Post on 17-Dec-2015
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:
001
ppp
r
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
%5.8010077.0
77.039.1 r
Return
Asset with dividend d is the dividend Return is
Multiple dividends d is the sum of dividends Return is
0
01p
pdpr
0
01p
pdpr
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
089.096.80
96.8061.057.87 r
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
219.020696
2069646115697 r
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
iii paV
10 0
Portfolio Return
Final value of the portfolio is
If there are no dividends the return is
n
iii paV
11 1
n
iii
n
iii
n
iii
pa
papa
r
1
11
0
01
Portfolio Return
Example Consider the portfolio in the table
The return on the portfolio is
Stock Holding Initial Price Final Price
A 100 2 3
B 200 3 2
C 150 1 2
052.0115032002100
115032002100215022003100
r
Portfolio Return
Including dividends The dividend payment from asset i is di
The return on the portfolio is
n
iii
n
iii
n
iiii
pa
padpa
r
1
11
0
01
Portfolio Return
Example
The return on the portfolio is
Stock Holding Initial Price Final Price
Dividend per Share
A 50 10 15 1
B 100 3 6 0
C 300 22 20 3
122.0
)22(300)3(100)10(50)22(300)3(100)10(50)320(300)6(100)115(50
r
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 26.18 0 30.83
Ford 17.31 0.40 11.07
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 The initial value of the portfolio is Proportion invested in asset i is
These proportions must sum to 1
iV0
0V
0
0V
VX
i
i
10
0
1 0
0
1
V
V
V
VX
N
i
iN
ii
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 short-sale 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
This is the standard method of calculation The scale (total value) of the portfolio does
not matter
N
iiirXr
1
Portfolio Proportions
Example Consider assets A, B, and C with returns
The initial proportions in the portfolio are
The return on the portfolio is
11
12 ,
31
332
,21
223 CBA rrr
950150
,950600
,950200 CBA XXX
%)2.5( 052.01950
150
3
1
950
600
2
1
950
200
r
Portfolio Proportions
Proportions must be recomputed at the start of each of the holding periods.
The initial value of the portfolio is
V0 = 100x10 + 200x8 = 2600 The portfolio proportions are
Stock Holding p(0) p(1) p(2) p(3)
A 100 10 15 12 16
B 200 8 9 11 16
138
26001600
0,135
26001000
0 BA XX
The portfolio return over the first year is
At the start of the second year the value of the portfolio is
V1 =100x15 + 200x9 = 3300
%)27( 269.08
89
13
8
10
1015
13
5
r
Portfolio Proportions
Portfolio Proportions
This gives the new portfolio proportions as
The return over the second period can be calculated to be
%)3( 03.09
911116
151512
115
r
116
33001800
1,115
33001500
1 BA XX
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
t
tT
rr
1