Asset Management
Lecture Two
I will more or less follow the structure
of the textbook “Investments” with a
few exceptions.
 These parts of the textbook are
omitted:

Part IV (fixed income)
 Part V (security analysis)
 Part VI (options and other derivatives)

Outline for today
Risk aversion and utility
 Estimating risk aversion
 Markowitz portfolio selection model
 How to find the efficient frontier and the
optimal risky portfolio with Excel

Risk Aversion and utility values


Risk aversion: a risk-averse investor will reject
a fair gamble.
1
Utility value U  E (r )  A 2
2

Risk-neutral investors


A=0
Risk lover

A<0
Risk Aversion and utility values
E(r)
A=4
U=1
A=2
U=0.5
σ
Risk Aversion and utility values
Portfolio L
Risk
E(r)=0.07
Aversion (A) σ=0.05
2
3.5
5
Portfolio M
E(r)=0.09
σ=0.10
Portfolio H
E(r)=0.13
σ=0.20
Risk Aversion and utility values
Portfolio L
Portfolio M
Portfolio H
Risk
E(r)=0.07
Aversion (A) σ=0.05
E(r)=0.09
σ=0.10
E(r)=0.13
σ=0.20
2
0.0675
0.0800
0.09
3.5
0.0656
0.0725
0.06
5
0.0638
0.0650
0.03
Certainty equivalent rate
Estimating A




Consider an
insurance policy with
a cost of v:
Probability
Outcome
p
-1
1-p
0
Expected return E (r )  p  (1)  (1  p)  0   p
 2 (r )  p  ( p  1) 2  (1  p)  p 2  p(1  p)
Variance
U  E (r )  1 A 2 (r )
Utility
2
  p  1 Ap(1  p)
2

-v=U
v  p(1  1 A(1  p))
2
risk premium v
p=0.001
A
p=0.01
0
v as a multiple v as a multiple
of p
of p
1
1
1
1.5
1.4950
2
1.9999
1.9900
3
2.4999
2.4850
4
2.9998
2.9800
5
3.4998
3.4750
return
Two-Security Portfolios with Various
Correlations
100%
Stock B
 = -1.0
100%
Stock A



 = 1.0
 = 0.2

Relationship depends on correlation coefficient
-1.0 <  < +1.0
If  = +1.0, no risk reduction is possible
If  = –1.0, complete risk reduction is possible
return
Markowitz portfolio selection model
minimum
variance
portfolio
Individual Assets
P
Markowitz portfolio selection model
n
E (rp )   wi E (ri )
i 1
n
n
   wi w j Cov(ri , rj )
2
p
i 1 j 1
return
Markowitz portfolio selection model
Indifference
curve
Capital market line
Market
portfolio
rf
Investors allocate their
money across the riskfree asset and the
market portfolio
Separation property:
the portfolio
manager offers the
same risky portfolio
to all investors

Investors borrow at the
risk-free rate and invest
in the market portfolio
Markowitz portfolio selection model

Sharpe ratio
Excess return / SD of excess return
 Reward to volatility
 The tangency portfolio has the highest Sharpe
ratio

return
Markowitz portfolio selection model
Indifference
curve
Capital market line
rf

Markowitz portfolio selection model

How to find the efficient frontier and the
optimal portfolio?
Find E(r) for each asset
 Find SD for each asset
 Find covariance between each pair of assets
 As a starting point, assume a weight for each
asset
 Use Excel Solver as an optimizer

Individual Homework




Construct a portfolio of assets with 5 financial assets
Explain briefly why you choose these assets for your
portfolio.
Use recent 36 monthly data to calculate E(r), var(r), and
cov.
Report for your minimum variance portfolio and the
tangency portfolio:




the weights of assets
expected return, SD and the Sharpe ratio
Repeat the exercise with no-short-sale constraint.
Due on Feb 13. Sent your excel file to Sérgio Gaspar
<[email protected]>