BAHACF3/BAHACFWP4

Portfolio Analysis
投资组合分析

Lecture 1 Introduction and Mathematical Foundations
第 1 讲引言和数学基础

Introduction
介绍

Module Learning Outcomes
模块学习成果

Learning Outcome 1
学习成果 1

Demonstrate a critical awareness of the theoretical underpinnings of modern portfolio theory and analysis and its application within modern financial institutions:
展示对现代投资组合理论和分析的理论基础及其在现代金融机构中的应用的批判性认识：

Learning Outcome 2
学习成果 2

Evaluate financial information in order to allocate funds across asset classes to form an optimal portfolio bearing in mind individual and investor preferences regarding return requirements and risk tolerance;
评估财务信息，以便在资产类别中分配资金，以形成最佳投资组合，同时牢记个人和投资者对回报要求和风险承受能力的偏好;

Learning Outcome 3
学习成果 3

Compare, appraise and apply the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT) to modern financial problems;
比较、评估和应用资本资产定价模型（CAPM） 和套利定价理论 （APT） 解决现代金融问题;

Learning Outcome 4
学习成果 4

Analyse appropriate benchmarks for evaluation of fund performance and critically appraise and apply appropriate performance measurement tools;
分析评估基金表现的适当基准，并批判性地评估和应用适当的绩效衡量工具;

Learning Outcome 5
学习成果 5

Appraise the behavioural characteristics of individual and institutional investors and demonstrate the ability to evaluate the relevance of behavioural finance within practical settings.
评估个人和机构投资者的行为特征，并展示在实际环境中评估行为金融学相关性的能力。

What is an Investment Portfolio?
什么是投资组合？

Combination of different asset types e.g. equity, bonds, property;
不同资产类型的组合，例如股票、债券、房地产;

Combination of different assets within each asset type e.g. shares in different companies;
每种资产类型中不同资产的组合，例如不同公司的股票;

Mix of domestic and overseas assets;
国内和海外资产的混合;

Mix of assets denominated in different currencies;
以不同货币计价的资产组合;

Mix of long-term and short-term assets;
长期和短期资产的组合;

Mix of risky and low-risk (or zero risk) assets.
风险和低风险（或零风险）资产的组合。

Some Key Questions

Why do people invest?
人们为什么投资？

What do people expect from an investment?
人们对投资有什么期望？

Why have different investors got different attitudes to their investment especially in relation to risk?
为什么不同的投资者 对他们的投资有不同的态度，尤其是在风险方面？

How do investors select individual investments?
投资者如何选择个人投资？

How do investors determine asset allocation within portfolios?
投资者如何确定投资组合中的资产配置？

Why do investors alter their portfolios on an ongoing basis?
为什么投资者不断改变他们的投资组合？

What criteria do investors apply in altering the assets they hold on an ongoing basis?
投资者在持续更改他们持有的资产时适用什么标准？

2. Mathematical foundations
2. 数学基础

There are three key measures that investors should be aware of in selecting investments
这是投资者在选择投资时应注意的关键措施

Return
返回

Risk (Variance)
风险 （方差）

Covariance/Correlation between asset types and individual assets
资产类型与单个资产之间的协方差/C或关系

2.1 Expected return on an asset
2.1 资产的预期回报

What level of return does an investor expect from an investment?
投资者期望从投资中获得的回报水平是多少？

This depends on the state of the economy.
这取决于经济状况。

Suppose there are four possible economic states
假设有四种可能的经济状态

The expected return is then the weighted average of all possible returns, where the weights are the probability of each outturn
预期回报是所有可能回报的加权平均值，其中权重是每次结果的概率

Economic      % Return
经济 % Return

State    Direct Co.   Odd Co.
国家直接 Co.Odd Co.

Depression    -20     5
抑郁症 20 5

Recession    10     20
衰退1020

Normal    30     -12
正常 30-12

Boom     50     9
吊杆509

Assume all economic states are equally likely (0.25)
假设ll 经济状态的可能性相同 （0.25）

Expected return
预期回报

Direct Co.
直接 Co.

0.25 x (-0.2) + 0.25 x (0.1) + 0.25 x (0.3) + 0.25 x (0.5) = 0.175
0.25 x （-0.2） + 0.25 x （0.1） + 0.25 x （0.3） + 0.25 x （0.5） = 0.175

Alternatively we could have (in this case) added up the four possible returns and divided by 4
或者，我们可以（在这种情况下）将四种可能的返回值相加并除以 4

-0.20 + 0.10 + .0.30 + 0.50  = 0.175

  4

Odd Co.
奇数 Co.

0.25 x (0.05) + 0.25 x (0.2) + 0.25 x (-0.12) + 0.25 x (0.09) = 0.055
0.25 x （0.05） + 0.25 x （0.2） + 0.25 x （-0.12） + 0.25 x （0.09） = 0.055

0.05 + 0.20 + -0.12 + 0.09  = 0.055
0.0 5 + 0。20 + -0.12 + 0。09 = 0.055

  4

2.2 Variance of an asset
2.2 资产的差异

Denoted by ²
用 ² 表示d

Measure of risk
风险度量

Square root is standard deviation 
平方根是标准差 

Calculated as weighted average of squared deviations from expected return, where weights are the probability of each one
计算为与预期回报的平方偏差的加权平均值，此处的 w权重是每个 1 的概率

Direct Co. => Expected return = 0.175
直接公司 => 预期回报 = 0.175

Economic  Return Deviation from Square of deviation
经济 ReturnDeviation 来自deviation 的 Square

State   Expected Return from Expected Return
State预期回报 来自 Expected Return

Depression  -0.2   -0.375  0.140625
抑郁 -0.2 -0.375 0.140625

Recession  0.1   -0.075  0.005625
衰退 0.1 -0.075 0.005625

Normal  0.3   0.125  0.015625
正常0.3 0.125 0.015625

Boom   0.5   0.325  0.105625
动臂 0.5 0.325 0.105625

² = 0.25 x 0.140624 +

0.25 x 0.005625 +

0.25 x 0.015625 +

 0.25 x 0.105625 =  0.066875

=  0.066875 =  0.258602
= ， 0.066875 = 0.258602

Odd Co.
奇数 Co.

² = 0.013225

=  0.013225 = 0.115
= ， 0.013225 = 0.115

Direct Co. has higher return but higher risk than Odd Co.
Direct Co. h的回报率较高，但风险高于 Odd Co。

2.3 Covariance and Correlation
2.3 协变和 C或相关

What is the relationship between the return on one asset and the return on another?
一种资产的回报与另一种资产的回报之间有什么关系？

In other words, if the return on one asset rises, does the return on the other asset rise or fall and does it rise or fall by a similar percentage each time?
换句话说，如果一种资产的回报率上升，另一种资产的回报率是上升还是下降，每次都是上升还是下降相似的百分比？

Covariance and correlation measure this relationship.
协方差和相关性衡量这种关系。

2.3.1 Covariance
2.3.1 协方差

Denoted by _xy
用 _xy 表示

Calculated as the weighted average of the product of the deviations from expected return for each asset, where weights are the probability of each economic state
计算为 每种资产与预期回报的偏差s 的乘积的加权平均值，其中权重是每种经济状态的概率

e.g. in the case of ‘Depression’, Direct Co. is -0.375 away from its average while Odd Co. is +0.005 away from its average.
例如，在“抑郁症”的情况下，Direct Co. 与其平均值相差 -0.375，而 Odd Co. 与其平均值相差 +0.005。

Multiply (-0.375) x (0.005) and multiply it by 0.25 = 0.00046875
乘以 （-0.375） x （0.005） 并乘以 0.25 = 0.00046875

Repeat for the other three economic states.
对其他 three economic states 重复。

Add the weighted product of square deviations
加上平方差的 加权结果

Here _do = -0.004875
这里 do = -0.004875

Intuition regarding covariance
关于协方差的直觉

If Odd Co. is generally above its average when Direct Co. is above its average and Odd Co. is generally below its average when Direct Co. is below its average, then the products (of deviations) will generally be positive and so covariance will be positive
如果奇数 Co. is 通常在 Direct Co. is 高于其平均值时 a 达到其平均值，而奇数 Co. is 通常在 Direct Co. s 低于其平均值，则（偏差的）乘积通常为正，因此协方差为正

If the deviations tend to take opposite signs, then the covariance will be negative
如果偏差倾向于采用相反的符号，则协方差将为负

In the above example, there is no definite pattern and the covariance is close to zero
在上面的例子中，没有确定的模式，协方差接近于零

The problem with covariance is that it can take on any value => difficult to interpret what the value means => correlation solves this problem
协方差的 p roblem 是它可以取任何值 => 难以解释该值的含义 => 相关性解决了这个问题

2.3.2 Correlation
2.3.2 关联

Denoted by _xy
用 _xy 表示

Correlation is a ‘standardised’ version of covariance
相关性是协方差的“标准化版本”

      _xy

_xy  =   --------------------------

    _x x _y

Note that in the equation above x is the multiplication sign
请注意，在上面的方程中 x 是乘号

Since _x and _y are always positive, then _xy will have the same sign as _xy
由于 _x 和 _y 总是正数，因此 _xy 将与 _xy 具有相同的符号

It can be shown that _xy is bounded by the values +1 and -1 i.e.
可以证明 _xy 以值 +1 和 -1 为界，即

    -1 ≤ _xy ≤ +1
-1 ≤_xy≤ +1

+1 => perfect positive correlation
+1 => 完全正相关

-1 => perfect negative correlation
-1 => 完全负相关

In the above example _xy = - 0.1639261
在上面的例子中_xy = - 0.1639261

2.4 Portfolio of different assets
2.4 不同资产的投资组合

Objective is normally to build a portfolio with the highest expected return and the lowest level of risk
目标通常是构建具有最高预期回报和最低风险水平的投资组合

e.g. from previous data
例如，从以前的数据

A portfolio made up of 100% Direct Co. has
由 100% DirectCo. h组成的投资组合

E(R_p) = .175 and _p = .2586
E（R_p） = .175 和 _p = .2586

A portfolio made up of 100% Odd Co. has
由 100% Odd Co. h as 组成的投资组合

E(R_p) = .055 and _p = .115
E（R_p） = .055 和 _p = .115

What would the E(R_p) and _p be if we built a portfolio with Direct Co. and Odd Co. in some proportions ?
如果我们在 Direct Co. 和 Odd Co. 的投资组合中按一定比例划分，E（R_p） 和 _p 会是多少？

2.4.1 Expected Return on a Portfolio
2.4.1 P 作品集的预期回报

Weighted average of expected returns on individual securities => weights are the proportion of total investment invested in individual securities
单个证券预期回报的加权平均数 => 权重是投资于单个证券的总投资的比例

   E(R_p) =  x_i E(R_i)
E（R， _p） =  x E（R）

     where  x_i = 1
其中  x_i = 1

Example:
示例：

€600 invested in Direct Co. and €400 invested in Odd Co.
600 欧元投资于 DirectCo。 400 欧元投资于 Odd Co。

E(R_p) = 0.6 x .175 + 0.4 x .055 = .127
E（R_p） = 0.6 x .175 + 0.4 x .055 = .127

2.4.2 Variance and standard deviation of a portfolio
2.4.2 投资组合的方差和标准差

Depends on:
取决于：

Variance of constituent stocks;
成分股的方差;

Proportion invested in each stock;
投资于每只股票的比例;

Number of assets;
资产数量;

Correlation/covariance between assets.
资产之间的相关性/协方差。

Two asset case:
两种资产案例：

Direct Co. and Odd Co.
Direct Co. 和 Odd Co.

²_p = x²_d.²_d + x²_o.²_o + 2x_d.x_o._do
²_p =^{x 2}_d.²_d + x²_o.²_o + 2x_{d.x o}._do

where x_{d +} x_o = 1
w这里x_{d +} x_o = 1

This can be re-written in a more normal format (i.e. in the format that matches the available data). Just replace the yellow with the green
这可以以更正常的格式（即与可用数据匹配的格式）重写。只需将黄色替换为绿色

²_p = x²_d.²_d + x²_o.²_o + 2x_d.x_o.ρ_do_d_o
²_p = ^{x 2}_d.²_d+^{x 2}_o.²_o + 2x_{d.x o}.ρ_do_d_o

where x_d + x_o = 1
其中x_d+ x_o = 1

Example:
示例：

Using previous data
使用以前的数据

Variance of portfolio
投资组合的差异

= (0.6)² (0.066875) + (0.4⁾² (0.013225) + 2 (0.6) (0.4) (-0.004875)
= （0.6）² （0.066875） + （0.4^）2 （0.013225） +2 （0.6） （0.4） （-0.004875）

= 0.023851

or
或

= (0.6)² (0.066875) + (0.4⁾² (0.013225) + 2(0.6)(0.4)(0.2586)(0.115)(- 0.1639261)
= （0.6）² （0.066875） + （0.4^）2 （0.013225） + 2（0.6）（0.4）（0.2586）（0.115）（- 0.1639261）

= 0.023851

_p= √0.023851 = 0.154438
_p= √0.023851 = 0.154438

Standard deviation of portfolio made up of
投资组合的标准差

  60% Direct Co. and 40% Odd Co.
60% 直接 Co. 和 40% 奇数 Co.

From above = 0.154438
从上方 = 0.154438

If we calculate the weighted average of the standard deviations of the two assets
如果我们计算 两种资产的标准差的加权平均值

(0.6) x (0.2586) + (0.4) x (0.115) = 0.20116
（0.6）x （0.2586） + （0.4） x（0.1， 1， 5） = 0.20116

The difference between 0.20116 and 0.154438 is due to diversification
0.20116 和 0.154438 之间的差异是由于多元化

The diversification effect
多元化效应

Extent of the diversification depends on 
多元化的程度取决于 

As long as  < 1, the standard deviation of a portfolio of two securities is less than the weighted average of the standard deviation of the individual securities
只要  < 1，两种证券的投资组合的标准差小于单个证券的标准差的加权平均值

2.5 Conclusion regarding portfolio variance
2.5 关于投资组合方差的结论

For given asset variances, the lower is the value of the correlation between the two assets, the lower will be the portfolio variance
对于给定的资产差异，两种资产之间的相关性值越低，投资组合差异就越小

Since a correlation of +1 is the maximum possible, the portfolio variance will be at its highest in this case
由于 +1 的相关性是最大可能，因此在这种情况下，投资组合方差将处于最高水平

Since a correlation of -1 is the minimum possible, the portfolio variance will be at its lowest in this case
由于 -1 的相关性是可能的最小值，因此在这种情况下，投资组合方差将处于最低水平

The common sense message is that if two assets are negatively correlated i.e. as one increases, the other decreases, then by combining them in a portfolio the overall variability is reduced
常识性信息是，如果两种资产呈负相关，即随着一种资产的增加，另一种资产的减少，那么通过将它们组合在一个投资组合中，整体可变性会降低

The above is also true so long as the assets are not perfectly positively correlated
只要资产不是完全正相关，上述情况也是如此

3. Impact of different levels of correlation on Portfolio Risk
3. 不同相关性 ls 对投资组合 Risk 的影响

Below are three portfolios made up of two assets in each case.
以下是三种投资组合，每种资产由两种资产组成。

The tables show the impact of combining the assets in different weights.
这些表格显示了将不同权重的资产组合在一起的影响。

Portfolio 1: Asset A and Asset B
投资组合 1：资产 A 和资产 B

Assets A and B have low positive correlation.
资产 A 和 B 的正相关度较低。

Asset A has a low standard deviation
资产 A 的标准差较低

Asset B has a much higher standard deviation
资产 B 的标准差要高得多

Combining them in a portfolio can lead to a maximum risk reduction of 23.29% (based on different allocations measured in 10% ranges)
将它们组合在一个投资组合中，最高可降低 23.29% 的风险（基于在 10% 范围内衡量的不同配置）

Portfolio 2: Asset A and Asset C
投资组合 2：资产 A 和资产 C

Assets A and C have low negative correlation.
资产 A 和 C 的负相关度较低。

Asset A has a low standard deviation
资产 A 的标准差较低

Asset C has a higher standard deviation
资产 C 具有较高的标准差

Combining them in a portfolio can lead to a maximum risk reduction of 37.53% (based on different allocations measured in 10% ranges)
将它们组合在一个投资组合中可以导致最大风险降低 3 7.53%（基于在 10% 范围内衡量的不同配置）

Portfolio 3: Asset B and Asset C
投资组合 3：资产 B 和资产 C

Assets B and C have a stronger negative correlation.
资产 B 和 C 具有更强的负相关性。

Both assets have relatively high standard deviations
两种资产的标准差都相对较高

Combining them in a portfolio can lead to a maximum risk reduction of 52.57% (based on different allocations measured in 10% ranges)
将它们组合在一个投资组合中，最高可降低 52.57% 的风险（基于在 10% 范围内衡量的不同配置）

4. Minimum and Maximum Diversification Impact
4. Minimum 和 Maximum 多元化影响

4.1 Minimum Diversification Impact
4.1 M最小多元化影响

If the correlation between two assets is = +1, then the portfolio standard deviation equals the weighted average of the individual standard deviations, i.e. there is no additional gain in risk reduction.
如果两个评估ts 之间的相关性为 = +1，则投资组合标准差等于单个标准差的加权平均值，即风险降低没有额外的收益。

4.2 Maximum Diversification Impact
4.2 Maximum 多元化影响

If the correlation between two assets is = - 1, then a combination of assets can be found such that the portfolio standard deviation equals zero i.e. risk can be reduced to zero. Clearly this is a theoretical extreme.
如果两个评估ts 之间的相关性为 = - 1，则可以找到资产组合，使得投资组合标准差等于零，即风险可以降低到零。显然，这是一个理论上的极端。

5. Multi-Asset Portfolio
5. 多元资产投资组合

Section 2 above discussed the expected return and standard deviation of a two-asset portfolio. The formulae for calculating these are:
上文第 2 节讨论了双资产投资组合的预期回报和标准差。计算这些的公式是：

Expected Return on a Portfolio
投资组合的预期回报

E(R_p) =  x_i E(R_i)
E（R， _p） =  x E（R）

     where  x_i = 1
其中  x_i = 1

Standard Deviation of returns on a Portfolio
标准 D对 P 作品集的回报

  _______________________

_p = √x²_i.²_i + x²_j.²_j + 2x_i.x_j._ij
_p = √x².² + x².² + 2x.x._ij

where x_i + x_j = 1
其中x+ x= 1

Note:
注意：

Correlation of return between two assets,  _ij
两种资产之间的回报相关性，_ij

     _ij

_ij  =  ----------------------- => _ij= _{ij x}_i x _j

   _i x _j

where
哪里

_i= Standard deviation of returns on Asset
= 资产回报的标准差i

_j= Standard deviation of returns on Asset j
=资产 j 回报的标准差

_ij= Covariance of returns of Asset i and Asset j
_ij=资产 和资产 j 的回报 s 的协方差

The above formulae can be extended to any number of assets. However if the number of assets exceeds three, the mathematics becomes somewhat cumbersome. The relevant formulae for a three-asset portfolio are:
上述公式可以扩展到任意数量的资产。但是，如果资产数量超过 3，则数学运算会变得有些麻烦。三资产投资组合的相关公式为：

Expected Return on a Portfolio
投资组合的预期回报

E(R_p) =  x_i E(R_i)
E（R， _p） =  x E（R）

     where  x_i = 1
其中  x_i = 1

Standard Deviation of returns on a three-asset Portfolio (i, j and k)
标准 D三资产 P投资组合的回报 （， j 和 k）

  ____________________________________________________

_p = √x²_i.²_i + x²_j.²_j + x²_k.²_k + 2x_i.x_j._ij + 2x_i.x_k._ik + 2x_j.x_k._jk
_p = √x².² + x².²+x²_千米。²_k + 2x.x._ij + 2x.x _k._ik + 2x.x _k._jk

where x_i + x_j + x_k = 1
其中x +x+x_k= 1

Exercise:
锻炼： 

This will be done in a practical session but is presented here to allow you to consider it in advance.
这将在实践课程中完成，但在此处介绍是为了让您提前考虑。

Suppose you have the following information about 3 asset classes
假设您有以下有关 3 个资产类别的信息

Correlation Matrix
关联矩阵

Calculate the expected return and standard deviation of the following portfolios
计算以下投资组合的预期回报和标准差

Percentage invested in
投资百分比