Fin 2200 – CORPORATION FINANCE
Sum 2007
Assignment #6 key
1.
Professors A. Dua
Bell Canada Enterprises (BCE) was priced at $34 per share one year ago when you bought
300 shares. It just paid a dividend of $5.50 per share. Your accountant has determined that
you received a dollar return of $2.60 per share on your investment. Determine
a)
the current price per share.
$ return = dividend return + share price return
2.60 = 5.50 + (P1 – 34)
P1 = 2.60 – 5.50 + 34
P1 = $31.10
b)
the percent return on your investment.
% return = dividend yield + capital gains (losses) yield
= 5.50/34 + (31.10 – 34)/34
= 7.647059%
c)
what BCE’s risk premium was, if the risk-free rate over the year was 4.5%
Risk premium = actual return – risk-free rate
= 7.647059% – 4.5%
= 3.147059%
2.
In the last five years, Canadian Publishers Inc. (CPI) had the following returns:
Year
Return
1
64%
2
-32%
3
27%
Determine
a)
the mean (arithmetic average) annual return.
R1  R 2  R 3  R 4  R 5
n
64%  ( 32%)  27%  ( 13%)  44%

5
 18%
R
b)
the 5-year holding period return.
R H  (1  R1 )(1  R 2 )(1  R 3 )(1  R 4 )(1  R 5 )  1
 (1.64)(1  .32)(1.27)(1  .13)(1.44)  1
 77.434565%
4
-13%
5
44%
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
c)
Page 2 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
the 5-year holding period return expressed as an effective rate per year.
Effective annual R = (1.77434565)1/5 – 1
= 12.152161%
d)
which of the above returns best indicates how a 5 year investment in CPI performed.
Both b) and c). c) is probably preferred since it is an annual rate as opposed to
b) which is a 5-year effective rate.
e)
the variance of the sample of yearly returns.
(R 1  R ) 2  (R 2  R ) 2  (R 3  R ) 2  (R 4  R ) 2  (R 5  R ) 2
n1
2
(0.64  0.18)  ( 0.32  0.18) 2  (0.27  0.18) 2  ( 0.13  0.18) 2  (0.44  0.18) 2

51
0.6334

4
 0.15835
var iance 
f)
the standard deviation of the sample of yearly returns.
  0.15835  0.39793216  39.793216%
3.
Suppose market returns on Canadian stock and Canadian T-bill returns over the most
recent 6-year period are as follows:
Year
1
2
3
4
5
6
Canadian Common Stock
-1.43%
-8.61%
10.81%
22.18%
19.47%
26.58%
Canadian T-bills
6.65%
5.31%
3.44%
3.18%
4.09%
4.33%
Current T-bill rates are 3%. Common shares for Mercury Aerospace Limited trade on the
TSE, and, because of the risk inherent in the space technology sector, investors expect a
21% return. Determine
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
a)
Page 3 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
the historical market risk premium.
 1.43%  8.61%  10.81%  22.18%  19.47%  26.58%
6
 11.5%
R stocks 
6.65%  5.31%  3.44%  3.18%  4.09%  4.33%
6
 4.5%
R Tbills 
Historical risk premium = average stock return – average T-bill return
= 11.5% - 4.5%
= 7%
b)
the  of Mercury’s stock.
R Mercury  current R f   Mercury (historical risk premium )
0.21  0.03   Mercury (0.07)
 Mercury  2.57
4.
Use the data below to calculate the items indicated.
State of Economy Probability
Boom
0.25
Normal
0.60
Recession
0.15
Stock #1’s Return Stock #2’s Return Market Return
0.35
0.264
0.28
0.25
-0.04
0.11
0.15
-0.08
-0.04
Determine each of the following:
a)
E[R1].
E[R1] = 0.25(0.35)+0.6(0.25)+0.15(0.15) = 0.26 or 26%
b)
E[R2].
E[R2] = 0.25(0.264)+0.6(-0.04)+0.15(-0.08) = 0.03 or 3%
c)
E[RM].
E[RM] = 0.25(0.28)+0.6(0.11)+0.15(-0.04) = 0.13 or 13%
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
d)
Page 4 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
1
1  [0.25(0.35  0.26)2  0.6(0.25  0.26)2  0.15(0.15  0.26)2 ] 2  0.06244998
1
e)
2
 2  [0.25(0.264  0.03)2  0.6( 0.04  0.03)2  0.15( .08  0.03)2 ] 2  0.13580869
1
f)
M
 M  [0.25(0.28  0.13)2  0.6(0.11  0.13)2  0.15( 0.04  0.13)2 ] 2  0.10099505
1
g)
12
12  .25(.35  .26)(.264  .03)  .6(.25  .26)( .04  .03)  .15(.15  .26)( .08  .03)
 0.0075
h)
12
12
0.0075

 0.88430361
1 2 (0.06244998)( 0.13580869)
12 
i)
1M
1M  .25(.35  .26)(.28  .13)  .6(.25  .26)(.11  .13)  .15(.15  .26)( .04  .13)
 0.0063
j)
1M
1 M 
k)
1
1 
l)
 1M
0.0063

 0.99886813
1 M (0.06244998)( 0.10099505)
11M 0.06244998(0.99886813)

 0.61764527
M
0.10099505
M
M  1 exactly
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
5.
Page 5 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
Use the data below to answer the following problems regarding portfolios:
Security
1
2
3
a)
Expected Return
0.25
0.15
0.11
Standard Deviation
0.36
0.14
0.00
Assume 12 = -0.60 and your portfolio consists of security 1 and of security 2.
i)
Determine the expected return of your portfolio if your portfolio consists of
70% security 1 and 30% security 2.
E[Rp] = x1E[R1]+x2E[R2]
= 0.7(0.25) + 0.3(0.15)
= 0.22 or 22%
ii)
Determine the expected variance of your portfolio if your portfolio consists of
70% security 1 and 30% security 2.
 p  x1  1  x 2  2  2x1 x 2 12 1 2
2
2
2
2
2
 0.7 2 (0.36) 2  0.3 2 (0.14) 2  2(0.7)( 0.3)( 0.6)( 0.36)( 0.14)
 0.0525672
iii)
Determine, using calculus or a formula, portfolio weights that would give the
minimum standard deviation portfolio. [Hint for calculus users: Establish a
portfolio variance equation in X1 (or X2) only; take its derivative with respect
to X1 (or X2); set the derivative equal to zero and solve for X1 (or X2); then
solve for X2 (or X1).
Formula:
2
 2  12 1  2
x1  2
2
 1   2  212 1 2
(0.14) 2  ( 0.6)( 0.36)( 0.14)
(0.36) 2  (0.14) 2  2( 0.6)( 0.36)( 0.14)
 23.769554%
x2 = 1 – x1 = 1 – 0.2376554 = 0.76230446 or 76.230446%

Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
Page 6 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
Calculus:
 p  x 1  1  x 2  2  2x1 x 2 12 1  2
2
2
2
2
2
x 2  1  x1
 p  x 1  1  (1  x1 ) 2  2  2x 1 (1  x 1 )12 1  2
2
2
2
2
 p  x 1 (0.36) 2  (1  x1 ) 2 (0.14) 2  2x1 (1  x1 )( 0.6)( 0.36)( 0.14)
2
2
 p  0.20968x 1  0.09968x 1  0.0196
2
 p
2
2
x 1
 0.41936x 1  0.09968
0.41936x1  0.09968  0
x1  23.769554%
x 2  76.230446%
iv)
Determine the expected return and the standard deviation of the minimum
variance portfolio in iii) above.
E[Rp] = x1E[R1]+x2E[R2]
= 0.23769554(0.25) + 0.7623044(0.15)
= 0.17376955 or 17.376955%
 p  x1  1  x 2  2  2x1 x 2 12 1  2
2
2
2
2
2
 (.23769554) 2 (.36) 2  (.7623044) 2 (.14) 2  2(.23769554)(. 7623044)( .6)(. 36)(.14)
 0.0078368
 p  0.08805257
v)
Graph, using non-negative weights, the feasible set for the above portfolio
showing E[Rp] on the vertical axis and σp on the horizontal axis. (Use at least
five points.)
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
b)
Page 7 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
Assume 12 = +1.00 and your portfolio consists of security 1 and of security 2.
Graph, using non-negative weights, the feasible set for this portfolio showing E[Rp]
on the vertical axis and σp on the horizontal axis.
Risk v. Return
0.30
Expected return
0.25
0.20
0.15
0.10
0.05
0.00
0.00
0.10
0.20
0.30
0.40
Standard deviation
c)
Assume 12 = -1.00 and your portfolio consists of security 1 and of security 2.
i)
Determine the portfolio weights that would give you the minimum standard
deviation portfolio. (Hint: No calculus is necessary.)
Quadratic Equation:
2
2
2
2
2
 p  x 1  1  x 2  2  2x 1 x 2 12 1 2
x 2  1  x1
 p  x 1  1  (1  x1 ) 2  2  2x 1 (1  x1 )12 1 2
2
2
2
2
0  x 1 (0.36) 2  (1  x 1 ) 2 (0.14) 2  2x 1 (1  x 1 )( 1)( 0.36)( 0.14)
2
0  0.25x 1  0.14x1  0.0196
2
x 1  28%
x 2  72%
Formula:
2
x1 
1   2
0.14
0.36  0.14
 28%

x 2  72%
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
ii)
Page 8 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
Determine the standard deviation of the portfolio in i) above.
zero
iii)
Determine the expected return from the portfolio in i) above.
E[Rp] = x1E[R1]+x2E[R2]
= 0.28(0.25) + 0.72(0.15)
= 0.178 or 17.8%
iv)
Determine two sets of portfolio weights that would result in a portfolio with p
= 0.09.
 p  x1 1  x 2  2  2x1x 2121 2
2
2
2
2
2
x 2  1  x1
 p  x1 1  (1  x1 )2  2  2x1 (1  x1 )121 2
2
2
2
2
(0.09)2  x1 (0.36)2  (1  x1 )2 (0.14)2  2x1 (1  x1 )( 1)( 0.36)( 0.14)
2
0  0.25x1  0.14x1  0.0115
2
x1  46%, x 2  54%
x1  10%, x 2  90%
v)
Which of the above portfolios (in part iv) you would recommend to a rational
risk-averse investor. Explain.
E[Rp] = x1E[R1]+x2E[R2]
= 0.46(0.25) + 0.54(0.15)
= 0.199 or 19.9%
E[Rp] = x1E[R1]+x2E[R2]
= 0.1(0.25) + 0.9(0.15)
= 0.145 or 14.5%
Investors would prefer the 46%-54% portfolio since they get a higher return
than they do with the 10%-90% portfolio at the same degree of risk.
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
vi)
Page 9 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
Graph, using non-negative weights, the feasible set for the above portfolio
showing E[Rp] on the vertical axis and σp on the horizontal axis.
Return vs. Risk
0.30
Expected Return
0.25
0.20
0.15
0.10
0.05
0.00
0.00
0.10
0.20
0.30
0.40
Standard deviation
d)
Assume your portfolio consists of security 1 and of security 3. What portfolio
weights would provide a return of 27.8%?
E[R p ]  x1E[R 1 ]  x 3 E[R 3 ]
0.278  x1 0.25  (1  x1 )0.11
x1  120%
x 2  20%
6.
Use the incomplete data below to answer the following questions.
Security/Portfolio
T-bills (risk-free asset)
Market Portfolio
Portfolio Y
Expected Return E[R]
0.04
0.12
?
Net Dollar Value Owned
?
?
$1,000
T-bills and the Market Portfolio are the only two components of Portfolio Y. T-bills may be
held long or short (i.e., T-bills may have positive or negative weights in Portfolio Y).
a)
$400 of Portfolio Y is invested in the market portfolio. Determine the following:
i)
XT-bills
XT-bills = 1 – 400/1,000 = 0.6
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
ii)
Page 10 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
XMarket
XMarket = 400/1,000 = 0.4
iii)
Y
 Y  xT T  xM M
 0.6(0)  0.4(1)
 0.4
iv)
E[RY]
E[R Y ]  x T R T  x M R M
 0.6(0.04)  0.4(0.12)
 0.024  0.048
 7.2%
OR
E[R Y ]  R f   Y ( R M  R f )
 0.04  0.4(0.12  0.04)
 7.2%
It is also possible to calculate E[RY] from the first method above and then use
CAPM to calculate βY in iii) above.
b)
Ignore part a). When constructing portfolio Y, you shorted $800 worth of T-bills.
Determine the following:
i)
XT-bills
XT-bills = -800/1,000 = -0.8
ii)
XMarket
XMarket = 1 – (-0.8) = 1.8
iii)
Y
 Y  x T T  x M  M
 0.8(0)  1.8(1)
 1.8
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
iv)
Page 11 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
E[RY]
E[R Y ]  x T R T  x M R M
 0.8(0.04)  1.8(0.12)
 0.032  0.216
 18.4%
OR
E[R Y ]  R f   Y ( R M  R f )
 0.04  1.8(0.12  0.04)
 18.4%
It is also possible to calculate E[RY] from the first method above and then use
CAPM to calculate βY in iii) above.
c)
Ignore parts a) and b). Assume portfolio Y has an expected return of 26%. Determine
the following:
i)
XT-bills
E[R Y ]  XT R T  XM R M
0.26  XT (0.04)  (1  XT )0.12)
XT  1.75
ii)
XMarket
XMarket = 1 – (-1.75) = 2.75
OR
E[R Y ]  XT R T  XM R M
0.26  (1  XM )( 0.04)  XM 0.12)
XT  2.75
iii)
Y
E[R Y ]  R f   Y ( R M  R f )
0.26  0.04   Y (0.12  0.04)
 Y  2.75
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
iv)
Page 12 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
Dollar amount in T-bills
Dollar amount in T-bills = -1.75(1,000) = -$1,750
v)
Dollar amount in the market
Dollar amount in the market = 2.75(1,000) = $2,750
vi)
Explain what the answers to parts i) to v) mean.
Negative weights and negative dollar amounts indicate that T-bills have been
“borrowed” or “short-sold” A weight of 2.75 for the market portfolio
indicates that more than 100% of portfolio Y is in the market. A β Y of 2.75
indicates portfolio Y has 2.75 times the systematic risk of the market.
d)
Ignore parts a) to c). If the market portfolio has a  = 0.32, and Portfolio Y’s
proportion invested in T-bills is 25%, then determine the following:
i)
Y
 p  x 1  1  x 2  2  2x 1 x 2  12  1  2
2
2
2
2
2
 0.25 2 (0) 2  0.75 2 (0.32) 2  2(0.25)( 0.75)( 0)( 0)( 0.32)
 p  0.24
ii)
Y
 Y  XT T  XM M
 0.25(0)  0.75(1)
 0.75
iii)
YM
Y 
 YM  Y
M
 YM (0.24)
0.32
 1 exactly
0.75 
 YM
Notes:
1.
XM = βM for all portfolios composed of a market portfolio and a riskfree security.
Fin 2200 – CORPORATION FINANCE
Term 2 2006/2007
Assignment #6 key
2.
Page 13 of 13
Professors A. Dua, J. Falk, A. Paseka, R. Scott
ρYM = 1 for all portfolios composed of the market portfolio and a riskfree security as long as XM > 1.