwjs.t. ∑i=1nwi=1μw=wTμi=w1μ1+⋯+wnμn=μˉ
min\,\sigma^2=\sum_{i,j=1}^nV_{ij}w_iw_j\\ s.t.\,\sum_{i=1}^nw_i=1\\
\mu_w=w^T\mu_i=w_1\mu_1+\cdots+w_n\mu_n=\bar\muminσ2=i,j=1∑n​Vij​wi​wj​s.t.i=1∑n
​wi​=1μw​=wTμi​=w1​μ1​+⋯+wn​μn​=μˉ​

Right and non systematic risks eliminated by portfolio , Systematic risk cannot be eliminated

Unsystematic risk is the unique risk of an enterprise , For example, enterprises fall into legal disputes , strike , New product development failure , wait . It can be called dispensable risk , Specific risks , Specific asset risk . Unsystematic risk is mainly reduced through decentralization , As a result, there will be little unsystematic risk in a portfolio of many assets .

System risk refers to the risk taken by the whole market , Such as the economic boom , Changes in the overall market interest rate level and other risks arising from changes in the overall market environment . It can be called undivided risk , market risk . Systemic risk affects all assets , Can't be removed by decentralization .

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