７ -1
第７章风险资产与无风险资产之间的资本配置
Capital Allocation Between the Risky Asset
and the Risk-Free Asset
７ -2
7.1 风险资产与无风险资产之间的资本配置
7.2 无风险资产
7.3 一种风险资产与一种无风险资产的资产组合
7.4 风险忍让与资产配置
7.5 消极策略：资本市场线风险资产与无风险资产组合的资本配置
Allocating Capital Between Risky& Risk Free Assets
７ -3
控制资产组合风险最直接的方法是：部分资产投资于短期国库券和其他安全的货币市场证券，部分投资于有风险的资产上。
这一资本配置决策是资产配置选 择的一个例子－这种选择面向广泛的投资类型，而不是只在某类资产中选择特定的证券。绝大多数投资专家认为，资产配置是资产组合构架中最重要的部分。
风险资产与无风险资产组合的资本配置
Allocating Capital Between Risky& Risk Free Assets
７ -4
The most straightforward way to control the risk of the
portfolio is through the fraction of the portfolio invested in
Treasury bills and other safe money market securities
versus risky assets,This capital allocation decision is an
example of an asset allocation choice — a choice among
broad investment classes,rather than among the specific
securities within each asset class,Most investment
professionals consider asset allocation the most important
part of portfolio construction,
风险资产与无风险资产组合的资本配置
Allocating Capital Between Risky& Risk Free Assets
７ -5
问题 Issues
检查风险／收益平衡
Examine risk/return tradeoff.
风险厌恶度的差异将影响风险资产与无风险资产组合的资本配置
Demonstrate how different degrees of risk
aversion will affect allocations between
risky and risk free assets.
风险资产与无风险资产组合的资本配置
Allocating Capital Between Risky & Risk Free
Assets (cont.)
７ -6
rf = 7%?rf = 0%
E(rp) = 15%?p = 22%
y = % in p (1-y) = % in rf
举例
Example
７ -7
E(rc) = yE(rp) + (1 - y)rf
rc = 全部或组合收益 complete or combined portfolio
For example,y =,75
E(rc) =,75(.15) +,25(.07)
=,13 or 13%
投资组合预期收益
Expected Returns for Combinations
７ -8
多种组合
Possible Combinations
E(r)
E(rp) = 15%
rf = 7%
22%0
P
F
c
E(rc) = 13% C
７ -9
pc =
Since r
f
y
= 0,then?
* Rule 4 in Chapter 6
*
多种组合方差
Variance on the Possible Combined Portfolios
７ -10
c =,75(.22) =,165 or 16.5%
If y =,75,then
c = 1(.22) =,22 or 22%
If y = 1
c = (.22) =,00 or 0%
If y = 0
无杠杆作用的组合
Combinations Without Leverage
７ -11
在无风险收益率和投资股票间选择
Borrow at the Risk-Free Rate and invest in
stock.
采用 50％为比例杠杆
Using 50% Leverage,
rc = (-.5) (.07) + (1.5) (.15) = 0.19
c = (1.5) (.22) =,33
杠杆资本配置线作用
Using Leverage with Capital Allocation Line
７ -12
资本配置线
CAL (Capital Allocation Line)
E(r)
E(rp) = 15%
rf = 7%
p = 22%0
P
F
) S = 8/22
E(rp) - rf = 8%
资本配置线
７ -13
资本配置线,它表示投资者的所有可行的风险收益组合。它的斜率 S，等于选择的资产组合每增加一单位标准差上升的期望收益，换句话说，就是每单位额外风险的额外收益的测度。基于这一原因，该斜率也可称为酬报与波动性比率。
This straight line is called the capital allocation line (CAL),
It depicts all the risk–return combinations available to
investors,The slope of the CAL,denoted S,equals the
increase in the expected return of the complete portfolio
per unit of additional standard deviation—in other words,
incremental return per incremental risk,For this reason,the
slope also is called the reward-to-variability ratio.
资本配置线
CAL (Capital Allocation Line)
７ -14
高借款率资本配置线
CAL with Higher Borrowing Rate
E(r)
9%
7% ) S =,36
) S =,27
P
p = 22%
７ -15
假定投资预算为 300 000 美元，我们的投资者另外借 120
000 美元，把所有可用资金全部投入风险资产中。这是一个风险资产的杠杆头寸，因为它有部分资金来自借贷。在例子中
y=420 000/300 000=1.4
1 - y = 1 - 1,4 = - 0,4，这反映出无风险资产是空头，即一个借入头寸。投资者不是以 7 % 利率借出，而是借入。
资产组合收益率分布仍旧展现出相同的酬报与波动性比率：
E (r C ) = 7% + (1.4 X 8%) = 18.2%
σC = 1.4 X 22% = 30.8%
S= E(rC ) - rf / σC = 18.2 – 7/30.8 = 0.36
杠杆头寸
leveraged position
７ -16
Suppose the investment budget is $300,000 and our investor
borrows an additional $120,000,investing the total available funds in
the risky asset,This is a leveraged position in the risky asset; it is
financed in part by borrowing,In that case
Y = 420,000 / 300,000= 1.4
and 1 – y =1 -1.4 = 0.4
reflecting a short position in the risk-free asset,which is a borrowing
position,Rather than lending at a 7% interest rate,the investor borrows
at 7%,The distribution of the portfolio rate of return still exhibits the same
reward-to-variability ratio:
E (r C ) = 7% + (1.4 X 8%) = 18.2%
σC = 1.4 X 22% = 30.8%
S= E(rC ) - rf / σC = 18.2 – 7/30.8 = 0.36
杠杆头寸
leveraged position
７ -17
非政府投资者不能以无风险利率借入资金，借款者的违约风险使得贷款者要求更高的贷款利率。
因此，非政府投资者的借款成本将超过贷出利率
rf=7%。假设借入利率 rf B =9%，则在借入资金的条件下，酬报与波动性比率，也就是资本配置线的 B斜率将为,[E(rP)-r B f ]/σP＝ 6/22＝ 0.27。因此，资本配置线将在点 P处被“弯曲”，如上图所示。在 P点左边，投资者以 7%借出,CAL的斜率为 0.36。在 P点右边，这里 y＞ 1，投资者以
9%借入额外资金，投资于风险资产，斜率为
0.27。
高借款率资本配置线
CAL with Higher Borrowing Rate
７ -18
Of course,nongovernment investors cannot borrow at the
risk-free rate,The risk of a borrower’s default causes
lenders to demand higher interest rates on loans,Therefore,
the nongovernment investor’s borrowing cost will exceed the
lending rate of rf =7%,Suppose the borrowing rate is rB f
=9%,Then in the borrowing range,the reward-to-variability
ratio,the slope of the CAL,will be [E(rP) – rB f]/ σp = 6/22
=0.27,The CAL will therefore be ―kinked‖ at point P,as
shown in Figure 7.3,To the left of P the investor is lending
at 7%,and the slope of the CAL is,36,To the right of P,
where y >1,the investor is borrowing at 9% to finance extra
investments in the risky asset,and the slope is,27.
高借款率资本配置线
CAL with Higher Borrowing Rate
７ -19
风险厌恶水平越高会导致选择较少风险的资产
Greater levels of risk aversion lead to larger
proportions of the risk free rate.
风险厌恶水平越低会导致选择较高风险的资产
Lower levels of risk aversion lead to larger
proportions of the portfolio of risky assets.
接受高风险高收益的意愿将导致杠杆
Willingness to accept high levels of risk for high
levels of returns would result in leveraged
combinations.
风险厌恶和配置
Risk Aversion and Allocation
７ -20
效用方程
Utility Function
U = E ( r ) -,005 A?2
Where
U = 效用 utility
E ( r ) = 资产或资产组合的预期收益
expected return on the asset or portfolio
A = 风险厌恶系数 coefficient of risk aversion
2 = 收益的方差 variance of returns
７ -21
风险厌恶和配置
Risk Aversion and Allocation
面对资本配置线的投资者现在必须从可行的选择集合中选出一个最优组合,这个选择需要风险与收益之间的一种替代关系 。 个人投资者风险厌恶的不同意味着在给定一个相等的机会集合 （ 无风险收益率和酬报与波动性比率 ） 下,不同投资者将选择不同的风险资产头寸 。 特别地讲,投资者越厌恶风险,越将选择较少风险的资产,并持有较多无风险的资产 。
７ -22
风险厌恶和配置
Risk Aversion and Allocation
The investor confronting the CAL now must choose
one optimal portfolio,C,from the set of feasible
choices,This choice entails a trade-off between
risk and return,Individual investor differences in
risk aversion imply that,given an identical
opportunity set (that is,a risk-free rate and a
reward-to-variability ratio),different investors will
choose different positions in the risky asset,In
particular,the more risk-averse investors will
choose to hold less of the risky asset and more of
the risk- free asset.
７ -23
风险厌恶和配置
Risk Aversion and Allocation
我们解释这个函数说，资产组合的效用随期望收益率上升而上升，随着方差上升而下降。这种变化关系的重要程度由风险厌恶系数 A决定。对风险中性的投资者,A＝ 0。更高水平的风险厌恶反映在更大的 A值上。一个投资者面对无风险利率为 rf 和期望收益为 E(rP)、标准差为 σP的风险资产组合,他将发现，对于 y的任何选择，整个资产组合期望收益由等式 7-1给出，这里我们重复其中的一部分：
E(rC)＝ rf＋ y[E(rP)-rf]
由等式 7-2，全部资产组合的方差为：
σc= y2 σP 2
７ -24
风险厌恶和配置
Risk Aversion and Allocation
We interpret this expression to say that the utility from a portfolio
increases as the expected rate of return increases,and it decreases
when the variance increases,The relative magnitude of these
changes is governed by the coefficient of risk aversion,A,For risk-
neutral investors,A=0,Higher levels of risk aversion are reflected in
larger values for A,
An investor who faces a risk-free rate,rf,and a risky portfolio with
expected return E(rP) and standard deviation σP will find that,for
any choice of y,the expected return of the complete portfolio is
given by equation 7.1:
E(rC)＝ rf＋ y[E(rP)-rf]
From equation 7.2,the variance of the overall portfolio is
σc = y2 σP 2
７ -25
风险厌恶和配置
Risk Aversion and Allocation
投资者试图通过选择风险资产的最优配置 y来使他的效用最大化。我们将问题一般写成下列形式：
The investor attempts to maximize utility,U,by
choosing the best allocation to the risky asset,y,
Max U = E (rc) - 0.005Aσ2c = rf + y[ E (rp) - rf ] -
0.005 Ay 2 σ 2p
这里,A是风险厌恶系数。
７ -26
风险厌恶和配置
Risk Aversion and Allocation
学过微积分的学生将记得，最大化问题的解决是利 用了一阶导数为零。对 U求一 阶导,令其为零，解出厌恶风险投资者的最优风险资产头寸的收益率 y*，具体的公式如下：
Students of calculus will remember that the
maximization problem is solved by setting the
derivative of this expression to zero,Doing so and
solving for y yields the optimal position for risk-
averse investors in the risky asset,y*,as follows:
y*= (E(rP) – rf ) / 0.01Aσ 2p
７ -27
风险厌恶和配置
Risk Aversion and Allocation
回到我们的数字例子 [rf＝ 7%,E(rP )＝ 15%,σ P=22%]中，具有风险厌恶系数
A＝ 4 的投资者的最优解为
Going back to our numerical example [rf＝ 7%,E(rP )＝ 15%,σP=22%],the
optimal solution for an investor with a coefficient of risk aversion A＝ 4 is
y* = （ 15 – 7）／（ 0.01 ＊ 4 ＊ 22 2 ）＝ 0.41
投资者将以投资预算的 41%投资于风险资产,59%投资于无风险资产。有 41%
投资于风险资产，则整个资产组合的收益率将有如下的期望收益和标准差：
The investor will invest 41% of the investment budget in the risky asset and
59% in the risk-free asset,With 41% invested in the risky portfolio,the rate of
return of the complete portfolio will have an expected return and standard
deviation as follows:
E(rP)＝ 7+[0.41× (15-7)]＝ 10.28%
σ C＝ 0.41× 22＝ 9.02%
７ -28
风险偏好的资本市场线
CAL with Risk Preferences
E(r)
7%
P
贷款者 Lender
借款者 Borrower
p = 22%
贷款者与借款者相比有更大的Ａ（风险厌恶系数）
The lender has a larger A when compared to the
borrower
７ -29
小结把基金从风险资产组合移至无风险资产是降低风险的最简单方式。
Shifting funds from the risky
portfolio to the risk-free asset is
the simplest way to reduce risk,
７ -30
小结短期国库券仅在名义期限上提供了一种完全的无风险资产。因此，短期国库券实际收益率的标准差比其他资产，譬如长期债券和普通股股票更小。
所以，为了分析的目的，我们把短期国库券考虑作为无风险资产。除短期国库券以外，货币市场基金拥有相对安全的债权，譬如商业票据和银行存单。但是，它们有违约风险。这一风险与绝大多数其他风险资产的风险相比较是很小的。为方便起见，我们经常把货币市场基金当作无风险资产。
T-bills provide a perfectly risk-free asset in nominal terms
only,Nevertheless,the standard deviation of real rates on
short-term T-bills is small compared to that of other as- sets
such as long-term bonds and common stocks,so for the purpose of
our analysis we consider T-bills as the risk-free asset,Money
market funds hold,in addition to T-bills,short-term relatively
safe obligations such as CP and CDs,These entail some default
risk,but again,the additional risk is small relative to most
other risky assets,For convenience,we often refer to money
market funds as risk-free assets.
７ -31
小结投资者的风险资产组合可以由它的酬报与波动性比率 S＝ [E(rP)-
rf]/ σ P表示。这个比率也是资本配置线的斜率，当画图时，这条线是从无风险资产联到风险资产。所有风险资产与无风险资产的组合都在这条线上。当其他条件相等时,投资者偏好一条斜率较陡的资本配置线
，因为这表明它对任一风险水平有更高的期望 收益。如果借入利率高于贷出利率，资本配置线将在风险资产点处“被弯曲”。
An investor’s risky portfolio (the risky asset) can be
characterized by its reward-to- variability ratio,
S＝ [E(rP)-rf]/σ P This ratio is also the slope of the CAL,the
line that,when graphed,goes from the risk-free asset through
the risky asset,All combinations of the risky asset and the
risk-free asset lie on this line,Other things equal,an
investor would prefer a steeper-sloping CAL,because that
means higher expected return for any level of risk,If the
borrowing rate is greater than the lending rate,the CAL will
be,kinked” at the point of the risky asset.
７ -32
小结投资者的风险厌恶程度可以由他的无差异曲线的斜率表示。无差异曲线说明,在任何期望收益和风险水平上，为弥补一个附加的标准差，需要有一风险溢价。风险 厌恶程度较高的投资者有更陡的无差异曲线，也就是说，他们在面临更大的风险时要求有更多的风险溢价补偿。
The investor’s degree of risk aversion is
characterized by the slope of his or her indifference
curve,Indifference curves show,at any level of
expected return and risk,the required risk premium
for taking on one additional percentage of standard
deviation,More risk-averse investors have steeper
indifference curves; that is,they require a greater
risk premium for taking on more risk.
７ -33
小结风险资产的最优头寸 y*，与风险溢价成正比，与方差和风险厌恶程度成反比。
y* = (E(rP)– rf)/0.01Aσ 2p
用图形表示，这个资产组合处于无差异曲线与资本配置线相切的切点。
The optimal position,y*,in the risky asset,is
proportional to the risk premium and inversely
proportional to the variance and degree of risk
aversion:
y* = (E(rP)– rf)/0.01Aσ 2p
Graphically,this portfolio represents the point at
which the indifference curve is tangent to the CAL.
７ -34
小结消极投资策略不考虑进行证券分析，把目标放在投资无风险资产与一个风险资产的广泛组合（譬如标准普尔 500股票的资产组合）。如果在 1997年
，投资者用标准普 尔 500指数的平均历史收益和标准差代表它的期望收益和标准差，那么已发行股票的价值将意味着对于普通投资者而言，有约为
A＝ 2.96的风险厌恶程度。与其他的研究相联系，我们估计风险厌恶系数在
2.0至 4.0之间。
A passive investment strategy disregards security analysis,
targeting instead the risk-free asset and a broad portfolio of
risky assets such as the S&P 500 stock portfolio,If in 1999
investors took the mean historical return and standard deviation
of the S&P 500 as proxies for its expected return and standard
deviation,then the values of outstanding assets would imply a
degree of risk aversion of about A=3.0 for the average investor,
This is in line with other studies,which estimate typical
risk aversion in the range of 2.0 through 4.0.
第７章风险资产与无风险资产之间的资本配置
Capital Allocation Between the Risky Asset
and the Risk-Free Asset
７ -2
7.1 风险资产与无风险资产之间的资本配置
7.2 无风险资产
7.3 一种风险资产与一种无风险资产的资产组合
7.4 风险忍让与资产配置
7.5 消极策略：资本市场线风险资产与无风险资产组合的资本配置
Allocating Capital Between Risky& Risk Free Assets
７ -3
控制资产组合风险最直接的方法是：部分资产投资于短期国库券和其他安全的货币市场证券，部分投资于有风险的资产上。
这一资本配置决策是资产配置选 择的一个例子－这种选择面向广泛的投资类型，而不是只在某类资产中选择特定的证券。绝大多数投资专家认为，资产配置是资产组合构架中最重要的部分。
风险资产与无风险资产组合的资本配置
Allocating Capital Between Risky& Risk Free Assets
７ -4
The most straightforward way to control the risk of the
portfolio is through the fraction of the portfolio invested in
Treasury bills and other safe money market securities
versus risky assets,This capital allocation decision is an
example of an asset allocation choice — a choice among
broad investment classes,rather than among the specific
securities within each asset class,Most investment
professionals consider asset allocation the most important
part of portfolio construction,
风险资产与无风险资产组合的资本配置
Allocating Capital Between Risky& Risk Free Assets
７ -5
问题 Issues
检查风险／收益平衡
Examine risk/return tradeoff.
风险厌恶度的差异将影响风险资产与无风险资产组合的资本配置
Demonstrate how different degrees of risk
aversion will affect allocations between
risky and risk free assets.
风险资产与无风险资产组合的资本配置
Allocating Capital Between Risky & Risk Free
Assets (cont.)
７ -6
rf = 7%?rf = 0%
E(rp) = 15%?p = 22%
y = % in p (1-y) = % in rf
举例
Example
７ -7
E(rc) = yE(rp) + (1 - y)rf
rc = 全部或组合收益 complete or combined portfolio
For example,y =,75
E(rc) =,75(.15) +,25(.07)
=,13 or 13%
投资组合预期收益
Expected Returns for Combinations
７ -8
多种组合
Possible Combinations
E(r)
E(rp) = 15%
rf = 7%
22%0
P
F
c
E(rc) = 13% C
７ -9
pc =
Since r
f
y
= 0,then?
* Rule 4 in Chapter 6
*
多种组合方差
Variance on the Possible Combined Portfolios
７ -10
c =,75(.22) =,165 or 16.5%
If y =,75,then
c = 1(.22) =,22 or 22%
If y = 1
c = (.22) =,00 or 0%
If y = 0
无杠杆作用的组合
Combinations Without Leverage
７ -11
在无风险收益率和投资股票间选择
Borrow at the Risk-Free Rate and invest in
stock.
采用 50％为比例杠杆
Using 50% Leverage,
rc = (-.5) (.07) + (1.5) (.15) = 0.19
c = (1.5) (.22) =,33
杠杆资本配置线作用
Using Leverage with Capital Allocation Line
７ -12
资本配置线
CAL (Capital Allocation Line)
E(r)
E(rp) = 15%
rf = 7%
p = 22%0
P
F
) S = 8/22
E(rp) - rf = 8%
资本配置线
７ -13
资本配置线,它表示投资者的所有可行的风险收益组合。它的斜率 S，等于选择的资产组合每增加一单位标准差上升的期望收益，换句话说，就是每单位额外风险的额外收益的测度。基于这一原因，该斜率也可称为酬报与波动性比率。
This straight line is called the capital allocation line (CAL),
It depicts all the risk–return combinations available to
investors,The slope of the CAL,denoted S,equals the
increase in the expected return of the complete portfolio
per unit of additional standard deviation—in other words,
incremental return per incremental risk,For this reason,the
slope also is called the reward-to-variability ratio.
资本配置线
CAL (Capital Allocation Line)
７ -14
高借款率资本配置线
CAL with Higher Borrowing Rate
E(r)
9%
7% ) S =,36
) S =,27
P
p = 22%
７ -15
假定投资预算为 300 000 美元，我们的投资者另外借 120
000 美元，把所有可用资金全部投入风险资产中。这是一个风险资产的杠杆头寸，因为它有部分资金来自借贷。在例子中
y=420 000/300 000=1.4
1 - y = 1 - 1,4 = - 0,4，这反映出无风险资产是空头，即一个借入头寸。投资者不是以 7 % 利率借出，而是借入。
资产组合收益率分布仍旧展现出相同的酬报与波动性比率：
E (r C ) = 7% + (1.4 X 8%) = 18.2%
σC = 1.4 X 22% = 30.8%
S= E(rC ) - rf / σC = 18.2 – 7/30.8 = 0.36
杠杆头寸
leveraged position
７ -16
Suppose the investment budget is $300,000 and our investor
borrows an additional $120,000,investing the total available funds in
the risky asset,This is a leveraged position in the risky asset; it is
financed in part by borrowing,In that case
Y = 420,000 / 300,000= 1.4
and 1 – y =1 -1.4 = 0.4
reflecting a short position in the risk-free asset,which is a borrowing
position,Rather than lending at a 7% interest rate,the investor borrows
at 7%,The distribution of the portfolio rate of return still exhibits the same
reward-to-variability ratio:
E (r C ) = 7% + (1.4 X 8%) = 18.2%
σC = 1.4 X 22% = 30.8%
S= E(rC ) - rf / σC = 18.2 – 7/30.8 = 0.36
杠杆头寸
leveraged position
７ -17
非政府投资者不能以无风险利率借入资金，借款者的违约风险使得贷款者要求更高的贷款利率。
因此，非政府投资者的借款成本将超过贷出利率
rf=7%。假设借入利率 rf B =9%，则在借入资金的条件下，酬报与波动性比率，也就是资本配置线的 B斜率将为,[E(rP)-r B f ]/σP＝ 6/22＝ 0.27。因此，资本配置线将在点 P处被“弯曲”，如上图所示。在 P点左边，投资者以 7%借出,CAL的斜率为 0.36。在 P点右边，这里 y＞ 1，投资者以
9%借入额外资金，投资于风险资产，斜率为
0.27。
高借款率资本配置线
CAL with Higher Borrowing Rate
７ -18
Of course,nongovernment investors cannot borrow at the
risk-free rate,The risk of a borrower’s default causes
lenders to demand higher interest rates on loans,Therefore,
the nongovernment investor’s borrowing cost will exceed the
lending rate of rf =7%,Suppose the borrowing rate is rB f
=9%,Then in the borrowing range,the reward-to-variability
ratio,the slope of the CAL,will be [E(rP) – rB f]/ σp = 6/22
=0.27,The CAL will therefore be ―kinked‖ at point P,as
shown in Figure 7.3,To the left of P the investor is lending
at 7%,and the slope of the CAL is,36,To the right of P,
where y >1,the investor is borrowing at 9% to finance extra
investments in the risky asset,and the slope is,27.
高借款率资本配置线
CAL with Higher Borrowing Rate
７ -19
风险厌恶水平越高会导致选择较少风险的资产
Greater levels of risk aversion lead to larger
proportions of the risk free rate.
风险厌恶水平越低会导致选择较高风险的资产
Lower levels of risk aversion lead to larger
proportions of the portfolio of risky assets.
接受高风险高收益的意愿将导致杠杆
Willingness to accept high levels of risk for high
levels of returns would result in leveraged
combinations.
风险厌恶和配置
Risk Aversion and Allocation
７ -20
效用方程
Utility Function
U = E ( r ) -,005 A?2
Where
U = 效用 utility
E ( r ) = 资产或资产组合的预期收益
expected return on the asset or portfolio
A = 风险厌恶系数 coefficient of risk aversion
2 = 收益的方差 variance of returns
７ -21
风险厌恶和配置
Risk Aversion and Allocation
面对资本配置线的投资者现在必须从可行的选择集合中选出一个最优组合,这个选择需要风险与收益之间的一种替代关系 。 个人投资者风险厌恶的不同意味着在给定一个相等的机会集合 （ 无风险收益率和酬报与波动性比率 ） 下,不同投资者将选择不同的风险资产头寸 。 特别地讲,投资者越厌恶风险,越将选择较少风险的资产,并持有较多无风险的资产 。
７ -22
风险厌恶和配置
Risk Aversion and Allocation
The investor confronting the CAL now must choose
one optimal portfolio,C,from the set of feasible
choices,This choice entails a trade-off between
risk and return,Individual investor differences in
risk aversion imply that,given an identical
opportunity set (that is,a risk-free rate and a
reward-to-variability ratio),different investors will
choose different positions in the risky asset,In
particular,the more risk-averse investors will
choose to hold less of the risky asset and more of
the risk- free asset.
７ -23
风险厌恶和配置
Risk Aversion and Allocation
我们解释这个函数说，资产组合的效用随期望收益率上升而上升，随着方差上升而下降。这种变化关系的重要程度由风险厌恶系数 A决定。对风险中性的投资者,A＝ 0。更高水平的风险厌恶反映在更大的 A值上。一个投资者面对无风险利率为 rf 和期望收益为 E(rP)、标准差为 σP的风险资产组合,他将发现，对于 y的任何选择，整个资产组合期望收益由等式 7-1给出，这里我们重复其中的一部分：
E(rC)＝ rf＋ y[E(rP)-rf]
由等式 7-2，全部资产组合的方差为：
σc= y2 σP 2
７ -24
风险厌恶和配置
Risk Aversion and Allocation
We interpret this expression to say that the utility from a portfolio
increases as the expected rate of return increases,and it decreases
when the variance increases,The relative magnitude of these
changes is governed by the coefficient of risk aversion,A,For risk-
neutral investors,A=0,Higher levels of risk aversion are reflected in
larger values for A,
An investor who faces a risk-free rate,rf,and a risky portfolio with
expected return E(rP) and standard deviation σP will find that,for
any choice of y,the expected return of the complete portfolio is
given by equation 7.1:
E(rC)＝ rf＋ y[E(rP)-rf]
From equation 7.2,the variance of the overall portfolio is
σc = y2 σP 2
７ -25
风险厌恶和配置
Risk Aversion and Allocation
投资者试图通过选择风险资产的最优配置 y来使他的效用最大化。我们将问题一般写成下列形式：
The investor attempts to maximize utility,U,by
choosing the best allocation to the risky asset,y,
Max U = E (rc) - 0.005Aσ2c = rf + y[ E (rp) - rf ] -
0.005 Ay 2 σ 2p
这里,A是风险厌恶系数。
７ -26
风险厌恶和配置
Risk Aversion and Allocation
学过微积分的学生将记得，最大化问题的解决是利 用了一阶导数为零。对 U求一 阶导,令其为零，解出厌恶风险投资者的最优风险资产头寸的收益率 y*，具体的公式如下：
Students of calculus will remember that the
maximization problem is solved by setting the
derivative of this expression to zero,Doing so and
solving for y yields the optimal position for risk-
averse investors in the risky asset,y*,as follows:
y*= (E(rP) – rf ) / 0.01Aσ 2p
７ -27
风险厌恶和配置
Risk Aversion and Allocation
回到我们的数字例子 [rf＝ 7%,E(rP )＝ 15%,σ P=22%]中，具有风险厌恶系数
A＝ 4 的投资者的最优解为
Going back to our numerical example [rf＝ 7%,E(rP )＝ 15%,σP=22%],the
optimal solution for an investor with a coefficient of risk aversion A＝ 4 is
y* = （ 15 – 7）／（ 0.01 ＊ 4 ＊ 22 2 ）＝ 0.41
投资者将以投资预算的 41%投资于风险资产,59%投资于无风险资产。有 41%
投资于风险资产，则整个资产组合的收益率将有如下的期望收益和标准差：
The investor will invest 41% of the investment budget in the risky asset and
59% in the risk-free asset,With 41% invested in the risky portfolio,the rate of
return of the complete portfolio will have an expected return and standard
deviation as follows:
E(rP)＝ 7+[0.41× (15-7)]＝ 10.28%
σ C＝ 0.41× 22＝ 9.02%
７ -28
风险偏好的资本市场线
CAL with Risk Preferences
E(r)
7%
P
贷款者 Lender
借款者 Borrower
p = 22%
贷款者与借款者相比有更大的Ａ（风险厌恶系数）
The lender has a larger A when compared to the
borrower
７ -29
小结把基金从风险资产组合移至无风险资产是降低风险的最简单方式。
Shifting funds from the risky
portfolio to the risk-free asset is
the simplest way to reduce risk,
７ -30
小结短期国库券仅在名义期限上提供了一种完全的无风险资产。因此，短期国库券实际收益率的标准差比其他资产，譬如长期债券和普通股股票更小。
所以，为了分析的目的，我们把短期国库券考虑作为无风险资产。除短期国库券以外，货币市场基金拥有相对安全的债权，譬如商业票据和银行存单。但是，它们有违约风险。这一风险与绝大多数其他风险资产的风险相比较是很小的。为方便起见，我们经常把货币市场基金当作无风险资产。
T-bills provide a perfectly risk-free asset in nominal terms
only,Nevertheless,the standard deviation of real rates on
short-term T-bills is small compared to that of other as- sets
such as long-term bonds and common stocks,so for the purpose of
our analysis we consider T-bills as the risk-free asset,Money
market funds hold,in addition to T-bills,short-term relatively
safe obligations such as CP and CDs,These entail some default
risk,but again,the additional risk is small relative to most
other risky assets,For convenience,we often refer to money
market funds as risk-free assets.
７ -31
小结投资者的风险资产组合可以由它的酬报与波动性比率 S＝ [E(rP)-
rf]/ σ P表示。这个比率也是资本配置线的斜率，当画图时，这条线是从无风险资产联到风险资产。所有风险资产与无风险资产的组合都在这条线上。当其他条件相等时,投资者偏好一条斜率较陡的资本配置线
，因为这表明它对任一风险水平有更高的期望 收益。如果借入利率高于贷出利率，资本配置线将在风险资产点处“被弯曲”。
An investor’s risky portfolio (the risky asset) can be
characterized by its reward-to- variability ratio,
S＝ [E(rP)-rf]/σ P This ratio is also the slope of the CAL,the
line that,when graphed,goes from the risk-free asset through
the risky asset,All combinations of the risky asset and the
risk-free asset lie on this line,Other things equal,an
investor would prefer a steeper-sloping CAL,because that
means higher expected return for any level of risk,If the
borrowing rate is greater than the lending rate,the CAL will
be,kinked” at the point of the risky asset.
７ -32
小结投资者的风险厌恶程度可以由他的无差异曲线的斜率表示。无差异曲线说明,在任何期望收益和风险水平上，为弥补一个附加的标准差，需要有一风险溢价。风险 厌恶程度较高的投资者有更陡的无差异曲线，也就是说，他们在面临更大的风险时要求有更多的风险溢价补偿。
The investor’s degree of risk aversion is
characterized by the slope of his or her indifference
curve,Indifference curves show,at any level of
expected return and risk,the required risk premium
for taking on one additional percentage of standard
deviation,More risk-averse investors have steeper
indifference curves; that is,they require a greater
risk premium for taking on more risk.
７ -33
小结风险资产的最优头寸 y*，与风险溢价成正比，与方差和风险厌恶程度成反比。
y* = (E(rP)– rf)/0.01Aσ 2p
用图形表示，这个资产组合处于无差异曲线与资本配置线相切的切点。
The optimal position,y*,in the risky asset,is
proportional to the risk premium and inversely
proportional to the variance and degree of risk
aversion:
y* = (E(rP)– rf)/0.01Aσ 2p
Graphically,this portfolio represents the point at
which the indifference curve is tangent to the CAL.
７ -34
小结消极投资策略不考虑进行证券分析，把目标放在投资无风险资产与一个风险资产的广泛组合（譬如标准普尔 500股票的资产组合）。如果在 1997年
，投资者用标准普 尔 500指数的平均历史收益和标准差代表它的期望收益和标准差，那么已发行股票的价值将意味着对于普通投资者而言，有约为
A＝ 2.96的风险厌恶程度。与其他的研究相联系，我们估计风险厌恶系数在
2.0至 4.0之间。
A passive investment strategy disregards security analysis,
targeting instead the risk-free asset and a broad portfolio of
risky assets such as the S&P 500 stock portfolio,If in 1999
investors took the mean historical return and standard deviation
of the S&P 500 as proxies for its expected return and standard
deviation,then the values of outstanding assets would imply a
degree of risk aversion of about A=3.0 for the average investor,
This is in line with other studies,which estimate typical
risk aversion in the range of 2.0 through 4.0.
