CHAPTER 3
Optional Risk-free and Risky, and Risky Portfolio
1
4.1 Asset Allocation across Portfolios
• Capital Allocation
• Choice between risky and risk-free assets
• Risky asset: Stocks and corporate bonds
•The Risk-Free Asset
•Treasury bonds (still affected by inflation)
•Price-indexed government bonds
•Money market instruments effectively risk-free
•Risk of CDs and commercial paper is miniscule
compared to most assets
Portfolio Asset Allocation: Expected Return and Risk
Expected Return of the Complete Portfolio
E (rC ) = y  E (rp ) + (1 − y)  r f
where E (rC ) = Expected Return of the complete portfolio
E (rp ) = Expected Return of the risky portfolio
rf = Return of the risk free asset
y = Percentage assets in the risky portfolio
Standard Deviation of the Complete Portfolio
 C = y  p
where
 C = Standard deviation of the complete portfolio
 P = Standard deviation of the risky portfolio
4.4 Portfolio Asset Allocation: Expected Return and Risk
• Example, risk-free, rf=7%, E(rp) =15%, p=22%,
• Risk premium= 15%-7%=8%
• If you decide to invest half in risk-free and half in portfolio
P. Your complet expected return and SD of the portfolio P:
E(rc) = 0.5×15 +0.5(7) = 11% and
 c = y   p = 0.5  22 = 11%
Slope =
E (rp ) − rf
p
15 − 7 8
=
=
= 0.36
22
22
Figure4.1 Investment Opportunity Set
CAL: Plot of risk-return combinations available by varying
allocation between risky and risk-free
4.3 Risk Aversion and Capital Allocation
• The utility function:
U = E(r) -0.5A2
• Portfolios receives higher utility scores, is more attractive.
• The expected return of the complete portfolio is
E(rc) = rf + y[E(rp) – rf ]
and c = y p, then the variance of the overall
portfolio is
2c= y22p
Dr. Lay SAU
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• Max U = E(rc) -0.5Ac2
= rf + y[E(rp) – rf ] - 0.5Ay22p
To solve this, we set the derivate of this expression to zero
Max U’ = [E(rp) – rf ] – 2× 0.5Ay 2p = 0
Y*= make Utility function optimal
A= coefficient of risk aversion, ranges from 2 to 4. The larger A
is higher risk aversion
Dr. Lay SAU
7
Going back to numerical example, rf=7%,
E(rp) =15%, p=22%. Investor with coefficient of risk
aversion A=4 is
0.15 − 0.07
y =
= 0.41
2
4  0.22
If you have $100,000, you should invest:
•0.41 x $100,000 =$41,000 in risky portfolio
(equities) and bonds, and
• 0.59 x $100,000 = $59,000 in T-bill.
Dr. Lay SAU
8
Table 4.1 Utility Levels for Various Position in Risky Asset (y) for an
Investor with Risk Aversion A=4
y
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
E(rc)
0.07
0.078
0.086
0.094
0.102
0.110
0.118
0.126
0.134
0.142
0.150
c
0.0
0.022
0.044
0.066
0.088
0.110
0.132
0.154
0.176
0.198
0.220
U
0.0700
0.0770
0.0821
0.0853
0.0865
0.0858
0.0832
0.0786
0.0720
0.0636
0.0532
Dr. Lay SAU
• E (rc) = y15% + (1-y)7%
• c = y22 + (1-y) 0
• U = E(rc) -0.5(4)c2
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Dr. Lay SAU
10
4.4 Passive Strategies and the Capital Market Line
• Passive Strategy
•Investment policy that avoids security analysis, not
attempt to identify miss-priced securities. Indexing has
become a popular strategy for passive investors.
• Capital Market Line (CML)
•a passive strategy using T-bill and the broad stock
market index as the risky portfolio generates and
investment opportunity set that is represented by the
CML.
4.5 Passive Strategies and the Capital Market Line
• Cost and Benefits of Passive Investing
• Passive investing is inexpensive and simple
• Expense ratio of active mutual fund averages 1%
• Expense ratio of hedge fund averages 1%-2%,
plus 10% of returns above risk-free rate
• Active management offers potential for higher
returns
• Historical data, based on 1926 to 2005, approximately 75% invested
risky assets.
• We assume this portfolio has the same reward–risk characteristics
that the S&P 500 has exhibited since 1926, that is, a risk premium
of 8.4% and standard deviation of 20.5%. Substituting these value in
equation 3.5, we obtain
y =
E (rp ) − rf
A
2
p
(3.5)
8.4%
Y =
= 0.75
2
A  20.5

•
Which implies a coefficient of risk aversion, A = 2.7.
•
This means that passive investors allocate their
investment budget among instrument according to
their risk aversion (in this case, A =2.7).
•
A broad range of studies, A ranges from 2.00 to 4.00
Dr. Lay SAU
14
4.6 Covariance and Correlation Review
Day
X Return
Y Return
1
1.1
1.7
3
4.2
2.1
4.9
1.4
4.1
0.2
1.30
2.5
3.74
2
3
4
5
Mean
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4.7 Portfolios of Two Risky Assets
• Portfolios of two risky assets principles can be applied to portfolios of
many assets.
• The rate of return on a portfolio consist of bond fund, D and stock fund, E:
𝑟𝑝 = 𝑤𝐷 𝑟𝐷 +𝑤𝐸 𝑟𝐸
(6.1)
where wD and wE are proportions invested in bond D and stock E,
respectively. rD and rE are rates of return
on bond D and stock E,
respectively.
𝐸(𝑟𝑝 ) = 𝑤𝐷 𝐸(𝑟𝐷 )+𝑤𝐸 𝐸(𝑟𝐸 )
(6.2)
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• The variance of the two-asset portfolio is
Using example 4.3 in Excel to illustrate this formula.
❖Note: The Covariance of a variable with itself is the variance of that variable:
2𝑝 = 𝑤𝐷2 2𝐷 +𝑤𝐸2 2𝐸 +2𝑤𝐷 𝑤𝐸 𝐶𝑜𝑣 𝑟𝐷 𝑟𝐸
(6.3)
❖Correlation of two assets:
Cov(𝑟𝐷 ,𝑟𝐸)
Corr(𝑟𝐷 , 𝑟𝐸 ),  =
𝐷 𝐸
or Cov(𝑟𝐷 , 𝑟𝐸 )=  ∗ 𝐷 𝐸
(6.4)
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• Therefore, equation (6.3) can be express as:
2𝑝 = 𝑤𝐷2 2𝐷 +𝑤𝐸2 2𝐸 +2𝑤𝐷 𝑤𝐸 ∗  ∗ 𝐷 𝐸
•
In case of perfect positive correlation, DE=1
2𝑝 = (𝑤𝐷 𝐷 + 𝑤𝐸 𝐸 )2 and 𝑝 = 𝑤𝐷 𝐷 + 𝑤𝐸 𝐸
•
(6.5)
(6.6)
In case of perfect negative correlation, DE= -1
2𝑝 = (𝑤𝐷 𝐷 − 𝑤𝐸 𝐸 )2 and 𝑝 = 𝑤𝐷 𝐷 − 𝑤𝐸 𝐸
(6.7)
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• When  =-1, a perfectly hedged position can be obtained by choosing
the portfolio proportion to solve system equation.
𝑤𝐷 + 𝑤𝐸 =1
(1)
𝑤𝐷 𝐷 − 𝑤𝐸 𝐸 = 0
(2)
(1) we get 𝑤𝐸 =1-𝑤𝐷 , then substitute this into equation (2), we get:
𝑤𝐷 𝐷 − (1−𝑤𝐷 )𝐸 = 0
𝑇ℎ𝑒𝑛, 𝑤𝐷 𝐷 − 𝐸 + 𝑤𝐷 𝐸 = 0,  𝑤𝐷 (𝐷 + 𝐸 )= 𝐸 . Therefore:
𝐸
𝐷
𝑤𝐷 =
and 𝑤𝐸 =
𝑜𝑟 = 1 − 𝑤𝐷
 𝐷 + 𝐸
 𝐷 + 𝐸
(6.8)
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Correlation Effects
Possible value of correlation 1,2 : -1.0    +1.0
•
If  = 1.0, perfectly positively correlated, no advantage of hedging
•
If  = - 1.0, perfectly negatively correlated, maximum hedging advantage
•
See next example:
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Example: Portfolio Risk and Return
Debt
Equity
Expected return, E(r)
Standard deviation, 
Covariance, Cov(rD,rE)
Correlation coefficient, DE
8%
12%
13%
20%
• 𝐸(𝑟𝑝 )=8𝑤𝐷 +13𝑤𝐸
(6.9), and we have
72
0.30
2𝑝 = 𝑤𝐷2 2𝐷 +𝑤𝐸2 2𝐸 +2𝑤𝐷 𝑤𝐸 ∗  ∗ 𝐷 𝐸
(6.5)
2𝑝 = 122 𝑤𝐷2 +202 𝑤𝐸2 +2𝑤𝐷 𝑤𝐸 ∗ 0.30 ∗ 12 ∗ 20
(6.10) and then,
𝑝 =
2𝑝
(6.11)
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2𝑝 = 122 𝑤𝐷2 +202 𝑤𝐸2 +2𝑤𝐷 𝑤𝐸 ∗ 0.30 ∗ 12 ∗ 20
(6.10) and then,
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Figure 3.3 Portfolio Standard Deviation as a Function of Investment Proportions
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• Note: At DE = -1, the minimum weight in bond funds is 0.625 and
equity funds 0.375 (equation 3.12)
𝐸
𝐷
𝑤𝐷 =
and 𝑤𝐸 =
𝑜𝑟 = 1 − 𝑤𝐷
 𝐷 + 𝐸
 𝐷 + 𝐸
𝑤𝐷 =
20
12+20
(6.8)
= 0.625 and the 𝑤𝐸 = 1-0.625 =0.375
• Look at the solid curve in Figure 3.3 for DE=0.3 (assumed value).
The weight increases from 0 to 1, p first fall with the initial
diversification from bond to stocks, then rises again.
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1. Minimum Variance Portfolio Calculation
• Table 3.1 At a given DE=0.30, the portfolio weights result in minimum
standard deviation of 11.45%, at wD =0.82 and wE =0.18. How to
calculate?
2𝐸−𝐶𝑜𝑣(𝑟𝐷 ,𝑟𝐸)
𝑤𝑚𝑖𝑛 𝐷 = 2 2
(6.11)
𝐷+𝐸−2𝐶𝑜𝑣(𝑟𝐷 ,𝑟𝐸)
For our example (DE=0.30)
𝑤𝑚𝑖𝑛 𝐷
202 −72
= 2 2
12 +20 −2(72)
= 0.82, and 𝑤𝐸 =1-0.82 =0.18
• 𝐸(𝑟𝑝 )=8𝑤𝐷 +13𝑤𝐸 =8(0.82)+13(0.18) =8.9%
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4.8 The Optimal Risky Portfolio with Two Risky Assets and a
Risk- Free Asset
Figure 3.5 shows two CALs drawn from rf = 5%
• The first possible CAL drawn through portfolio A, which invested 82% in
bonds and 18% in stocks (Table 3.1 bottom panel), with E(rp) =8.9% and
p=11.45%
• The slope of the CAL combining with T-bill and with the minimum-variance
of portfolio ( the reward-to-variability or Sharpe ratio) is
𝑆𝐴 =
𝐸 𝑟𝐴 −𝑟𝑓
𝐴
=
8.9−5
=
11.5
0.34
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Figure 3.5 The Opportunity Set of the Debt and Equity Funds and
Two Feasible CALs
27
• Now consider the CAL that uses portfolio B invests 70% in bonds and
30% in stocks. Its E(rp)=9.5% and p=11.70%. Slope B is
𝐸(𝑟𝐵 ) = 𝑤𝐷 𝐸(𝑟𝐷 )+𝑤𝐸 𝐸(𝑟𝐸 )
(6.2)
𝐸(𝑟𝐵 ) = 0.70 ∗ 8 + 0.30 ∗ 13= 9.5%
𝐵 = [122 𝑤𝐷2 +202 𝑤𝐸2 +2𝑤𝐷 𝑤𝐸 ∗ 0.30 ∗ 12 ∗ 20]0.5
(6.10) and then,
𝐵 = [122 ∗0.702 +202 ∗ 0.702 +2(0.70 ∗ 0.30) ∗ 0.30 ∗ 12 ∗ 20]0.5 =11.70
𝑆𝐵 =
𝐸 𝑟𝐵 −𝑟𝑓
𝐵
=
9.5−5
=
11.70
0.38
Thus, portfolio B dominates A. We can continue to yield the CAL with the
highest feasible reward-to-variability ratio labeled P in Figure 3.6 is the
optimal risky portfolio to mix with T-bills.
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2. Weight of the Optimal Risky Portfolio
• In the case of two risky assets, the solution for the weight of the
optimal risky portfolio, P, can be shown to be as follow:
For example, using our data, the solution for the optimal portfolio is:
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The expected return and standard deviation of this optimal portfolio, P, is
𝐸(𝑟𝑝 ) = 𝑤𝐷 𝐸(𝑟𝐷 )+𝑤𝐸 𝐸(𝑟𝐸 )
(6.2)
𝑃 = [122 𝑤𝐷2 +202 𝑤𝐸2 +2𝑤𝐷 𝑤𝐸 ∗ 𝐶𝑜𝑣(𝑟𝐷 𝑟𝐸 )]0.5
(6.10)
𝑃 = [122 ∗0.42 +202 ∗ 0.62 +2∗0.4 ∗ 0.6(72)]0.5 =14.2%
𝑇ℎ𝑒𝑛, 𝑆𝑃 =
𝐸 𝑟𝑃 −𝑟𝑓
𝑃
=
11−5
=
14.2
0.42
The slope P has exceeded any slopes we have considered.
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31
32
Optimal Complete Portfolio with T-bill Example:
• An investor with A = 4 would take a position in Portfolio P of
• Thus, 74.39% in Portfolio P, and 25.61% in T-bill.
• While, portfolio P consists of 40% in bonds, ywD=0.4×0.7439=29.76%
and 60% in stocks, ywE=0.6 ×0.7439 = 44.63%.
• Graphical solution of this asset allocation problem is presented in
Figures 4.7 and 4.8.
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If you have $100,000, you should invest:
• 0.2561 x $100,000 = $25,610 in T-bill
• 0.7439 x $100,000 = $74,390 in portfolio P, which consists of:
• 0.2976 x $100,000 = $29,760 in bonds
• 0.4463 x $100,000 = $44,630 in stocks
E(rc) = wfrf +wDrD + wErE
E(rc) = (0.2561x 5)+(0.2976 x 8)+(0.4463 x13) = 9.46%
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Figure 3.7 Determination of the Optimal Overall Portfolio
• Note: Slope C and P are the same.
E(rc)=9.46% and c=10.57%
𝑆𝐶 =
𝑆𝑃 =
𝐸 𝑟𝑃 −𝑟𝑓
𝑐
𝐸 𝑟𝑃 −𝑟𝑓
𝑃
=
9.46−5
=
10.57
=
11−5
=
14.2
0.42
0.42
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Figure 3.8 The Proportions of the Optimal Overall Portfolio
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4.9 Markowitz Portfolio Selection Model
• Two methods of computing the efficient set of risky portfolio:
• First using minimization program. The points marked by squares are result
of a variance minimization program (Figure 3.11).
• Second method, draw a vertical line that represents the SD constraint. We
then consider all portfolio plot on this line (have the same SD) and choose
the one with the highest expected return.
• Repeating this procedure for many vertical lines (levels of SD) give us the
points marked by cycles (Figure 4.11) that trace the upper portion of the
minimum-variance, the efficient frontier emerges.
• All portfolios that lie on the minimum-variance frontier from the global
minimum-variance portfolio and upward provide the best risk-return
combination
37
Figure 3.11The Efficient Portfolio Set
38
39
Figure 3.9 The Minimum-Variance Frontier of Risky Assets
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