Chapter5 Risk and Return: Past and Prologue
1)
Group Project: Investment Portfolio
Goal: To Learn The Basic Tools And Skills In Investment Analysis
Assets Available: Stocks, Bonds, Commodities
Portfolio Selection Strategies: Risk and Reward Specification
Performance Calculation and Analysis:
Presentation in the last class before the final
2a)
review of chapter 4 HW
2b)
Survey
3)
Chapter 5 Risk and Return: Past and Prologue
Return definition:
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
Multi-period: n period
Average Return and geometric return
Average Return= (r1+r2+..+rn)/n
Geometric return: g
Terminal Value of investment: TV=(1+r1)(1+r2)..(1+rn)
g= TV1/n -1
P
DPP
HPR
0
101
Example:
A portfolio manager made 100% in the first yr and lost 50% in the second yr:
t=1 r=100%
t=2 r=-50%
start with $100, t=1 (100%), V1= $200, t=2 (-50%), V2= $100
to calculate g:
TM=(1+r1)(1+r2)=(1+1)(1-.5)=2*.5=1
g= TV1/n -1= (1)1/2-1=1-1=0, so g is better measurement of performance for this portfolio manager.
Portfolio Expected Return and Risk
P: portfolio composition
y: proportion of investment budget
1- y: proportion in risk free asset
rf: rate of return on risk-free asset
rp: actual rate of return
E(rp): expected rate of return
σp: standard deviation
E(rC): return on complete portfolio
E(rC) = yE(rp) + (1 − y)rf
σC = yσrp
Sharpe Ratio (also called Reward to Variability Ratio) for portfolio= Risk Premium/σ
• Capital Allocation Line (CAL)
Plot of risk-return combinations available by varying allocation between risky and risk-free
E(rC) = yE(rp) + (1 − y)rf
5%
12%
15%
7%
7/15
σC = yσrp + (1 − y) σrf
when y=0
E(rC) = rf
σC = 0
when y=1
E(rC) = E(rp)
σC = σrp
example:
rf=5%, E(rp)=12%, σrp=15%,
Q: what y will give you 10% for the combined portfolio?
E(rC) = yE(rp) + (1 − y)rf
10%= y*12%+ (1-y)5%
10%= y*12%+ 5%-y*5%
5%=7% y
Y=5%/7%=.714=71.4%
Q: what y will give your client 10% risk for the combined portfolio?
σC = yσrp
.1=y*.15
y= 66.66%
Higher volatility==higher σ
Risk Aversion and Capital Allocation
• y: Preferred capital allocation
degree of risk aversion ( A )–measures the price of risk she demands from the complete portfolio in
which her entire wealth is invested.
Example: p-12
a. Allocating 70% of the capital in the risky portfolio P, and 30% in risk-free asset,
the client has an expected return on the complete portfolio calculated by adding
up the expected return of the risky proportion (y) and the expected return of the
proportion (1 – y) of the risk-free investment:
E(rC) = y E(rP) + (1 – y) rf
= (0.7 0.17) + (0.3 0.07) = 0.14 or 14% per year
The standard deviation of the portfolio equals the standard deviation of the risky fund
times the fraction of the complete portfolio invested in the risky fund:
C = y P = 0.7 0.27 = 0.189 or 18.9% per year
b. The investment proportions of the client’s overall portfolio can be calculated by the
proportion of risky portfolio in the complete portfolio times the proportion allocated in
each stock.
Security
Investment
Proportions
T-Bills 30.0%
Stock A 0.7 27% = 18.9%
Stock B 0.7 33% = 23.1%
Stock C 0.7 40% = 28.0%
c. We calculate the reward-to-variability ratio (Sharpe ratio) using Equation 5.14.
For the risky portfolio:
S =
Portfolio Risk Premium
tandard eviation of Portfolio cess Return
=
(rP － rf
P
=
1 －
“” /”
For the client’s overall portfolio:
S =
(r － rf
=
1 －
1
= 0.3704
Chapter6 EFFICIENT DIVERSIFICATION
1) Project Group
s
2) Review of HW for chapter 5
3) EFFICIENT DIVERSIFICATION
Asset Allocation with Two Risky Assets
For the two asset portfolio:
E(rp)=W1*r1+ W2*r2
E(rp) = Expected Return on Two Risky Assets
W1 = Proportion of funds in Security 1 (weight, percentage)
W2 = Proportion of funds in Security 2
r1 = Expected return on Security 1
r2 = Expected return on Security 2
1
2
= Variance of Security 1
2
2
= Variance of Security 2
Cov(r1,r2) = Covariance of returns for Security 1 and Security 2
Cov(r1,r2)= ρ*1*2
Relevant formulas for n securities are as follows:
In the two-asset case it is fairly easy to calculate the minimum variance weight with the
following equations:
Once the weights are known, the minimum variance portfolio expected return and risk can be
calculated
),(2
2121
2
2
2
2
2
1
2
1
2
rrCovWWWW
p
portfolio the in securities #
n ;rW)rE(
n
1i
iip
Wi
i=1
n
= 1Wi
i=1
n
WiWi
i=1i=1
n
= 1
n
1I
n
1J
JIJI
2
p
)]r,Cov(r W[Wσ
1. Example:
The parameters of the opportunity set are:
E(rS) = 15%, E(rB) = 9%, S = 32%, B = 23%, = 0.15, rf = 5.5%
From the standard deviations and the correlation coefficient we generate the covariance matrix [note
that Cov(rS, rB) = SB]:
Bond
s
Stocks
Bonds 529.0 110.4
Stocks 110.4 1024.0
The minimum-variance portfolio proportions are:
wMin(S) =
－
－
=
－
－
= .3142
wMin(B) = 1 – .3142 = .6858
The mean and standard deviation of the minimum variance portfolio are:
E(rMin) = ( .3142 15%) + ( .6858 9%) 10.89%
Min = [
+
+ 2 wS wB Cov(rS, rB)]1/2
= [( .3142
2
1024) + ( .6858
2
529) + (2 .3142 .6858 110.4)]1/2
= 19.94%
