Investment Analysis and
Portfolio Management
Lecture 2
Gareth Myles
Return

Return



The reason for holding a security is to benefit from
the return it offers
The holding period return is the proportional
increase in value measured over the holding period
Asset with no dividend



Initial wealth V0 is the purchase price p(0)
Final wealth V1 is the selling price p(1)
Return is:
p1  p0
r
p0
Return

Example




The price of Lastminute.com stock trading in
London on May 29 2002 was 0.77
The price at close of trading on May 28 2003 was
1.39
No dividends were paid
The return for the year of this stock is given by
r
1.39  0.77
100  80.5%
0.77
Return

Asset with dividend



d is the dividend
Return is
p1  d  p0
r
p0
Multiple dividends


d is the sum of dividends
Return is
p1   d  p0
r
p0
Return

Example




The price of IBM stock trading in New York on May 29
2002 was $80.96
The price on May 28 2003 was $87.5.
A total of $0.61 was paid in dividends over the year in
four payments of $0.15, $0.15, $0.15 and $0.16
The return over the year on IBM stock was
r
87.57  0.61  80.96
 0.089
80.96
Portfolio Return

Two methods

(i) The initial and final values of the portfolio can be
determined, dividends added to the final value, and
the return computed

(ii) The prices and payments of the individual
assets, and the holding of those assets, can be
used directly
Portfolio Return

Total value




A portfolio of 200 General Motors stock and 100 IBM
stock is purchased for $20,696 on May 29 2002
The value of the portfolio on May 28 2003 was
$15,697
A total of $461 in dividends was received
The return over the year on the portfolio was
15697  461  20696
r
 0.219
20696
Portfolio Return

Individual assets





Consider a portfolio of n assets
The quantity of asset i in the portfolio is ai
Initial price of asset i is pi(0)
Final price of asset i is pi(1)
Initial value of the portfolio is
n
V0   ai pi 0 
i 1
Portfolio Return
Final value of the portfolio is
n
V1   ai pi 1
i 1
 If there are no dividends the return is

n
n
 ai pi 1   ai pi 0
r  i 1
i 1
n
 ai pi 0
i 1
Portfolio Return

Example

Consider the portfolio in the table
Stock
A
B
C
Holding
100
200
150
Initial Price
2
3
1
Final Price
3
2
2
The return on the portfolio is

100  3  200  2  150  2  100  2  200  3  150 1
r
100  2  200  3  150 1
 0.052

Portfolio Return

Including dividends


The dividend payment from asset i is di
The return on the portfolio is
n
n
 ai  pi 1  di    ai pi 0
r  i 1
i 1
n
 ai pi 0
i 1
Portfolio Return

Example
Stock Holding Initial Price Final
Price
A
50
10
15
Dividend per
Share
1
B
C
0
3
100
300

3
22
6
20
The return on the portfolio is

50(15  1)  100(6)  300(20  3)  50(10)  100(3)  300(22)
r
50(10)  100(3)  300(22)
 0.122
Short Selling

Short selling means holding a negative
quantity
 Short 100 shares of Ford stock means that the
holding of Ford is given by – 100
 Dividends count against the return since they
are a payment that has to be made
 Example

On June 3 2002 a portfolio is constructed of 200
Dell stocks and a short sale of 100 Ford stocks.
The prices on these stocks on June 2 2003, and
the dividends paid are given in the table
Short Selling
Stock
Initial Price Dividend
Final Price
Dell
Ford
26.18
17.31
30.83
11.07
0
0.40
The return over the year on this portfolio is
r = [200 x 30.83 + (-100) x 11.47 – (200 x 26.18 + (-100) x 17.31)]
(200 x 26.18 + (-100) x 17.31)
= 0.43 (43%)
Portfolio Proportions

The proportion of the portfolio invested in each
asset can also be used to find the return
 Value of the investment in asset i is V i
0
 The initial value of the portfolio is V0
 Proportion invested in asset i is
Xi 

V0i
V0
These proportions must sum to 1
N
NVi V
 Xi   0  0  1
i 1
i 1V0 V0
Portfolio Proportions

If asset i is short-sold, its proportion is
negative so Xi < 0
 Example

A portfolio consists of a purchase of 100 of stock A
at $5 each, 200 of stock B at $3 each, and a shortsale of 150 of stock C at $2 each

The total value of the portfolio is
V0 =100 x 5 + 200 x 3 – 150 x 2 = 800
The portfolio proportions are
XA =5/8, XB = 6/8, XC = -3/8

Portfolio Proportions

Return

The return on a portfolio is the weighted average
of the returns on the individual assets in the
portfolio
N
r   X i ri
i 1

This is the standard method of calculation
 The scale (total value) of the portfolio does
not matter
Portfolio Proportions

Example

Consider assets A, B, and C with returns
3 2 1
23
1
2 1
rA 
 , rB 
  , rC 
1
2
2
3
3
1

The initial proportions in the portfolio are
200
600
150
XA 
, XB 
, XC 
950
950
950

The return on the portfolio is
200  1  600  1  150
1  0.052 (5.2%)
r
 
  
950  2  950  3  950
Portfolio Proportions

Proportions must be recomputed at the start of
each of the holding periods.
Stock Holding
p(0)
p(1)
p(2)
p(3)
A
100
10
15
12
16
B
200
8
9
11
16

The initial value of the portfolio is
V0 = 100x10 + 200x8 = 2600
 The portfolio proportions are
1000 5
1600 8
X A 0 
 , X B 0 

2600 13
2600 13
Portfolio Proportions

The portfolio return over the first year is
5 15  10 8 9  8
r

13 10
13 8
 0.269 (27%)

At the start of the second year the value of the
portfolio is
V1 =100x15 + 200x9 = 3300
Portfolio Proportions

This gives the new portfolio proportions as
1500 5
1800 6
X A 1 
 , X B 1 

3300 11
3300 11

The return over the second period can be
calculated to be
5 12  15 6 11  9
r

11 15
11 9
 0.03 (3%)
Mean Return

Mean return is the average of past returns
 Observe the return on an asset (or portfolio)
for periods 1,2,3,...,T
 Let rt be the observed return in period t
 The mean return is
T r
r  t
t 1 T
Mean Return

Example

Consider the following returns observed over
10 years
Year
1 2 3 4 5
6 7 8 9
Return (%) 4 6 2 8 10 6 1 3 4

The mean return is
r = 4 + 6 + 2 + 8 + 10 + 6 + 1 + 4 + 3 + 6
10
= 5%
10
6