6-1
第 6章风险与风险厌恶
Risk and Risk Aversion
6-2
风险与风险厌恶
Risk and Risk Aversion
6.1风险与风险厌恶
6.2 资产组合风险
6-3
风险与风险厌恶
Risk and Risk Aversion
风险的存在意味着可能产生一个以上的结果，
单一前景是指将某一初始财富被用于投资时面临着风险，这一投资机会只产生两种可能的结果。
The presence of risk means that more than
one outcome is possible,A simple prospect is
an investment opportunity in which a certain
initial wealth is placed at risk,and there are
only two possible outcomes,
6-4
W = 100
初始投资额收益 W1 = 150 利润 Profit = 50
收益 W2 = 80 利润 Profit = -20概率 2,1-p =,4
预期收益 E(W) = pW1 + (1-p)W2 =0.6 (150) +0.4(80) = 122
预期盈利 (expected profit)=122-100=22
方差 s2 (variance) = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2
=0.6 (150-122)2 + 0.4(80-122)2 = 1,176,000
标准差 s = 34,293
风险 -不确定收益
Risk - Uncertain Outcomes
6-5
W1 = 150 Profit = 50
W2 = 80 Profit = -201-p =,4100
风险投资 Risky Inv.
无风险投资 Risk Free T-bills Profit = 5
预期盈利 (The expected profit) = 122-100=22
风险溢价 Risk Premium =风险投资预期盈利 22 – 国库券 盈利 5 = 17
风险投资与无风险投资
Risky Investments with Risk-Free Investment
6-6
投资者的风险观 Investor’s view of risk
– 风险厌恶型 Risk Averse
– 风险中性 Risk Neutral
– 风险爱好型 Risk Seeking
效用价值 Utility
效用价值方程 Utility Function
U = E ( r ) -,005 A s 2
A 是投资者风险厌恶系数
A measures the degree of risk aversion
风险厌恶与效用价值
Risk Aversion & Utility
6-7
风险厌恶与价值,投资举例
Risk Aversion and Value,Using the Sample Investment
U = E ( r ) -,005 A s 2
=,22 -,005 A (34%) 2
风险厌恶度 效用值
Risk Aversion A Value
High 5 -6.90
3 4.66
Low 1 16.22
T-bill = 5%
6-8
风险厌恶、风险与收益的权衡
Equilibrium of Risk Aversion,Risk and return
如果证券 A可以无风险的获得回报率为 10％，
而证券 B以 50％的概率获得 20％的收益，
50％的概率的收益为 0，你将选择哪一种证券？
对于一个风险规避的投资者，虽然证券 B的期望收益为 10％，但它具有风险，而证券 A
的无风险收益为 10％，显然证券 A优于证券
B。
6-9
均值方差标准（ Mean-variance criterion)
若投资者是风险厌恶的，则对于证券 A和证券 B，当且仅仅当
22
ABss?
时成立则该投资者认为,A占优于 B”，从而该投资者是风险厌恶性的。
6-10
占优原则
Dominance Principle
1
2 3
4
预期收益 Expected Return
方差或标准差 Variance or Standard Deviation
2占优 dominates 1; 有更高收益 has a higher return
2占优 dominates 3; 有更低风险 has a lower risk
4占优 dominates 3; 有更高收益 has a higher return
6-11
效用和差异曲线
Utility and Indifference Curves
表现投资者对收益和风险权衡的意愿
Represent an investor’s willingness to trade-off return
and risk.
效用数值可以看成是对资产组合排序的一种方法。风险 -
收益曲线越吸引人，资产组合的效用值也就越高。预期收益越高，资产组合得到的效用数值越大，而波动性强的资产组合，其效用数值也低。
The utility score may be viewed as a means of ranking
portfolios,Higher utility values are assigned to portfolios
with more attractive risk-return profiles,Portfolios receive
higher utility scores for higher expected returns and
lower scores for higher volatility.
6-12
效用函数（ Utility function）的例子假定一个风险规避者具有如下形式的效应函数
2( ) 0,0 0 5U E r A s=?
– 其中,A为投资者风险规避的程度。
– 若 A越大，表示投资者越害怕风险，在同等风险的情况下，越需要更多的收益补偿。
– 若 A不变，则当方差越大，效用越低。
6-13
确定性等价收益率（ Certainly equivalent
rate）
为使无风险资产与风险资产具有相同的效用而确定的无风险资产的报酬率，称为风险资产的确定性等价收益率。
由于无风险资产的方差为 0，因此，其效用
U就等价于无风险回报率，因此,U就是风险资产的确定性等价收益率。
6-14
例如：对于风险资产 A，其效用为
2
( ) 0,0 0 5
1 0 % 0,0 0 5 4 4
2%
U E r A s=?
=
=
它等价于收益（效用）为 2％的无风险资产
( ) 2 %fU E r==
结论：只有当风险资产的确定性等价收益至少不小于无风险资产的收益时，这个投资才是值得的。
6-15
Standard Deviation
回报
return
标准差2
6-16
风险厌恶型投资者的无差异曲线
Indifference Curves
预期收益 Expected Return
标准差 Standard Deviation
增加效用 Increasing Utility
6-17
风险中性投资者的无差异曲线
Risk neutral,Indifference Curves
风险中性型的投资者对风险无所谓，只关心投资收益。
期望收益 Expected Return
标准差 Standard Deviation
6-18
风险偏好投资者的无差异曲线
Risk Seeking,Indifference Curves
期望收益 Expected Return
标准差 Standard Deviation
风险偏好型的投资者将风险作为正效用的商品看待，当收益降低时候，
可以通过风险增加得到效用补偿。
6-19
资产组合风险
Portfolio Risk
到现在为止，我们的讨论一直集中在个人全部资产组合的风险与收益上。这样的资产组合是由各种类型的资产组成的，除了在金融市场上的直接投资外，投资者还持有养老基金、以储蓄形式进行的人寿保险计划、住宅，还有并非最不重要的是他们自身技能带来的获利能力（人力资本）。
Investor portfolios are composed of diverse types of
assets,In addition to direct investment in financial
markets,investors have stakes in pension funds,life
insurance policies with savings components,homes,and
not least,the earning power of their skills (human
capital),
6-20
资产组合风险
Portfolio Risk
在评估一个资产组合的风险时，投资者必须考虑到资产收益之间的相互作用。从 根本上说，比如当资产组合的另一部分情况很糟时，通过签定保险合约，交一大笔保 险金可以降低风险。当资产组合中的一部分资产，如房屋或工厂遭受巨大损失时，购 买的火险就派上了用场。
这两种资产（住宅与保险）收益的相互抵消形式稳定了整个资产组合的风险。投资于补偿形式的资产，使之抵消我们可能遇到的某种风险称之为套期保值。
Investors must take account of the interplay between asset returns
when evaluating the risk of a portfolio,At a most basic level,for
example,an insurance contract serves to reduce risk by providing a
large payoff when another part of the portfolio is faring poorly,A fire
insurance policy pays off when another asset in the portfolio—a
house or factory,for example—suffers a big loss in value,The
offsetting pattern of returns on these two assets (the house and the
insurance policy) stabilizes the risk of the overall portfolio,Investing in
an asset with a payoff pattern that offsets exposure to a particular
source of risk is called hedging.
6-21
期望收益
Expected Return
规则 1 在任何情况下，资产的平均或预期收益就是其收益的概率加权平均值。
Rule 1,The return for an asset is the
probability weighted average return in all
scenarios.
=
s
srsPrE )()()(
6-22
期望收益
Expected Return
Normal Year for Sugar Abnormal Year
Bullish Bearish
Stock Market Stock Market Sugar Crisis
糖生产的正常年份 异常年份名 称 股市的牛市 股市的熊市 糖的生产危机概率 Probability 0.50 0.30 0.2
收益率 (%) 25 10 -25
（ 贝斯特 ·凯迪 ）
凯恩收益率 (%) 1 -5 35
E(r贝斯特 ·凯迪 )＝ (0.5× 25)＋ (0.3× 10)＋ 0.2(-25)＝ 10.5%
E(r凯恩 )＝ (0.5× 1)＋ (0.3× -5)＋ 0.2(35)＝ 6%
6-23
收益的方差
Variance of Return
规则 2 资产收益的方差是预期收益的偏差的平方的期望值。
Rule 2,The variance of an asset’s return is the expected
value of the squared deviations from the expected return.
本例中
σ2贝斯特 ·凯迪 =0.5(25-10.5)2+0.3(10-10.5)2+0.2(-25-10.5) 2
=357.25
σ =(357.25)1/2 = 18.9 %。
])()()[( 22=
s
rEsrsPs
6-24
资产组合的收益率
Return on a Portfolio
规则 3 资产组合的收益率是构成资产组合的每个资产收益率的加权平均值，资产组合的构成比例为权重。
Rule 3,The rate of return on a portfolio is a
weighted average of the rates of return of each
asset comprising the portfolio,with the portfolio
proportions as weights.
rp = W1r1 + W2r2
W1 = 资金在证券 1中的比例 Proportion of funds in Security 1
W2 = 资金在证券 2中的比例 Proportion of funds in Security 2
r1 = 证券 1的预期收益 Expected return on Security 1
r2 = 证券 2的预期收益 Expected return on Security 2
6-25
资产组合的收益率
Return on a Portfolio
在本例中，休曼埃克斯的资产组合中 50%投资于贝斯特 ·凯迪公司的股票,50 % 投资于国库券。因此有
The Humanex portfolio is 50% invested in Best stock
and 50% invested in risk-free bills,Therefore,
E(r休曼埃克斯 )＝ 0.5E(r贝斯特 ·凯迪 )＋ 0.5r国库券
＝ (0.5× 10.5)＋ (0.5× 5)＝ 7.75%
6-26
组合风险与无风险资产
Portfolio Risk with Risk-Free Asset
规则 4 当一个风险资产与一个无风险资产相组合时，资产组合的标准差等于风险资产的标准差乘以该资产组合投资于这部分资产上的比例。
Rule 4,When a risky asset is combined with a
risk-free asset,the portfolio standard deviation
equals the risky asset’s standard deviation
multiplied by the portfolio proportion invested in
the risky asset.
ss r i s k y a s s e tr i s k y a s s e tp w?=
6-27
组合风险与无风险资产
Portfolio Risk with Risk-Free Asset
在本例中，休曼埃克斯的资产组合中 50%投资于贝斯特 ·凯迪公司的股票,50 %投资于国库券。因此有
The Humanex portfolio is 50% invested in Best
stock and 50% invested in risk-free bills,
Therefore,
σ休曼埃克斯 ＝ 0.5σ贝斯特 ·凯迪 ＝ 0.5× 18.9＝ 9.45%
6-28
协方差
Covariance
协方差的定义为：
Cov(r贝斯特 ·凯迪,r凯恩 )＝ Σ Pr( s )[r贝斯特 ·凯迪 (s)-E(r
贝斯特 ·凯迪 )][r凯恩 (s)-E(r凯恩 )]
The covariance is defined as,
Cov(rBest,rKane)= Σ Pr(s)[rBest(s)-E(rBest)]
[rKane(s)-E(rKane)]
6-29
协方差
Covariance
我们引用协方差与相关性的概念来量化资产的套期保值或分散化。 协方差 (covariance)测度的是两个风险资产收益的相互影响的方向与程度。正的协方差意味着资产收益同向变动；负的协方差表明它们朝相反的方向变动，譬如，
贝斯特 ·凯迪公司股票与糖凯恩公司股票的关系就是反方向变动的。
To quantify the hedging or diversification potential of an
asset,we use the concepts of covariance and correlation,
The covariance measures how much the returns on two
risky assets move in tandem,A positive covariance
means that asset returns move together,A negative
covariance means that they vary inversely,as in the
case of Best and SugarKane,
6-30
协方差
Covariance
在本例中，由于在下面的表中归纳出当 E(r贝斯特 ·凯迪 )＝
10.5%,E(r凯恩 )＝ 6%时每一情景的收益，我们可以利用 6-4
式来计算协方差。两种股票的协方差为：
Cov(r贝斯特 ·凯迪,r凯恩 )＝ 0.5(25 -10.5)(1-6)＋ 0.3(10-10.5)(-5-6)＋ 0.2(-25 -
10.5)(35-6)＝ -240.5
负的协方差证实了糖凯恩公司股票对贝斯特 ·凯迪公司股票具有的套期保值作用。糖凯恩公司股票的收益与贝斯特 ·凯迪公司股票是呈反方向变动的。
The negative covariance confirms the hedging quality of
SugarKane stock relative to Best Candy,SugarKane’s
returns move inversely with Best’s,
6-31
相关系数
correlation coefficient
相关系数（ correlation coefficient）是比协方差更简便的计算方法。它把协方差的值放在 -1（完全负相关）和＋ 1（完全正相关）之间。两个变量的相关系数等于它们的协方差除以标准差。用希腊字母 ρ 代表相关系数，我们有
ρ (贝斯特 ·凯迪，糖凯恩 )＝ [Cov(r贝斯特 ·凯迪,r凯恩 )]/( σ贝斯特 ·凯迪 σ凯恩 )
＝ -240.5/(18.9× 14.73)＝ -0.86
6-32
相关系数
correlation coefficient
An easier statistic to interpret than the covariance is the
correlation coefficient,which scales the covariance to a
value between -1 (perfect negative correlation) and 1
(perfect positive correlation),The correlation coefficient
between two variables equals their covariance divided by
the product of the standard deviations,Denoting the
correlation by the Greek letter ρ,we find that
ρ (Best,SugarKane) =[cov(rBest,rSugarKane)] /
(σBest* σSugarKane)
=-240.5/(18.9X14.73)＝ -0.86
6-33
规则 5,方差分别是 s12和 s22 的两个风险资产以 w1 和 w2 的权重构成一个资产组合,该资产组合的方差 sp2 为：
Rule 5,When two risky assets with variances s12 and s22,
respectively,are combined into a portfolio with portfolio
weights w1 and w2,respectively,the portfolio variance is
given by:
sp2 = w12s12 + w22s22 + 2W1W2 Cov(r1r2)
Cov(r1r2) =证券 1和证券 2收益的协方差
=Covariance of returns for Security 1 and Security 2
组合风险
Portfolio Risk
6-34
例 计算资产组合方差贝斯特 ·凯迪公司股票与糖凯恩公司股票的权重相等，
w1＝ w2＝ 0.5，
σ 贝斯特 ·凯迪 ＝ 18.9%，
σ 凯恩 ＝ 14.73%，
Cov(r贝斯特 ·凯迪,r凯恩 )＝ -240.5，
我们得到：
σ２ P＝ (0.52× 18.92)+ (0.52× 14.732)+[2× 0.5× 0.5× (-240.5)]＝
23.3
所以,σP＝ (23.3)1/2＝ 4.83%
组合风险
Portfolio Risk
6-35
在本例中 Best 和 糖 Kane用相同的权重
In this example,with equal weights in Best and
SugarKane,
w1=w2=0.5,
σ Best =18.9%,σ Kane=14.73%,
Cov(rBest,rKane)=-240.5,
we find that
σ２ P＝ (0.52× 18.92)+ (0.52× 14.732)+[2× 0.5× 0.5× (-240.5)]
＝ 23.3
So that σP＝ (23.3)1/2＝ 4.83%
组合风险
Portfolio Risk
6-36
投机是为风险溢价而进行的风险投资。风险溢价要大到足以补偿风险厌恶型投资者的投资风险。
Speculation is the undertaking of a risky
investment for its risk premium,The risk
premium has to be large enough to
compensate a risk-averse investor for the risk
of the investment,
小结
6-37
公平游戏是风险溢价为零的冒险前景。风险厌恶型投资者不会参加这类活动。
A fair game is a risky prospect that has a
zero-risk premium,It will not be undertaken
by a risk-averse investor.
小结
6-38
投资者对预期收益与资产组合的波动性的偏好可以用效用函数来表示。效用函数越大，预期回报越高；
效用函数越小，资产组合方差越大。投资者的风险厌恶程度越强，对风险妨碍就越大。我们可以用无差异曲线图来描述这些偏好
Investors’ preferences toward the expected return
and volatility of a portfolio may be ex- pressed by a
utility function that is higher for higher expected
returns and lower for higher portfolio variances,
More risk-averse investors will apply greater
penalties for risk,We can describe these
preferences graphically using indifference curves.
小结
6-39
确定等价值概括了风险厌恶型投资者对风险资产组合的需求。确定等价收益率是一种可以确切得到的与风险组合有相同效用的值。
The desirability of a risky portfolio to a risk-
averse investor may be summarized by the
certainty equivalent value of the portfolio,The
certainty equivalent rate of return is a value
that,if it is received with certainty,would yield
the same utility as the risky portfolio.
小结
6-40
套期保值是购买一种风险资产以降低资产组合的风险。套期保值资产与原有资产组合收益的负相关性使得套期保值资产的波动性具有降低风险的特性。当一种套期保值资产与原有资产组合完全负相关时，它就是一种理想的套期保值工具，
其作用相当于资产组合的保险合约。
Hedging is the purchase of a risky asset to reduce the
risk of a portfolio,The negative correlation between
the hedge asset and the initial portfolio turns the
volatility of the hedge asset into a risk-reducing
feature,When a hedge asset is perfectly negatively
correlated with the initial portfolio,it serves as a
perfect hedge and works like an insurance contract on
the portfolio.
小结
第 6章风险与风险厌恶
Risk and Risk Aversion
6-2
风险与风险厌恶
Risk and Risk Aversion
6.1风险与风险厌恶
6.2 资产组合风险
6-3
风险与风险厌恶
Risk and Risk Aversion
风险的存在意味着可能产生一个以上的结果，
单一前景是指将某一初始财富被用于投资时面临着风险，这一投资机会只产生两种可能的结果。
The presence of risk means that more than
one outcome is possible,A simple prospect is
an investment opportunity in which a certain
initial wealth is placed at risk,and there are
only two possible outcomes,
6-4
W = 100
初始投资额收益 W1 = 150 利润 Profit = 50
收益 W2 = 80 利润 Profit = -20概率 2,1-p =,4
预期收益 E(W) = pW1 + (1-p)W2 =0.6 (150) +0.4(80) = 122
预期盈利 (expected profit)=122-100=22
方差 s2 (variance) = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2
=0.6 (150-122)2 + 0.4(80-122)2 = 1,176,000
标准差 s = 34,293
风险 -不确定收益
Risk - Uncertain Outcomes
6-5
W1 = 150 Profit = 50
W2 = 80 Profit = -201-p =,4100
风险投资 Risky Inv.
无风险投资 Risk Free T-bills Profit = 5
预期盈利 (The expected profit) = 122-100=22
风险溢价 Risk Premium =风险投资预期盈利 22 – 国库券 盈利 5 = 17
风险投资与无风险投资
Risky Investments with Risk-Free Investment
6-6
投资者的风险观 Investor’s view of risk
– 风险厌恶型 Risk Averse
– 风险中性 Risk Neutral
– 风险爱好型 Risk Seeking
效用价值 Utility
效用价值方程 Utility Function
U = E ( r ) -,005 A s 2
A 是投资者风险厌恶系数
A measures the degree of risk aversion
风险厌恶与效用价值
Risk Aversion & Utility
6-7
风险厌恶与价值,投资举例
Risk Aversion and Value,Using the Sample Investment
U = E ( r ) -,005 A s 2
=,22 -,005 A (34%) 2
风险厌恶度 效用值
Risk Aversion A Value
High 5 -6.90
3 4.66
Low 1 16.22
T-bill = 5%
6-8
风险厌恶、风险与收益的权衡
Equilibrium of Risk Aversion,Risk and return
如果证券 A可以无风险的获得回报率为 10％，
而证券 B以 50％的概率获得 20％的收益，
50％的概率的收益为 0，你将选择哪一种证券？
对于一个风险规避的投资者，虽然证券 B的期望收益为 10％，但它具有风险，而证券 A
的无风险收益为 10％，显然证券 A优于证券
B。
6-9
均值方差标准（ Mean-variance criterion)
若投资者是风险厌恶的，则对于证券 A和证券 B，当且仅仅当
22
ABss?
时成立则该投资者认为,A占优于 B”，从而该投资者是风险厌恶性的。
6-10
占优原则
Dominance Principle
1
2 3
4
预期收益 Expected Return
方差或标准差 Variance or Standard Deviation
2占优 dominates 1; 有更高收益 has a higher return
2占优 dominates 3; 有更低风险 has a lower risk
4占优 dominates 3; 有更高收益 has a higher return
6-11
效用和差异曲线
Utility and Indifference Curves
表现投资者对收益和风险权衡的意愿
Represent an investor’s willingness to trade-off return
and risk.
效用数值可以看成是对资产组合排序的一种方法。风险 -
收益曲线越吸引人，资产组合的效用值也就越高。预期收益越高，资产组合得到的效用数值越大，而波动性强的资产组合，其效用数值也低。
The utility score may be viewed as a means of ranking
portfolios,Higher utility values are assigned to portfolios
with more attractive risk-return profiles,Portfolios receive
higher utility scores for higher expected returns and
lower scores for higher volatility.
6-12
效用函数（ Utility function）的例子假定一个风险规避者具有如下形式的效应函数
2( ) 0,0 0 5U E r A s=?
– 其中,A为投资者风险规避的程度。
– 若 A越大，表示投资者越害怕风险，在同等风险的情况下，越需要更多的收益补偿。
– 若 A不变，则当方差越大，效用越低。
6-13
确定性等价收益率（ Certainly equivalent
rate）
为使无风险资产与风险资产具有相同的效用而确定的无风险资产的报酬率，称为风险资产的确定性等价收益率。
由于无风险资产的方差为 0，因此，其效用
U就等价于无风险回报率，因此,U就是风险资产的确定性等价收益率。
6-14
例如：对于风险资产 A，其效用为
2
( ) 0,0 0 5
1 0 % 0,0 0 5 4 4
2%
U E r A s=?
=
=
它等价于收益（效用）为 2％的无风险资产
( ) 2 %fU E r==
结论：只有当风险资产的确定性等价收益至少不小于无风险资产的收益时，这个投资才是值得的。
6-15
Standard Deviation
回报
return
标准差2
6-16
风险厌恶型投资者的无差异曲线
Indifference Curves
预期收益 Expected Return
标准差 Standard Deviation
增加效用 Increasing Utility
6-17
风险中性投资者的无差异曲线
Risk neutral,Indifference Curves
风险中性型的投资者对风险无所谓，只关心投资收益。
期望收益 Expected Return
标准差 Standard Deviation
6-18
风险偏好投资者的无差异曲线
Risk Seeking,Indifference Curves
期望收益 Expected Return
标准差 Standard Deviation
风险偏好型的投资者将风险作为正效用的商品看待，当收益降低时候，
可以通过风险增加得到效用补偿。
6-19
资产组合风险
Portfolio Risk
到现在为止，我们的讨论一直集中在个人全部资产组合的风险与收益上。这样的资产组合是由各种类型的资产组成的，除了在金融市场上的直接投资外，投资者还持有养老基金、以储蓄形式进行的人寿保险计划、住宅，还有并非最不重要的是他们自身技能带来的获利能力（人力资本）。
Investor portfolios are composed of diverse types of
assets,In addition to direct investment in financial
markets,investors have stakes in pension funds,life
insurance policies with savings components,homes,and
not least,the earning power of their skills (human
capital),
6-20
资产组合风险
Portfolio Risk
在评估一个资产组合的风险时，投资者必须考虑到资产收益之间的相互作用。从 根本上说，比如当资产组合的另一部分情况很糟时，通过签定保险合约，交一大笔保 险金可以降低风险。当资产组合中的一部分资产，如房屋或工厂遭受巨大损失时，购 买的火险就派上了用场。
这两种资产（住宅与保险）收益的相互抵消形式稳定了整个资产组合的风险。投资于补偿形式的资产，使之抵消我们可能遇到的某种风险称之为套期保值。
Investors must take account of the interplay between asset returns
when evaluating the risk of a portfolio,At a most basic level,for
example,an insurance contract serves to reduce risk by providing a
large payoff when another part of the portfolio is faring poorly,A fire
insurance policy pays off when another asset in the portfolio—a
house or factory,for example—suffers a big loss in value,The
offsetting pattern of returns on these two assets (the house and the
insurance policy) stabilizes the risk of the overall portfolio,Investing in
an asset with a payoff pattern that offsets exposure to a particular
source of risk is called hedging.
6-21
期望收益
Expected Return
规则 1 在任何情况下，资产的平均或预期收益就是其收益的概率加权平均值。
Rule 1,The return for an asset is the
probability weighted average return in all
scenarios.
=
s
srsPrE )()()(
6-22
期望收益
Expected Return
Normal Year for Sugar Abnormal Year
Bullish Bearish
Stock Market Stock Market Sugar Crisis
糖生产的正常年份 异常年份名 称 股市的牛市 股市的熊市 糖的生产危机概率 Probability 0.50 0.30 0.2
收益率 (%) 25 10 -25
（ 贝斯特 ·凯迪 ）
凯恩收益率 (%) 1 -5 35
E(r贝斯特 ·凯迪 )＝ (0.5× 25)＋ (0.3× 10)＋ 0.2(-25)＝ 10.5%
E(r凯恩 )＝ (0.5× 1)＋ (0.3× -5)＋ 0.2(35)＝ 6%
6-23
收益的方差
Variance of Return
规则 2 资产收益的方差是预期收益的偏差的平方的期望值。
Rule 2,The variance of an asset’s return is the expected
value of the squared deviations from the expected return.
本例中
σ2贝斯特 ·凯迪 =0.5(25-10.5)2+0.3(10-10.5)2+0.2(-25-10.5) 2
=357.25
σ =(357.25)1/2 = 18.9 %。
])()()[( 22=
s
rEsrsPs
6-24
资产组合的收益率
Return on a Portfolio
规则 3 资产组合的收益率是构成资产组合的每个资产收益率的加权平均值，资产组合的构成比例为权重。
Rule 3,The rate of return on a portfolio is a
weighted average of the rates of return of each
asset comprising the portfolio,with the portfolio
proportions as weights.
rp = W1r1 + W2r2
W1 = 资金在证券 1中的比例 Proportion of funds in Security 1
W2 = 资金在证券 2中的比例 Proportion of funds in Security 2
r1 = 证券 1的预期收益 Expected return on Security 1
r2 = 证券 2的预期收益 Expected return on Security 2
6-25
资产组合的收益率
Return on a Portfolio
在本例中，休曼埃克斯的资产组合中 50%投资于贝斯特 ·凯迪公司的股票,50 % 投资于国库券。因此有
The Humanex portfolio is 50% invested in Best stock
and 50% invested in risk-free bills,Therefore,
E(r休曼埃克斯 )＝ 0.5E(r贝斯特 ·凯迪 )＋ 0.5r国库券
＝ (0.5× 10.5)＋ (0.5× 5)＝ 7.75%
6-26
组合风险与无风险资产
Portfolio Risk with Risk-Free Asset
规则 4 当一个风险资产与一个无风险资产相组合时，资产组合的标准差等于风险资产的标准差乘以该资产组合投资于这部分资产上的比例。
Rule 4,When a risky asset is combined with a
risk-free asset,the portfolio standard deviation
equals the risky asset’s standard deviation
multiplied by the portfolio proportion invested in
the risky asset.
ss r i s k y a s s e tr i s k y a s s e tp w?=
6-27
组合风险与无风险资产
Portfolio Risk with Risk-Free Asset
在本例中，休曼埃克斯的资产组合中 50%投资于贝斯特 ·凯迪公司的股票,50 %投资于国库券。因此有
The Humanex portfolio is 50% invested in Best
stock and 50% invested in risk-free bills,
Therefore,
σ休曼埃克斯 ＝ 0.5σ贝斯特 ·凯迪 ＝ 0.5× 18.9＝ 9.45%
6-28
协方差
Covariance
协方差的定义为：
Cov(r贝斯特 ·凯迪,r凯恩 )＝ Σ Pr( s )[r贝斯特 ·凯迪 (s)-E(r
贝斯特 ·凯迪 )][r凯恩 (s)-E(r凯恩 )]
The covariance is defined as,
Cov(rBest,rKane)= Σ Pr(s)[rBest(s)-E(rBest)]
[rKane(s)-E(rKane)]
6-29
协方差
Covariance
我们引用协方差与相关性的概念来量化资产的套期保值或分散化。 协方差 (covariance)测度的是两个风险资产收益的相互影响的方向与程度。正的协方差意味着资产收益同向变动；负的协方差表明它们朝相反的方向变动，譬如，
贝斯特 ·凯迪公司股票与糖凯恩公司股票的关系就是反方向变动的。
To quantify the hedging or diversification potential of an
asset,we use the concepts of covariance and correlation,
The covariance measures how much the returns on two
risky assets move in tandem,A positive covariance
means that asset returns move together,A negative
covariance means that they vary inversely,as in the
case of Best and SugarKane,
6-30
协方差
Covariance
在本例中，由于在下面的表中归纳出当 E(r贝斯特 ·凯迪 )＝
10.5%,E(r凯恩 )＝ 6%时每一情景的收益，我们可以利用 6-4
式来计算协方差。两种股票的协方差为：
Cov(r贝斯特 ·凯迪,r凯恩 )＝ 0.5(25 -10.5)(1-6)＋ 0.3(10-10.5)(-5-6)＋ 0.2(-25 -
10.5)(35-6)＝ -240.5
负的协方差证实了糖凯恩公司股票对贝斯特 ·凯迪公司股票具有的套期保值作用。糖凯恩公司股票的收益与贝斯特 ·凯迪公司股票是呈反方向变动的。
The negative covariance confirms the hedging quality of
SugarKane stock relative to Best Candy,SugarKane’s
returns move inversely with Best’s,
6-31
相关系数
correlation coefficient
相关系数（ correlation coefficient）是比协方差更简便的计算方法。它把协方差的值放在 -1（完全负相关）和＋ 1（完全正相关）之间。两个变量的相关系数等于它们的协方差除以标准差。用希腊字母 ρ 代表相关系数，我们有
ρ (贝斯特 ·凯迪，糖凯恩 )＝ [Cov(r贝斯特 ·凯迪,r凯恩 )]/( σ贝斯特 ·凯迪 σ凯恩 )
＝ -240.5/(18.9× 14.73)＝ -0.86
6-32
相关系数
correlation coefficient
An easier statistic to interpret than the covariance is the
correlation coefficient,which scales the covariance to a
value between -1 (perfect negative correlation) and 1
(perfect positive correlation),The correlation coefficient
between two variables equals their covariance divided by
the product of the standard deviations,Denoting the
correlation by the Greek letter ρ,we find that
ρ (Best,SugarKane) =[cov(rBest,rSugarKane)] /
(σBest* σSugarKane)
=-240.5/(18.9X14.73)＝ -0.86
6-33
规则 5,方差分别是 s12和 s22 的两个风险资产以 w1 和 w2 的权重构成一个资产组合,该资产组合的方差 sp2 为：
Rule 5,When two risky assets with variances s12 and s22,
respectively,are combined into a portfolio with portfolio
weights w1 and w2,respectively,the portfolio variance is
given by:
sp2 = w12s12 + w22s22 + 2W1W2 Cov(r1r2)
Cov(r1r2) =证券 1和证券 2收益的协方差
=Covariance of returns for Security 1 and Security 2
组合风险
Portfolio Risk
6-34
例 计算资产组合方差贝斯特 ·凯迪公司股票与糖凯恩公司股票的权重相等，
w1＝ w2＝ 0.5，
σ 贝斯特 ·凯迪 ＝ 18.9%，
σ 凯恩 ＝ 14.73%，
Cov(r贝斯特 ·凯迪,r凯恩 )＝ -240.5，
我们得到：
σ２ P＝ (0.52× 18.92)+ (0.52× 14.732)+[2× 0.5× 0.5× (-240.5)]＝
23.3
所以,σP＝ (23.3)1/2＝ 4.83%
组合风险
Portfolio Risk
6-35
在本例中 Best 和 糖 Kane用相同的权重
In this example,with equal weights in Best and
SugarKane,
w1=w2=0.5,
σ Best =18.9%,σ Kane=14.73%,
Cov(rBest,rKane)=-240.5,
we find that
σ２ P＝ (0.52× 18.92)+ (0.52× 14.732)+[2× 0.5× 0.5× (-240.5)]
＝ 23.3
So that σP＝ (23.3)1/2＝ 4.83%
组合风险
Portfolio Risk
6-36
投机是为风险溢价而进行的风险投资。风险溢价要大到足以补偿风险厌恶型投资者的投资风险。
Speculation is the undertaking of a risky
investment for its risk premium,The risk
premium has to be large enough to
compensate a risk-averse investor for the risk
of the investment,
小结
6-37
公平游戏是风险溢价为零的冒险前景。风险厌恶型投资者不会参加这类活动。
A fair game is a risky prospect that has a
zero-risk premium,It will not be undertaken
by a risk-averse investor.
小结
6-38
投资者对预期收益与资产组合的波动性的偏好可以用效用函数来表示。效用函数越大，预期回报越高；
效用函数越小，资产组合方差越大。投资者的风险厌恶程度越强，对风险妨碍就越大。我们可以用无差异曲线图来描述这些偏好
Investors’ preferences toward the expected return
and volatility of a portfolio may be ex- pressed by a
utility function that is higher for higher expected
returns and lower for higher portfolio variances,
More risk-averse investors will apply greater
penalties for risk,We can describe these
preferences graphically using indifference curves.
小结
6-39
确定等价值概括了风险厌恶型投资者对风险资产组合的需求。确定等价收益率是一种可以确切得到的与风险组合有相同效用的值。
The desirability of a risky portfolio to a risk-
averse investor may be summarized by the
certainty equivalent value of the portfolio,The
certainty equivalent rate of return is a value
that,if it is received with certainty,would yield
the same utility as the risky portfolio.
小结
6-40
套期保值是购买一种风险资产以降低资产组合的风险。套期保值资产与原有资产组合收益的负相关性使得套期保值资产的波动性具有降低风险的特性。当一种套期保值资产与原有资产组合完全负相关时，它就是一种理想的套期保值工具，
其作用相当于资产组合的保险合约。
Hedging is the purchase of a risky asset to reduce the
risk of a portfolio,The negative correlation between
the hedge asset and the initial portfolio turns the
volatility of the hedge asset into a risk-reducing
feature,When a hedge asset is perfectly negatively
correlated with the initial portfolio,it serves as a
perfect hedge and works like an insurance contract on
the portfolio.
小结
