Topic 2: Portfolio Theory
FINA 3702C
Investment Analysis and Portfolio Management
Arkodipta Sarkar
Learning Outcomes
• Four steps of Portfolio Investment
– Step 1: Assess risk tolerance
– Step 2: Estimate portfolio risk and return
– Step 3: Identify the opportunity set
– Step 4: Optimal asset allocation
Learning Outcomes
• Four steps of Portfolio Investment
– Step 1: Assess risk tolerance
– Step 2: Estimate portfolio risk and return
– Step 3: Identify the opportunity set
– Step 4: Optimal asset allocation
Example 1: Risk Aversion
• Initial Investment is $100
State
Probability
A
B
C
D
Good
0.5
105
110
115
130
Bad
0.5
105
85
95
90
E(Value)
105
97.5
105
110
Risk
Premium
0
-7.5
0
5
• Ranking:
Asset B < Asset C < Risk-free Asset A< Asset D
Step 1. Assessing Risk Tolerance
• Standard assumptions about risk preference:
–
–
–
–
–
Prefer more wealth than less wealth.
Prefer less risk to more risk.
More tolerant to risk as we become wealthier.
Experience more disutility from a decline in wealth than from an equal increase in wealth.
Prefer a certain outcome to an uncertain outcome of equal value.
Quantify Risk Tolerance
• Link the risk tolerance to utility if we only consider risk and return of financial assets
– A Simple Quadratic Utility Function
– 𝑈 = 𝐸 ( 𝑟 ) − 0.5 𝐴2
– E(r): expected return
– σ: standard deviation
– A: the degree of risk aversion
• Investor’s view of risk
– Risk Lover: A < 0
– Risk Neutral: A = 0
– Risk Averse: A > 0
Note: the parameter 0.5 in the utility function
here is to simplify the mathematical solution
Indifference Curve:
U = E(r) - 0.5 ×A×σ2
σ
A=2
U=0.05 U=0.1
E(r)
E(r)
Indifference Curve
U=0.15
E(r)
40%
35%
0.00
0.05
0.10
0.15
0.05
0.05
0.10
0.15
0.10
0.06
0.11
0.16
0.15
0.07
0.12
0.17
0.20
0.09
0.14
0.19
10%
0.25
0.11
0.16
0.21
5%
0.30
0.14
0.19
0.24
0%
0.35
0.17
0.22
0.27
0.40
0.21
0.26
0.31
0.45
0.25
0.30
0.35
0.50
0.30
0.35
0.40
Expected Return
30%
25%
20%
15%
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
Standard Deviation
U=0.05
U=0.1
U=0.15
Note: the utility level is computed in decimal place
using return and risk in decimal places. There is no
unit attached to it.
Example: Risk Aversion
• A portfolio has an expected rate of return of 20% and standard deviation of 30%. T-bills
offer a safe rate of return of 7%. Would an investor with a risk-aversion parameter A = 4
prefer to invest in T-bills or the risky portfolio?
Example: Risk Aversion
• A portfolio has an expected rate of return of 20% and standard deviation of 30%. T-bills
offer a safe rate of return of 7%. Would an investor with a risk-aversion parameter A = 4
prefer to invest in T-bills or the risky portfolio?
– A=4, risky portfolio: U=.20-(.5*4*.3^2)=.02
– A=4, T-bill: U=.07-(.5*4*0)=.07
Example: Risk Aversion
• A portfolio has an expected rate of return of 20% and standard deviation of 30%. T-bills
offer a safe rate of return of 7%. Would an investor with a risk-aversion parameter A = 4
prefer to invest in T-bills or the risky portfolio?
– A=4, risky portfolio: U=.20-(.5*4*.3^2)=.02
– A=4, T-bill: U=.07-(.5*4*0)=.07
• What if the investor has A = 2?
Example: Risk Aversion
• A portfolio has an expected rate of return of 20% and standard deviation of 30%. T-bills
offer a safe rate of return of 7%. Would an investor with a risk-aversion parameter A = 4
prefer to invest in T-bills or the risky portfolio?
– A=4, risky portfolio: U=.20-(.5*4*.3^2)=.02
– A=4, T-bill: U=.07-(.5*4*0)=.07
• What if the investor has A = 2?
– A=2,risky portfolio: U=.20-(.5*2*.3^2)=.11
Learning Outcomes
• Four steps of Portfolio Investment
– Step 1: Assess risk tolerance
– Step 2: Estimate portfolio risk and return
– Step 3: Identify the opportunity set
– Step 4: Optimal asset allocation
Risk vs. Return: Portfolios vs. Individual Stocks
Risk vs. Return: Portfolios vs. Individual Stocks
Formula Refresher
• Expectation:𝑬 𝒓 = σ𝒔 𝒑 𝒔 𝒓(𝒔)
• Variance: 𝝈 =
σ𝒔
• Covariance: 𝝈𝑨,𝑩 =
𝒑𝒔 𝒓𝒔 − 𝑬 𝒓
σ𝒔
𝟐
𝒑𝒔 𝒓𝑨 − 𝑬 𝒓𝑨
𝒓𝑩 − 𝑬 𝒓𝑩
Historical Trade-off between Risk and Return
• There is a general increasing relationship between historical volatility and average return
for large portfolios
– Volatility seems to be a reasonable measure of risk to evaluate a large portfolio
• There is no clear relationship between volatility and returns for individual stocks.
– Larger stocks have lower volatility overall
– Even the largest stocks are typically more volatile than a portfolio of large stocks
– All individual stocks have lower returns and/or higher risk than the portfolios
• Questions:
– Why do portfolios have same return, lower volatility than individual stocks?
– Why do we observe a positive risk-return relationship only for portfolios?
Portfolio Theory: A Two Asset Portfolio
• Example: Fire insurance
– Asset 1: Your house, worth $100,000
– Asset 2: Your insurance policy on the house
• Two states of the world:
– State 1: Your house burns down and retains no value. The insurance policy pays out
$100,000. The probability of state 1 is 10%.
– State 2: Your house does not burn down and retains its full value. The insurance policy does
not pay out.
• Question: What is the riskiness of each of these two assets (a) individually, (b)
together if held in a portfolio?
Portfolio Theory: A Two Asset Portfolio
State
House Payoffs
Insurance Payoffs
Total Payoffs
1
0
100,000
100,000
2
100,000
0
100,000
• Compute expected payoff and risk (standard deviation)
 p( s )r ( s )
– E Value House = 0.1 × $0 + 0.9 × $100,000 = $90,000
s
𝑆𝑡𝐷𝑒𝑣 Value House =
–
2
.1 × 0 − 90,000 + 0.9 × 100,000 − 90,000
= $30,000
2
=
 p (r − E(r))
2
s
s
s
Portfolio Theory: A Two Asset Portfolio
House Only
Insurance Only
Portfolio of both
Expected Payoff
90,000
10,000
100,000
Risk
30,000
30,000
0
• Important: expected value is additive, but risk is not.
• Perfect negative correlation gives perfect insurance (hedging).
• This hedging intuition holds for financial markets too.
Another Example: Why Buy Gold?
Asset
Return
Variability
Gold
8.8%
20.8
S&P 500
12.8%
18.3
• Coefficient of correlation between S&P 500 and Gold = −0.4
• Gold has a lower return and a higher variability than the S&P 500
• Is Gold a bad investment?
Portfolio Theory: A Two Asset Portfolio.
• Investment strategy
• a fraction of wGold% in gold, (100- wGold)% in the S&P500.
wGold
Return
Variability
100%
8.8%
20.8%
80%
9.6%
15.5%
60%
10.4%
11.7%
45.4%
11.0%
10.7%
40%
11.2%
10.8%
20%
12.0%
13.5%
0%
12.8%
18.3%
Hold only gold
Minimum variance portfolio
Hold only S&P500
Portfolio Theory: A Two Asset Portfolio
• Gold does not look good on its own.
– Lower return than the stock market (8.8% vs. 12.8%)
– Higher variability than the stock market (20.8% vs. 18.3%)
• BUT, adding it to a portfolio can reduce the total risk
– It acts as insurance against some forms of risk (inflation risk, … etc.)
– Technically speaking, the low correlation coefficient (here negative) with the S&P 500 makes
gold a reasonably good hedging instrument.
• Let us take a step back and derive the formulas to obtain the data in the
previous table.
Portfolio Theory: A Two Asset Portfolio
• Two assets: Suppose you invest a proportion 𝑤𝐺𝑜𝑙𝑑 of your wealth in gold and the rest
(𝑤𝑆&𝑃500 = 1- 𝑤𝐺𝑜𝑙𝑑 ) in the S&P500. The return on your portfolio:
෩ 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 = 𝐰𝐆𝐨𝐥𝐝 𝐑
෩ 𝐆𝐨𝐥𝐝 + (𝟏 − 𝐰𝐆𝐨𝐥𝐝 ) 𝐑
෩ 𝐒𝐏𝟓𝟎𝟎
𝐑
• The expected return on your portfolio is:
෩ 𝑷𝒐𝒓𝒕𝒇𝒐𝒍𝒊𝒐 ) = 𝒘𝑮𝒐𝒍𝒅 𝑬(𝑹
෩ 𝑮𝒐𝒍𝒅 ) + (𝟏 − 𝒘𝑮𝒐𝒍𝒅 )𝑬(𝑹
෩ 𝑺𝑷𝟓𝟎𝟎 )
• 𝑬(𝑹
• The expected return of a portfolio varies linearly with the portfolio weights.
Portfolio Theory: A Two Asset Portfolio
• The variance of the portfolio return is:
෩ 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 = 𝐰 𝟐 𝐆𝐨𝐥𝐝 𝐕𝐚𝐫 𝐑
෩ 𝐆𝐨𝐥𝐝 + 𝟏 − 𝐰𝐆𝐨𝐥𝐝 𝟐 𝐕𝐚𝐫 𝐑
෩ 𝐒𝐏𝟓𝟎𝟎 +
𝐕𝐚𝐫 𝐑
෩ 𝐆𝐨𝐥𝐝, 𝐑
෩ 𝐒𝐏𝟓𝟎𝟎 )
𝟐𝐰𝐆𝐨𝐥𝐝 𝟏 − 𝐰𝐆𝐨𝐥𝐝 𝐂𝐨𝐯(𝐑
• Or in short-hand notation
𝛔𝟐 𝐑෩ 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 = 𝐰 𝟐 𝐆𝐨𝐥𝐝 𝛔𝟐 𝐆𝐨𝐥𝐝 + (𝟏 − 𝐰𝐆𝐨𝐥𝐝 )𝟐 𝛔𝟐 𝐒𝐏𝟓𝟎𝟎 +
𝟐𝐰𝐆𝐨𝐥𝐝 𝟏 − 𝐰𝐆𝐨𝐥𝐝 𝛔𝐆𝐨𝐥𝐝,𝐒𝐏𝟓𝟎𝟎
𝛔𝐆𝐨𝐥𝐝, 𝐒𝐏𝟓𝟎𝟎 = 𝛒 𝐆𝐨𝐥𝐝, 𝐒𝐏𝟓𝟎𝟎 𝛔𝐆𝐨𝐥𝐝 𝛔𝐒𝐏𝟓𝟎𝟎
• The variance of the portfolio return does not vary linearly with portfolio weights.
Formula Snapshot
Portfolio Return:
rp = WA rA + WB rB
Portfolio Risk:
σp2 = WA2σA2 + WB2 σB2 + 2WAWBCov(rA,rB)
or
σp2 = WA2σA2 + WB2 σB2 + 2WAWBσAσBρAB
where Cov = covariance and ρAB = correlation coefficient between Asset A and Asset B’s returns.
When more than one asset is combined, some of the risks are eliminated. The amount of risks
that remain will depend on the covariance (or correlation).
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
0.44
0.48
0.52
0.56
0.60
0.64
0.68
0.72
0.76
0.80
0.84
0.88
0.92
0.96
1.00
Average Return / Standard
Deviation
Return & Risk of Stock+Gold Portfolios
Return is linear in w, risk is not!
0.25
0.20
Risk
0.15
0.10
0.05
0.00
Weight in Gold (w)
Portfolio Theory: A Two Asset Portfolio.
• The expected return of a portfolio varies linearly with the portfolio weights.
• The variance of the portfolio return does not vary linearly with the portfolio
weights.
– This is why diversification reduces risk.
– The crucial parameter is the correlation coefficient.
• We usually plot the previous picture in the mean-standard deviation space.
Plotting Mean vs. Std. Dev.
Expected
Return
Minimum
variance
portfolio
Efficient
Frontier
0% Gold
12.8%
11%
8.8%
100% Gold
10.65%
18.3% 20.8%
Standard
deviation
Interpreting the Mean-Std. Dev. Plot
• The efficient frontier—set of portfolios that maximise expected return for a
given level of standard deviation.
• Would you ever own Gold on its own? No.
– Gold only is strictly dominated by the S&P500.
– Similarly all portfolios between 100% Gold and the minimum variance portfolio are strictly
dominated.
• Where is the magic?
– The crucial parameter that is driving this diversification effect is the correlation between the
two stocks.
• Let us look at what happens when the correlation is equal to +1 or to -1.
Case of perfect positive correlation
Minimum
variance
portfolio
Expected
Return
12.8%
8.8%
rho = +1: No benefits from
diversification.
0% Gold
100% Gold
18.3% 20.8%
Standard
deviation
Case of perfect negative correlation
Minimum
variance
portfolio
Expected
Return
rho = -1: Maximum benefits
from diversification
(remember the house fire
insurance example).
Efficient
Frontier
0% Gold
12.8%
10.54%
8.8%
100% Gold
0%
18.3% 20.8%
Standard
deviation
Correlation between +1 and -1
(e.g., Gold & S&P)
rho = -1
Expected
Return
0% Gold
12.8%
rho = +1
rho = -0.4
8.8%
100% Gold
18.3% 20.8%
Standard
deviation
The Magic of Risk Reduction!
Asset A
Asset B
Asset C
Asset D
E(R)
10%
10%
10%
10%
SD
20%
20%
20%
20%
-
1.0
0.0
-1.0
(A+C)
(A+D)
10%
10%
10%
20%
14%
0%
Correlation
w. Asset A
Portfolio:
E(R) (%)
SD (%)
(A+B)
-
Portfolio Risk and Return of N Assets
Multiple Assets in a portfolio:
Portfolio Return:
N
rp =  wi ri
i =1
Portfolio Risk:
  12
cov12 cov13 ... cov1n   w1 



2
w
cov

cov
...
cov
2
21
2
23
2
n


N N

2
2
 port =  wi w j covij =  w1 w2 …L wn −1 wn   cov31 cov32  3
M
... cov3n   …



i =1 j =1
w
…
M
M
M
M
M
…
…
…
…

  n −1 
 cov
cov n 2 cov n 3 ...  n2   wn 
n1

= w12 12 + w22 22 + w32 32 +…L + wn2 n2 + 2 w1w2 cov12 + 2 w1w3 cov13 + 2 w2 w3 cov 23 +L…
(N) variance
(N2 ― N)/2 unique covariance
because covariance is symmetric, i.e., COV12=COV21
Portfolio Risk Diversification
• Portfolio diversification is the investment in several different asset classes or sectors
• Diversification is not just holding a lot of assets
• For example, if you own 50 internet stocks, you are not diversified. However, if you own
50 stocks that span 20 different industries, then you are clearly much more diversified.
• Diversification can substantially reduce the variability of returns without an equivalent
reduction in expected returns
• However, there is a minimum level of risk that cannot be diversified away and that is the
systematic risk.
Diversification Example
• Assume all stocks have
– 15% expected return
– 50% standard deviation
– all stocks’ returns are independent (i.e. ρi,j = 0)
• Consider investing equally in N stocks
• One can show that for the stock portfolio
0.5
E (rp ) = 0.15,  p =
N
Power of Diversification
• In this naïve diversification example, one can eliminate all risks by holding lots of
stocks (N large)
p
N
Correlation Effect
• However, in reality, stocks are correlated. It will be impossible to diversify all risk
away.
• One can diversify away idiosyncratic risk, but not systematic risk
p
Market Risk
N
Market vs. Firm-Specfiic Risk
Portfolio Risk Diversification
Number of Stocks in the Average SD of Annual
Portfolio
Portfolio Returns
Ratio of Portfolio SD to
Individual Stock SD
1
49.24%
1.00
10
23.93%
0.49
50
20.12%
0.41
100
19.69%
0.40
300
19.34%
0.39
500
19.27%
0.39
1000
19.21%
0.39
Table 1 in Meir Statman, “How Many Stocks Make a Diversified Portfolio?” Journal of Financial and
Quantitative Analysis 22 (September 1987), pp. 353–64. These figures are from were derived from E. J. Elton
and M. J. Gruber, “Risk Reduction and Portfolio Size: An Analytic Solution,” Journal of Business 50 (October
1977), pp. 415–37.
Diversification causes risk reduction without a
corresponding reduction in expected return.
Learning Outcomes
• Four steps of Portfolio Investment
– Step 1: Assess risk tolerance
– Step 2: Estimate portfolio risk and return
– Step 3: Identify the opportunity set
– Step 4: Optimal asset allocation
Learning Outcomes
• Four steps of Portfolio Investment
– Step 1: Assess risk tolerance
– Step 2: Estimate portfolio risk and return
– Step 3: Identify the opportunity set
➢ A: one risk-free asset and one risky asset
➢ B: two risky assets
➢ C: two risky assets and one risk-free asset
➢ D: N risky assets and one risk-free asset
– Step 4: Optimal asset allocation
Step 3A. Identify the Opportunity Set for One Risky and
One Risk-free Asset
• What is the possible return of a portfolio that consists of one risky asset A and one
risk-free asset B?
Given:
E(rA) = 15%
E(rB) = 7%
σA = 22%
σB = 0%
• What is the possible risk of this portfolio?
rp= WA *15% +(1- WA)*7%
σp= WA *22%
Graph: Opportunity Set of One Risky Asset and One RiskFree Asset
Return
15%
A
B
7%
22%
Risk (Standard Deviation)
Graph: Opportunity Set of One Risky Asset and One RiskFree Asset
Return
CAL
15%
A
B
It is called capital
allocation line
(CAL).
7%
22%
Risk (Standard Deviation)
Step 3B. Identify the Opportunity Set for
Two Risky Assets
Risk
Return
0%
13.4%
1.80%
5%
11.7%
2.58%
10%
10.1%
3.36%
15%
8.4%
4.14%
20%
6.7%
4.92%
25%
5.1%
5.70%
30%
3.4%
6.48%
35%
1.7%
7.26%
40%
0.0%
8.04%
45%
1.6%
8.82%
50%
3.3%
9.60%
55%
5.0%
10.38%
60%
6.7%
11.16%
65%
8.3%
11.94%
70%
10.0%
12.72%
75%
11.7%
13.50%
80%
13.3%
14.28%
85%
15.0%
15.06%
90%
16.7%
15.84%
95%
18.4%
16.62%
100%
20.0%
17.40%
After considering other portfolio weights, here
is the opportunity set.
20.0%
18.0%
16.0%
Portfolio Return
% in Firm A
14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
0.0%
5.0%
10.0%
15.0%
Portfolio Risk (standard deviation)
20.0%
25.0%
Example 7: The Efficient Set for Two Risky
Assets
20.0%
Efficient Frontier
18.0%
Portfolio Return
16.0%
14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
Portfolio Risk (standard deviation)
Note that some portfolios are “better” than others. They
have higher returns for the same level of risk or less. These
comprise the efficient frontier.
return
Example 8: The Efficient Set for Two
Risky Assets with Different Correlation
• The risk-return relation of
a two-security portfolio
depends on the
correlation coefficient:
-1.0 < ρ < +1.0
100%
stocks
 = -1.0
100%
bonds
• If ρ = +1.0, no risk
reduction is possible.
 = 1.0
 = 0.2
• If ρ = -1.0, maximum risk
reduction is possible.

• The smaller the
correlation, the greater
the risk reduction
potential.
Minimum variance portfolio (MVP) for two risky assets
 − Cov( rD , rE )
wD = 2
2
 D +  E − 2Cov( rD , rE )
wE = 1 − wD
2
E
For math gurus…
Objective function: minimize portfolio variance:
Minimize
 p2 = wD2  D2 + wE2 E2 + 2wD wE Cov(rD , rE )
Unknown: portfolio weights WD and WE
Hence: mathematically, it is equivalent to taking first order differentiation with respect to the above objective
function with respect to the weight WD, whereas WE = 1- WD
d p2 / dwD = 2wD D2 − 2(1 − wD ) E2 + 2(1 − 2 wD )Cov(rD , rE ) = 0
= wD ( D2 +  E2 − 2Cov(rD , rE )) =  E2 − Cov(rD , rE )
 E2 − Cov(rD , rE )
= wD = 2
 D +  E2 − 2Cov(rD , rE )
Minimum variance portfolio (MVP) for two risky assets
• The expected return and risk of Asset D and Asset E are summarized below.
E(r)
σ(r)
Risky Asset D 5%
15%
Risky Asset E 7%
20%
• Let the correlation between the two assets be -1. Calculate the MVP
Minimum variance portfolio (MVP) for two risky assets
• The expected return and risk of Asset D and Asset E are summarized below.
E(r)
σ(r)
Risky Asset D 5%
15%
Risky Asset E 7%
20%
• Let the correlation between the two assets be -1. Calculate the MVP
• Answer:
wD = 0.57 ; wE = 0.43
return
Step 3C. Identify the Opportunity Set for Two Risky
Assets and one Risk-free Asset
Asset F
Asset E
Tangency
Portfolio P
Asset D

With two risky assets and one risk-free asset, we can identify the opportunity set
after identifying the tangency portfolio P.
Math for Finding the Tangency Portfolio P for two
Risky Assets and one Risk-free Asset
• Intuition:
– Objective: Choose 𝑤𝐷 and 𝑤𝐸 such that they maximize Reward-to-variability ratio (i.e. Sharpe Ratio)
– Subject to: available investment option
• Mathematically: Choose 𝑤𝐷 and 𝑤𝐸 such that they maximize:
𝑆𝑝 = (𝐸(𝑟𝑝 ) − 𝑟𝑓 )/𝜎𝑝
• Subject to:
𝑟𝑝 = 𝑤𝐷 𝑟𝐷 + 𝑤𝐸 𝑟𝐸
𝜎𝑝2 = 𝑤𝐷2 𝜎𝐷2 + 𝑤𝐸2 𝜎𝐸2 + 2𝑤𝐷 𝑤𝐸 𝐶𝑜𝑣(𝑟𝐷 , 𝑟𝐸 )
𝑤𝐷 + 𝑤𝐸 = 1
wD =
( rD − rf ) E2 − ( rE − rf )Cov( rD , rE )
( rD − rf ) E2 + ( rE − rf ) D2 − ( rE − rf + rD − rf )Cov ( rD , rE )
wE = 1 − wD
For Math gurus…….
Sp =
wD rD + wE rE − rf
p
=
wD rD + wE rE − rf
wD2  D2 + wE2 E2 + 2wD wE cov(rD , rE )
=
wD rD + (1 − wD )rE − rf
wD2  D2 + (1 − wD ) 2  E2 + 2wD (1 − wD )cov(rD , rE )
let
h = wD2  D2 + (1 − wD )2  E2 + 2wD (1 − wD )cov(rD , rE )
FOC with respect to wD
S p
wD
=
(rD − rE )h − ( wD rD + (1 − wD )rE − rf )h'
h2
=0
Hence, you get the following equation (1)
1
(rD − rE )h 2 = ( wD rD + (1 − wD )rE − rf )[2wD D2 − 2(1 − wD ) E2 + 2(1 − 2wD )cov(rD , rE )]
2
where
(1)
11
[2wD D2 − 2(1 − wD ) E2 + 2(1 − 2wD )cov(rD , rE )]
2h
Then you substitute function h back and solve for wD .
h' =
You can therefore get the term for wD from equation (1)
(rD − rE )[ wD2  D2 + (1 − wD ) 2  E2 + 2wD (1 − wD )cov(rD , rE )] = ( wD rD + (1 − wD )rE − rf )[ wD D2 − (1 − wD ) E2 + (1 − 2 wD )cov( rD , rE )]
The good thing is that the term of wD2 drops out. Hence, with a bit of patience, you can get the below expression for wD :
wD =
(rD − rf ) E2 − (rE − rf )Cov(rD , rE )
(rD − rf ) E2 + (rE − rf ) D2 − (rE − rf + rD − rf )Cov(rD , rE )
wE = 1 − wD
Step 3D: Identify the Opportunity Set for N Risky
Assets and One Risk-free Asset (part 1)
Markowitz Efficient Frontier
13.50
2
13.00
Expected Return
12.50
12.00
11.50
11.00
10.50
3
MVP
Individual
Assets
4
10.00
9.50
9.00
0.00
2.00
4.00
6.00
8.00
1
10.00
12.00
14.00
16.00
Risk (Standard Deviation)
Consider a world with many risky assets; we can still identify the opportunity set, minimum variance
portfolio (MVP), and efficient frontier.
Step 3D: Identify the Opportunity Set for N Risky Assets and On
Risk-free Asset (part 2)
Markowitz Efficient Frontier
Tangency
portfolio
14.00
Expected Return
12.00
2
3
10.00
4
1
8.00
6.00
4.00
Rf
2.00
0.00
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
Risk (Standard Deviation)
In addition to risky securities, consider a world that also has a risk-free asset. In practice, we use the yield on the 1month, 3 month or 1-year government bond as the risk-free rate. Now an investor can allocate his money between the
risk free security and a portfolio of risky securities on the opportunity set. The key idea here is to get “Tangency
Portfolio”.
Learning Outcomes
• Four steps of Portfolio Investment
– Step 1: Assess risk tolerance
– Step 2: Estimate portfolio risk and return
– Step 3: Identify the opportunity set
– Step 4: Optimal asset allocation
Learning Outcomes
• Four steps of Portfolio Investment
– Step 1: Assess risk tolerance
– Step 2: Estimate portfolio risk and return
– Step 3: Identify the opportunity set
– Step 4: Optimal asset allocation
➢ A: one risk-free asset and one risky asset
➢ B: two risky assets
➢ C: two risky assets and one risk-free asset
➢ D: N risky assets and one risk-free asset
Step 4A: Graph of Optimal Complete Portfolio Allocation
of One Risky Asset and One Risk-free Asset
Step 4A: Math of Optimal Complete Portfolio Allocation of
One Risky Asset and One Risk-free Asset
• Objective: Choose WA to maximize: 𝑈 = 𝐸 (𝑟𝑝 ) – 0.5𝐴𝜎𝑝2
•
Subject to: 𝐸 𝑟𝑝 = 𝑤𝐴 𝐸 𝑟𝐴 + 1 − 𝑤𝐴 𝐸 𝑟𝐵
𝜎𝑃2 = 𝑤𝐴2 𝜎𝐴2
• Mathematical Derivation:
• 𝑀𝑎𝑥𝑈 = 𝐸 (𝑟𝑝 ) – 0.5𝐴𝜎𝑝2 = 𝑤𝐴 𝐸(𝑟𝐴 ) + (1 − 𝑤𝐴 )𝐸(𝑟𝐵 )– 0.5𝐴𝑤𝐴2 𝜎𝐴2
• First order differentiation with respect to 𝑤𝐴 (unknown)
𝑑𝑈/𝑑𝑤𝐴 = (𝐸(𝑟𝐴 ) – 𝐸(𝑟𝐵 )) − 𝜎𝐴2 𝐴𝑤𝐴 = 0
• Solution:
𝑾∗𝑨 = (𝑬(𝒓𝑨 ) – 𝑬(𝒓𝑩 )) / ( 𝑨 𝝈𝟐𝑨 )
Key Economic Takeaways:
• An investor who can tolerate risk (low value of A) will optimally invest a larger
proportion in the portfolio of risky assets (higher wA*).
• An increase in risky portfolio’s volatility (e.g., higher σ next month) will lower
allocation in the risky portfolio.
• An increase in risk premium will increase allocation in the risky portfolio
Step 4B: Graph of Optimal Complete Portfolio
Allocation with Two Risky Assets
U(1) U(2) U(3)
E(r)
S
P
Q
More
risk-averse
investor
Less
risk-averse
investor
St. Dev
Step 4B: Math of Optimal Complete Portfolio Allocation
with Two Risky Assets
• Objective: Choose 𝑤𝐷 and 𝑤𝐸 such that they maximize Investor Utility subject to the
available investment opportunity curve
• Objective: Choose 𝑤𝐷 and 𝑤𝐸 such that they
– maximize: 𝑈 = 𝐸 𝑟𝑝 − .5𝐴𝜎𝑝2
– subject to:
– 𝑟𝑝 = 𝑤𝐷 𝑟𝐷 + 𝑤𝐸 𝑟𝐸
– 𝜎𝑝2 = 𝑤𝐷2 𝜎𝐷2 + 𝑤𝐸2 𝜎𝐸2 + 2𝑤𝐷 𝑤𝐸 𝐶𝑜𝑣 𝑟𝐷 , 𝑟𝐸
– 𝑤𝐷 + 𝑤𝐸 = 1
• Solution:
rD − rE + A( E2 − Cov ( rD , rE ))
wD =
A( D2 +  E2 − 2Cov( rD , rE ))
wE = 1 − wD
Step 4C: Graph of Optimal Complete Portfolio Allocation
with 2 Risky Assets and one Risk-free Asset
➢ Now the individual chooses the appropriate mix between the tangency portfolio P (from two risky Asset D
and Asset E) and risk free asset F
E(r)
Investor 2
CAL
B
P
Investor 1
A
rf
D
F
E
Question: Which investor is more
risk averse, Investor 1 or Investor
2?
Step 4C: Graph of Optimal Complete Portfolio Allocation
with 2 Risky Assets and one Risk-free Asset
➢ Now the individual chooses the appropriate mix between the tangency portfolio P (from two risky Asset D
and Asset E) and risk free asset F
E(r)
Investor 2
CAL
B
P
Investor 1
A
rf
D
F
E
Question: Which investor is more
risk averse, Investor 1 or Investor
2?
Investor 1
Step 4D: Optimal Complete Portfolio with N Risky Assets
and One Risk-Free Asset
CAL
Return
Indifference
Curve
rP
rf
P
F
C is the
Optimal
Complete
Portfolio
Risk (Standard
Deviation)
σP
Opportunity
set of N risky
assets
Optimal risky
portfolio
Step 4D: Math of Optimal Complete Portfolio Allocation
for N Risky Assets and One Risk-Free Asset
• Intuition:
Objective: Choose y (the proportion in Tangency Portfolio P) it maximizes Investor Utility
Subject to: New CAL (P)
• Mathematically:
Choose y to Maximize 𝑈 = 𝐸 (𝑟𝐶 ) − .5 𝐴 𝜎𝑐2
Subject to:
𝐸 (𝑟𝐶 ) = 𝑟𝑓 + 𝑦 (𝐸(𝑟𝑝 ) – 𝑟𝑓 )
𝜎𝑐2 = 𝑦 2 𝜎𝑝2
Solution: 𝒀∗ =
𝑬 𝒓𝒑 – 𝒓𝒇
𝑨 𝝈𝟐𝒑
Markowitz Portfolio Selection Model
1952, Journal of Finance
• Key Assumptions
• If all investors have the same perception regarding the probability distribution of risky securities
(homogeneous expectations)
and
• if they can borrow and lend at the same rate (same 𝑟𝑓 )
then
• they will agree on the same risky portfolio M, i.e., all investors have the same capital allocation line
(going through 𝑟𝑓 and M).
Markowitz Portfolio Selection Model
1952, Journal of Finance
• Two funds separation theorem
– First task: All investors regardless of their degree of risk aversion will invest only in the same risky
portfolio “P”.
• Determination of the optimal risky portfolio is purely technical
• Need only two funds: money market fund + mutual fund with the optimal risky portfolio
– Second task: Consider three investors with different risk profiles – conservative, moderate, and
aggressive.
• Allocation of the complete portfolio to T-bills versus the risky portfolio depends on personal
preference. Investors will select the desired point along CAL based on their degree of risk aversion.
return
Markowitz Portfolio Theory
All investors have the same capital
allocation line. This line is the Capital
Market Line (CML).
Indifference Curve
Where a specific investor chooses to
invest along the CML depends on their
Optimal Risky Portfolio risk tolerance.
rf

The Separation Property states that the
market portfolio, M, is the same for all
investors— that is, the choice of the
market portfolio is separate from the
investors’ risk tolerance.
19
Example: Opportunity Set with Leverage
Given:
E(rA) = 15%
E(rB) = 7%
A = 22%
B = 0%
If you can borrow additional 50% of capital from the risk-free rate asset B
and invest in risky Asset A, what is the level of risk and return of this
portfolio?
Example: Opportunity Set with Leverage
Given:
E(rA) = 15%
E(rB) = 7%
A = 22%
B = 0%
If you can borrow additional 50% of capital from the risk-free rate asset B
and invest in risky Asset A, what is the level of risk and return of this
portfolio?
rp = (-0.5) (.07) + (1.5) (.15) = .19
σp = 1.5*0.22 = 0.33
Opportunity Set with Different Borrowing & Lending Rates
If you can borrow at 9% and lend at 7%. What would your CAL look like?
Return
15%
CAL
A
9%
7%
22%
Risk (Standard Deviation)
Summary
• Portfolio Theory: four steps of asset allocation
• Applications:
–
–
–
–
A: one risk-free asset and one risky asset;
B: two risky assets;
C: two risky assets and one risk-free asset;
D: N risky assets and one risk-free asset
• Concepts to be mastered: indifference curve; risk aversion; correlation; optimal risky
portfolio; optimal complete portfolio; Markowitz Portfolio Theory
• Readings: BKMJ Chapter 6 (excluding appendices); Chapter 7 (7.1 – 7.4)