FINA 3080:

Investment Analysis and Portfolio Management

Lecture 4 –
Portfolio Analyses and
The Markowitz Portfolio Selection Model

Chao Ying
 Recap from Last Lecture
• Expected vs. Realized Return & Risk

• Value at Risk (VaR)

• Risky, Riskfree Prospects & The Risk Premium

• Risk Preferences & Returns Utility

• Mean-Variance & Utility Indifference

2
 Announcements
• Homework 1 posted on Blackboard
• Team information on Blackboard; please check.
• Due Oct 11

• Exam conflict
• If you have notified us, you will hear from us
separately later

• No lecture next Tuesday (Oct 1)

3
 Outline of Today’s Lecture
• Risk & Return for Security Portfolios
• Allocation Decision Levels
• Complete Portfolio Return & Risk
• The Capital Allocation Line
• Risk Tolerance & Portfolio Choice
• Portfolios of Two Risky Assets
• Optimal Risky Portfolio
• Optimal Complete Portfolio
4
 Expected Portfolio Return
• Expected Portfolio Return, E(rP)
E(rP) = w1E(r1) + w2E(r2)

where w1 and w2 are portfolio weights, or the fraction

of wealth the investor puts into each security.
• E.g., Invest $25 in r1 and $75 in r2:
w1 = 25/100, w2 = 75/100

5
 Expected Portfolio Return
E(rP) = w1E(r1) + w2E(r2)
• w1 + w2 = 1
• Can w1 be negative? Can w2 be greater than 1?
• You have $100. Invest -$25 in r1 and $125 in r2
• What does that mean? Buy on margin•
or short sell

paybacklateilendmoneylintere.tl 6
 Portfolio Variance

• Portfolio Variance, P2 = E[rP – E(rP)]2

P2 = E[w1r1 + w2r2 – E(w1r1 + w2r2)]2

= E[w1r1 – E(w1r1) + w2r2 – E(w2r2)]2

= E[w1r1 – E(w1r1)]2 + E[w2r2 – E(w2r2)]2

+ 2E[(w1r1 – E(w1r1))(w2r2 – E(w2r2))]

7
 Portfolio Variance

P2 = Var(w1r1) + Var(w2r2) + 2Cov(w1r1,w2r2)

= w12Var(r1) + w22Var(r2) + 2w1w2Cov(r1,r2)

P2 = w1212 + w2222 + 2w1w212

where 12 = Cov(r1,r2), which we call
the covariance of the securities.

8
 Covariance and Correlation
• Measures if two variables move together
12 = Cov(r1,r2) = E[((r1-E(r1))((r2-E(r2))]
• Note that 22 = 2 = E[(r2-E(r2)]2 means variance is a
variable’s own covariance
• Correlation: A normalized measure of covariance
• 12 = 12
12
0 < 12 ≤  → Variables are positively correlated
12 =  → Variables uncorrelated
-1 ≤ 12 <  → Variables are negatively correlated 9
 Outline of Today’s Lecture
✓ Risk & Return for Security Portfolios
• Allocation Decision Levels
• Complete Portfolio Return & Risk
• The Capital Allocation Line
• Risk Tolerance & Portfolio Choice
• Portfolios of Two Risky Assets
• Optimal Risky Portfolio
• Optimal Complete Portfolio
10
 Allocation Decision Levels

treatalriskyassetsasawh.net
⑤
Complete Portfolio

Risky Assets Security

Optimal Selection
Weights
Stocks Optimal
between
Weights
Risky &
Bonds within
Riskfree Risky
Assets Assets
Others

Asset 1
Riskfree Assets onanyriskya.se
Allocation
④ Then ,
risky.is Hre
11
又一
 Outline of Today’s Lecture
✓ Risk & Return for Security Portfolios
✓ Allocation Decision Levels
• Complete Portfolio Return & Risk lriskylrisb.to
• The Capital Allocation Line
• Risk Tolerance & Portfolio Choice
• Portfolios of Two Risky Assets
• Optimal Risky Portfolio
• Optimal Complete Portfolio
12
 Complete Portfolio Return

E(rC) = yE(rP) + (1 – y)rf

→ E(rC) = rf + y[E(rP) – rf] linearcomb.in我们

Riskfree return: Risk

reward for inflation and Risk premium:
weight reward for taking
deferred consumption (quantity
without risk. risk (price of risk)
of risk
taken) 13
 Complete Portfolio Risk
C2 = y2P2 + (1 – y)2f2 + 2y(1 – y)Pf
constant 永远
But Pf = f2 = 0 by definition, so:

C2 = y2P2 → C = yP

• Suppose E(rP) = 15%, P = 22%, rf = 7%:

1. E(rC) = rf + y[E(rP) – rf] = 7% + y8%, and
2. C = y
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 Possible Complete Portfolios
Return
E(r) Unlevered Portfolio Levered Portfolio
(Margin buying) Capital
0y 0≤y≤1 y →1 CO
1<y 借钱 了 Allocation -
Line (CAL)
→ rf ≤ E(rC) ≤ E(rP) → E(rC) > E(rP)

linearcl.SI
→ 0 ≤ C ≤ P → C > P
P  Quantity of risk
E(rP)
•
15% P
E(rP) – rf = 8%  Risk Premium
rf S = [E(rP) – rf] ÷ P = 8/22

opeoniyhflAnytiskhowmuchrewardirisktreyoncanqe.tn
7% •F
S  Reward-to-variability
S  Sharpe Ratio
Risk
越 越好 
P = 22%
15
mon
高
 Outline of Today’s Lecture
✓ Risk & Return for Security Portfolios
✓ Allocation Decision Levels
✓ Complete Portfolio Return & Risk
• The Capital Allocation Line
• Risk Tolerance & Portfolio Choice
• Portfolios of Two Risky Assets
• Optimal Risky Portfolio
• Optimal Complete Portfolio
16
 showthelinesharp atio slower .int
The Capital Allocation Line
Return
E(r) Unlevered Portfolio Levered Portfolio
(Margin buying) Capital
0≤y≤1 1<y Allocation
Line (CAL)

If
borrowing
E(rP) rate is
•
15% P Sb = [E(rP) – rb] ÷ P higher,
i.e., rb > rf
rf S = [E(rP) – rf] ÷ P
7% •F = 8/22
erestīate 会
risktíe
Risk
P = 22% 
17
因为 youmaydefau.lt
高
比
 Outline of Today’s Lecture
✓ Risk & Return for Security Portfolios
✓ Allocation Decision Levels
✓ Complete Portfolio Return & Risk
✓ The Capital Allocation Line
• Risk Tolerance & Portfolio Choice
• Portfolios of Two Risky Assets
• Optimal Risky Portfolio
• Optimal Complete Portfolio
18
Risk Tolerance & Asset Allocation
• Given that:
E(rC) = rf + y[E(rP) – rf]
C = yP
UC = E[rC] – ½AC2

• UC = rf + y[E(rP) – rf] – ½A y2P2

• What is the Optimal Asset Allocation, y* = ?

3 difteient.ua 19
 Finding the Optimal Allocation
• How? Maximize Investor Utility (A=4)
Max UC = rf + y[E(rP) – rf] – ½A y2P2

• Find y at UC/y = 0 (First order condition)

i.e., [E(rP) – rf] – AyP2 = 0
• Hence: y* = [E(rP) – rf] ÷ AP2

• Numerically: y* = [0.15 – 0.07] ÷ 4(0.22)2

y* = 0.08 ÷ 4(0.0484) = 41.32%
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 Optimal Allocation: Graphically
0
2 0
UC = rf + y[E(rP) – rf] – ½A y P 2
17 1 0 ±
10%

U(y*)
8.65% •
C*
8%
Utility
U(y)
6%

4%
y* = [E(rP) – rf] ÷ AP2
U(y*) = rf + y*[E(rP) – rf] – ½A y*2P2
2%

y* = 0.4132 y
0%
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
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 CAt.tn 要会 draw

Optimal Allocation: Graphically

计算器 0 画
Return The Optimal Portfolio, C*, Utility Indifference
E(r) Curves (A=4)
involves y* = 41.32%. Capital
Its expected return, Allocation
E(rC*) = 10.28% and Line (CAL)
standard deviation,
C* = 9.02% 州 A 1 1 teasibley
-

㖄
E(rP)
15%
mquer •
P with utility of 8.65%
E(rC*)
I

• 切点
10.28% C*
rf •

tangib.ie/sameuti1itgline
7% •F Suboptimal Utility, U = 4.653%
Optimal Utility, U = 8.653%
再 往上 交点 , Unfeasible Utility, U = 12.653% Risk
C* = 9.02% P = 22% 
22
无
。
 Some more details about CAL
• Can y* be smaller than zero?
• No, because the expected return of the complete portfolio
would be lower than the riskfree rate
• What is the optimal complete portfolio if the borrowing rate is
higher than the riskfree rate (rb > rf)?
• Use rf to solve for y* first
Case 1: Unlevered (y* ≤ 1): not affected 不借钱
Case 2: Buying on margin (y* > 1): replace rf with rb and solve
for a new y* (call this y**)
• If y** > 1 you are fine (you buy on margin, borrowing at rb)
后
• But what if y* > 1 and y** < 1? That is, you want to buy on
margin if borrowing at rf (but can’t), and you don’t want to
buy on margin if borrowing at rb
此时 叫 为 最优 解 23
车
 Recap for CAL
Return
E(r)
结果 和 assumpt.im 不符
y* > 1

bshouldnotbeusedjcrbasinte.hr
y** < 1
呲 '

污
E(rP)
•
P

y=1 *
(当 y * < 1时 最优 解 )
rf •
F

Risk
P 
24
。
 Outline of Today’s Lecture
✓ Risk & Return for Security Portfolios
✓ Allocation Decision Levels
✓ Complete Portfolio Return & Risk
✓ The Capital Allocation Line
✓ Risk Tolerance & Portfolio Choice
• Portfolios of Two Risky Assets

| •
•
Optimal Risky Portfolio
Optimal Complete Portfolio
很多 的 州 的 怎么
arranqe
25
心
Portfolios of Two Risky Assets
• Say you can invest in two risky assets:

A bond fund (D) and a stock fund (E) with the

following annual expected return and risk
(standard deviation):

D E
E(r ) 0.08 0.13
 0.12 0.20
26
 Portfolios of Two Risky Assets
wD = bond fund weight, with E(rD) and D2
wE = stock fund weight, with E(rE) and E2
DE = covariance between these funds, thus:

E(rP) = wDE(rD) + wEE(rE)

P2 = wD2D2 + wE2E2 + 2wDwEDE
or: P = [wD2D2 + wE2E2 + 2wDwEDEDE]½
27
 Portfolio Risk vs. Security Risk

• Let’s see how expected portfolio return, E(r),

and risk, P, behave as we vary the portfolio
weights (wD, wE) from 0 to 1
i.e., (all stock/no bond → all bond/no stock)

• Consider correlations of DE = 1, 0.3, 0, -1

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 Portfolio Risk vs. Security Risk
(wD2D2 + wE2E2 + 2wDwE DE DE )½  DE  DE  DE  DE
1 0.3 0 -1
wD wE E(r P) P P P P
All E/No D 0.00 1.00 13% 20% 20% 20% 20%
0.10 0.90 13% 19% 18% 18% 17%
0.20 0.80 12% 18% 17% 16% 14%
0.30 0.70 12% 18% 15% 14% 10%
0.40 0.60 11% 17% 14% 13% 7%
0.50 0.50 11% 16% 13% 12% 4%
0.60 0.40 10% 15% 12% 11% 1%
0.70 0.30 10% 14% 12% 10% 2%
0.80 0.20 9% 14% 11% 10% 6%
0.90 0.10 9% 13% 12% 11% 9%
All D/No E 1.00 0.00 8% 12% 12% 12% 12%
V 29
5100k 和 tund 都
小
比
 Portfolio Risk vs. Security Risk
35%  = -1
(wD2D2 + wE2E2
P Portfolio
+ 2wDwE DE DE )½ =0
30% Standard
Deviation  = 0.3

25%
=1
exacttheso.me
20% •E

15%

• Minimum Variance
11.5% D
10.3% 10%

5%
Weight in
全 Bond 全 Stock Stock Fund
0.0% 0% • 0
-0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 wE
0.18 0.26 0.38
RBR 9 Return ? ) 30
 ④
Ecrp ) ④ ( E D) + l.EE )
↓

Outline of Today’s Lecture

↓ 1inear

✓ Risk & Return for Security Portfolios 已知

✓ Allocation Decision Levels

✓ Complete Portfolio Return & Risk
✓ The Capital Allocation Line
✓ Risk Tolerance & Portfolio Choice
✓ Portfolios of Two Risky Assets
• Optimal Risky Portfolio
• Optimal Complete Portfolio
31
二
 dosmtchangetheshapeAllcombinat.im
Portfolio Return vs. Portfolio Risk
14%
Portfolio
E(rP) Expected
13%
Return
•E
12%
 = -1
11%
=0 Portfolio
10% =1
•
 = 0.3 Opportunity Set
= All Possible
9%
Risky Portfolios
D (for  = 0.3)
8% •
7%

6%
P Portfolio
5% Standard
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% Deviation

32
片
 Find Optimal Risky Portfolio, P*
Allinvestorswincnoosesamehiqest.pt
• Find the Optimal Risky

snarpuogut.it
14%
Portfolio Portfolio, P*, with Highest
E(rP) Expected CAL(P*)
13%
Return
Reward-to-Variability, SP •E
12% 关 " CAL(A)
都 like.MN 5 点 (Not
11.0%11% •P* 有切
optimal)
10% SP* = [E(rP*) – rf] ÷ P*
= (11-5)/14.2 = 0.42
A
8.9% 9% •
SP* D Portfolio
8% •
Minimum Opportunity
7% Riskfree Variance Set
Portfolio Portfolio
6% SA = [E(rA) – rf] ÷ A
F = (8.9-5)/11.45 = 0.34 P Portfolio
5% • Standard
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% Deviation
11.45% 14.20%
33
无
 Optimal Risky Portfolio, P*

○ Portfolio
E(rP) Expected
Return
CAL(P*)
•E

AHinvestorchoose.
ìgertn
•P* ⑤ Efficient
Frontier
(Above A) retumForsames.de
A
• ⑦
D Portfolio
•
Minimum Opportunity
Riskfree Variance Set
Portfolio Portfolio

•
F ④ P Portfolio
① Standard
Deviation
34
 Outline of Today’s Lecture
✓ Risk & Return for Security Portfolios
✓ Allocation Decision Levels
✓ Complete Portfolio Return & Risk
✓ The Capital Allocation Line
✓ Risk Tolerance & Portfolio Choice
✓ Portfolios of Two Risky Assets
✓ Optimal Risky Portfolio
• Optimal Complete Portfolio
35
 mamriskytlrisR-freetreatauas.lt
Optimal Complete Portfolio, C*
Utility Indifference Curve (A = 4)
14%
Portfolio
E(rP) Expected CAL(P*)
13%
Return
•E
12%
Optimal
11% Complete •P*
Optimal Risky
Portfolio
Portfolio:
10%
9.46%
C*
• wD = 40% in bond Assets
9%
wE = 60% in stock
D

趑
8% •
7%

6%
F P Portfolio
5% • Standard
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% Deviation
10.56%
36
子
 Portfolio Weights Example (Cont’d)
Optimal Risky Portfolio P*(provided during hw/exam):
wD* = 40% in bond
wE* = 60% in stock
} given
so

麝
E(rP*) = wD* E(rD) + wE* E(rE) = 11%
P*2 = wD* 2D2 + wE* 2E2 + 2wD*wE* DE = 2.0164%
Now we can treat the portfolio P* as one risky asset.

• Solve y* = [E(rP*) – rf] ÷ AP*2

• y* = [0.11 – 0.05] ÷ 4(0.020164)

y* = 0.06 ÷ 0.080656 = 74.39%
37
 Optimal Complete Portfolio

• 74.39% in the Optimal

Stocks, Bonds, Risky Portfolio means
44.63% 29.76% 25.61% in the
Riskfree Asset

• 40% of the Optimal Risky

Portfolio in bonds (60% in
stocks) means
Riskfree, 29.76% (40% x 74.39%)
and 44.63% (60% x 74.39%)
25.61%
of the Complete portfolio is in
bonds and stocks.

38