1) The document contains 7 problems related to portfolio theory and asset pricing models. The problems calculate statistics such as expected returns, variances, standard deviations, and covariances for portfolios containing different securities. 2) Problem 1 provides the return probabilities and calculations of expected return, variance, standard deviation for a portfolio of 5 assets. Problem 2 does the same but for a 2 asset portfolio and calculates the portfolio return and standard deviation. 3) Subsequent problems analyze additional portfolios, market indexes, correlations between assets, and optimal weights to minimize risk. Calculations include betas, covariances, and utilizing formulas from portfolio theory.

1. Chapter 5: Risk and Return: Portfolio Theory and Assets Pricing Models
Problem 1
(ri - 0.119)2
Return (ri) Prob. (p) (ri) × p ×p
0.20 0.10 0.020 0.00066
0.18 0.45 0.081 0.00167
0.08 0.30 0.024 0.00046
0.00 0.05 0.000 0.00071
-0.06 0.10 -0.006 0.00320
Expected 0.119
Variance 0.00670
Stadev 0.08185
Problem 2
Security X Security Y
Return Prob. Return Prob.
2 2
(ri) (p) (ri) × p (ri - 0.11) × p (ri) (p) (ri) × p (ri - 0.205) × p
9
1 2 3 4 5 6 7 8 = (4) x (7)
0.30 0.10 0.030 0.0036 -0.20 0.05 -0.010 0.0082 -0.00038475
0.20 0.20 0.040 0.0016 0.10 0.25 0.025 0.0028 -0.00047250
0.10 0.40 0.040 0.0000 0.20 0.30 0.060 0.0000 0.00000600
0.05 0.20 0.010 0.0007 0.30 0.30 0.090 0.0027 -0.00034200
-0.10 0.10 -0.010 0.0044 0.40 0.10 0.040 0.0038 -0.00040950
Expected 0.110 0.205
Variance 0.0104 0.01748
Stadev 0.1020 0.13219
Covar -0.00160275
Portfolio return = 0.110 × 0.5 + 0.205 × 0.5 = 0.1575 or 15.75% 0.1575
Portfolio standard deviation =
0.0785
0.0104× .52 + 0.01748 .52 + 2 × .5 × .5 × −0.00160275 0.0785 7.85%
× = or
Problem 3
Security P Market Portfolio
Prob. Return
(p) (ri) (ri) × p (ri - 0.11)2 × p Return (ri) (ri) × p (ri - 0.205)2 × p
(1) (2) (3) (4) (5) (7) (8) (9)
0.30 0.30 0.090 0.0068 - 0.10 -0.030 0.0188 -0.01125
0.30 0.20 0.060 0.0008 0.20 0.060 0.0008 0.00075
0.40 0.00 0.000 0.0090 0.30 0.120 0.0090 -0.00900
Expected 0.150 0.150
Variance 0.0165 0.0285
Stadev 0.1285 0.1688
Covar -0.0195
Beta -0.684
Problem 4
STDEV Y STDEV M Corr YM
(%) (%)

2. 20 15 0.7 0.4 -0.25
Beta of Y
Beta (β) = (corrYM σY 0.93 0.53 -0.33
2
σM)/σ M
Problem 5
New Security
RN (%) Prob., p RN x p (%) (RN-ER)^2 x
p.
40 0.3 12 108.3
30 0.4 12 32.4
-10 0.3 -3 288.3
ER 21
Variance 429.0
STDEV 20.71
ER (%) 21
Portfolio return, RP (%) 18
Portfolio STDEV, σP (%) 25
Correlation between new security and portfolio 0.25
New Portfolio return (%) 21 × 0.05 + 18 × 0.95 18.15
New Portfolio STEV (%)
20.71 2 × .05 2 + 25 2 × .95 2 + 2 × .05 × .95 × .25 × 21 .71 × 25 24.03
Problem 6
Sunrise Sunset Sunrise Sunset Sunrise Sunset
Prob., p Return Return Expected Expected ret. Deviation x Deviation x Covar.
ret. p p
(1) (2) (3) (4)=(1x2) (5)=(1x3) (6)=[(2- (7)=[(3- (8)=[(2-
ER)^2×1] ER)^2×1] ER)][(3-
ER)]×1
0.1 32 30 3.20 3.00 44.94 63.50 53.42
0.2 20 17 4.00 3.40 16.93 29.77 22.45
0.4 14 6 5.60 2.40 4.10 0.58 1.54
0.2 -5 -12 -1.00 -2.40 49.93 56.45 53.09
0.1 -10 -16 -1.00 -1.60 43.26 43.26 43.26
10.80 4.80 159.16 193.56 173.76
Stdev 12.62 13.91
Sunrise Sunset
STDEV STDEV Covariance Correlation
12.62 13.91 173.76 0.99
Correlation is found as follows:
173.76 173.76
Correlation = = = 0.99
12.62 × 13.91 175.54
Problem 7
P Q
E( r ) 18 15
STDEV 23 19
Correlation 0 -1.00
Covariance corrPQ x σP x σQ 0 -437

3. Optimum
weight of P 0.406 0.452
Optimum
weight of Q 0.594 0.548
Optimum weight of P is calculated usin the following formula:
2
σ Q − cov ar PQ
w* = 2
P 2
σ P + σ Q - 2 cov ar PQ
cov ar PQ = 0 , then
19 2 − 0 361
w* =
P = = 0.406
23 + 19 2 − 2 × 0
2
890
w*
Q = 1 − 0.406 = 0.594