Question

In: Statistics and Probability

In this exercise, we examine the effect of combining investments with positively correlated risks, negatively correlated...

In this exercise, we examine the effect of combining investments with positively correlated risks, negatively correlated risks, and uncorrelated risks. A firm is considering a portfolio of assets. The
portfolio is comprised of two assets, which we will call ''A" and "B." Let X denote the annual rate of return from asset A in the following year, and let Y denote the annual rate of return from asset B in the following year. Suppose that
E(X) = 0.15 and E(Y) = 0.20,
SD(X) = 0.05 and SD(Y) = 0.06,
and CORR(X, Y) = 0.30.
(a) What is the expected return of investing 50% of the portfolio in asset A and 50% of the portfolio in asset B? What is the standard deviation of this return?
(b) Replace CORR(X, Y) = 0.30 by CORR(X, Y) = 0.60 and answer the questions in part (a). Do the same for CORR(X, Y) = 0.60, 0.30, and 0.0.
(c) (Spreadsheet Exercise). Use a spreadsheet to perform the following analysis. Suppose that the fraction of the portfolio that is invested in asset B is f, and so the fraction of the portfolio that is invested in asset A is (1 f). Letting f vary from f = 0.0 to f = 1.0 in increments of 5% (that is, f = 0.0, 0.05, 0.10, 0.15, . . . ), compute the mean and the standard deviation of the annual rate of return of the portfolio (using the original data for the problem). Notice that the expected return of the portfolio varies (linearly) from 0.15 to 0.20, and the standard deviation of the return varies (non-linearly) from 0.05 to 0.06. Construct a chart plotting the standard deviation as a function of the expected return.
(d) (Spreadsheet Exercise). Perform the same analysis as in part (c) with CORR (X, Y) = 0.30 replaced by CORR(X, Y) = 0.60, 0.0, 0.30, and 0.60.

Solutions

Expert Solution

a)

E(X) 0.15 sd(X) 0.05
E(Y) 0.2 sd(Y) 0.06
r 0.3
a)
expected return 0.175
sd 0.044441

formula

E(X) 0.15 sd(X) 0.05
E(Y) 0.2 sd(Y) 0.06
r 0.3
expected return =0.5*B1+0.5*B2
sd =SQRT(0.5^2 * D1^2 + 0.5^2 * D2^2 + 2* 0.5 *0.5 * B4*D1*D2)

b)

r 0.6
expected return 0.175
sd 0.049244289
r 0
expected return 0.175
sd 0.039051248

c)

w1 w2 mean sd
0 1 0.2 0.06
0.05 0.95 0.1975 0.057799
0.1 0.9 0.195 0.055705
0.15 0.85 0.1925 0.053728
0.2 0.8 0.19 0.051884
0.25 0.75 0.1875 0.050187
0.3 0.7 0.185 0.048652
0.35 0.65 0.1825 0.047294
0.4 0.6 0.18 0.04613
0.45 0.55 0.1775 0.045175
0.5 0.5 0.175 0.044441
0.55 0.45 0.1725 0.04394
0.6 0.4 0.17 0.043681
0.65 0.35 0.1675 0.043666
0.7 0.3 0.165 0.043898
0.75 0.25 0.1625 0.044371
0.8 0.2 0.16 0.045078
0.85 0.15 0.1575 0.046008
0.9 0.1 0.155 0.047149
0.95 0.05 0.1525 0.048485
1 0 0.15 0.05

formula

E(X) 0.15 sd(X) 0.05
E(Y) 0.2 sd(Y) 0.06
r = 0.3
w1 w2 mean sd
0 =1-A6 =A6*$B$1+B6*$B$2 =SQRT(A6^2*$D$1^2 + B6^2 *$D$2^2+ 2*A6*B6*0.3*$D$1*$D$2)
=0.05+A6 =1-A7 =A7*$B$1+B7*$B$2 =SQRT(A7^2*$D$1^2 + B7^2 *$D$2^2+ 2*A7*B7*0.3*$D$1*$D$2)
=0.05+A7 =1-A8 =A8*$B$1+B8*$B$2 =SQRT(A8^2*$D$1^2 + B8^2 *$D$2^2+ 2*A8*B8*0.3*$D$1*$D$2)
=0.05+A8 =1-A9 =A9*$B$1+B9*$B$2 =SQRT(A9^2*$D$1^2 + B9^2 *$D$2^2+ 2*A9*B9*0.3*$D$1*$D$2)
=0.05+A9 =1-A10 =A10*$B$1+B10*$B$2 =SQRT(A10^2*$D$1^2 + B10^2 *$D$2^2+ 2*A10*B10*0.3*$D$1*$D$2)
=0.05+A10 =1-A11 =A11*$B$1+B11*$B$2 =SQRT(A11^2*$D$1^2 + B11^2 *$D$2^2+ 2*A11*B11*0.3*$D$1*$D$2)
=0.05+A11 =1-A12 =A12*$B$1+B12*$B$2 =SQRT(A12^2*$D$1^2 + B12^2 *$D$2^2+ 2*A12*B12*0.3*$D$1*$D$2)
=0.05+A12 =1-A13 =A13*$B$1+B13*$B$2 =SQRT(A13^2*$D$1^2 + B13^2 *$D$2^2+ 2*A13*B13*0.3*$D$1*$D$2)
=0.05+A13 =1-A14 =A14*$B$1+B14*$B$2 =SQRT(A14^2*$D$1^2 + B14^2 *$D$2^2+ 2*A14*B14*0.3*$D$1*$D$2)
=0.05+A14 =1-A15 =A15*$B$1+B15*$B$2 =SQRT(A15^2*$D$1^2 + B15^2 *$D$2^2+ 2*A15*B15*0.3*$D$1*$D$2)
=0.05+A15 =1-A16 =A16*$B$1+B16*$B$2 =SQRT(A16^2*$D$1^2 + B16^2 *$D$2^2+ 2*A16*B16*0.3*$D$1*$D$2)
=0.05+A16 =1-A17 =A17*$B$1+B17*$B$2 =SQRT(A17^2*$D$1^2 + B17^2 *$D$2^2+ 2*A17*B17*0.3*$D$1*$D$2)
=0.05+A17 =1-A18 =A18*$B$1+B18*$B$2 =SQRT(A18^2*$D$1^2 + B18^2 *$D$2^2+ 2*A18*B18*0.3*$D$1*$D$2)
=0.05+A18 =1-A19 =A19*$B$1+B19*$B$2 =SQRT(A19^2*$D$1^2 + B19^2 *$D$2^2+ 2*A19*B19*0.3*$D$1*$D$2)
=0.05+A19 =1-A20 =A20*$B$1+B20*$B$2 =SQRT(A20^2*$D$1^2 + B20^2 *$D$2^2+ 2*A20*B20*0.3*$D$1*$D$2)
=0.05+A20 =1-A21 =A21*$B$1+B21*$B$2 =SQRT(A21^2*$D$1^2 + B21^2 *$D$2^2+ 2*A21*B21*0.3*$D$1*$D$2)
=0.05+A21 =1-A22 =A22*$B$1+B22*$B$2 =SQRT(A22^2*$D$1^2 + B22^2 *$D$2^2+ 2*A22*B22*0.3*$D$1*$D$2)
=0.05+A22 =1-A23 =A23*$B$1+B23*$B$2 =SQRT(A23^2*$D$1^2 + B23^2 *$D$2^2+ 2*A23*B23*0.3*$D$1*$D$2)
=0.05+A23 =1-A24 =A24*$B$1+B24*$B$2 =SQRT(A24^2*$D$1^2 + B24^2 *$D$2^2+ 2*A24*B24*0.3*$D$1*$D$2)
=0.05+A24 =1-A25 =A25*$B$1+B25*$B$2 =SQRT(A25^2*$D$1^2 + B25^2 *$D$2^2+ 2*A25*B25*0.3*$D$1*$D$2)
=0.05+A25 =1-A26 =A26*$B$1+B26*$B$2 =SQRT(A26^2*$D$1^2 + B26^2 *$D$2^2+ 2*A26*B26*0.3*$D$1*$D$2)

d)

r 0
w1 w2 mean sd
0 1 0.2 0.06
0.05 0.95 0.1975 0.057054798
0.1 0.9 0.195 0.054230987
0.15 0.85 0.1925 0.051548521
0.2 0.8 0.19 0.049030603
0.25 0.75 0.1875 0.046703854
0.3 0.7 0.185 0.044598206
0.35 0.65 0.1825 0.042746345
0.4 0.6 0.18 0.041182521
0.45 0.55 0.1775 0.039940581
0.5 0.5 0.175 0.039051248
0.55 0.45 0.1725 0.038538941
0.6 0.4 0.17 0.038418745
0.65 0.35 0.1675 0.038694315
0.7 0.3 0.165 0.039357337
0.75 0.25 0.1625 0.040388736
0.8 0.2 0.16 0.041761226
0.85 0.15 0.1575 0.043442491
0.9 0.1 0.155 0.045398238
0.95 0.05 0.1525 0.047594643
1 0 0.15 0.05
r 0.6
w1 w2 mean sd
0 1 0.2 0.06
0.05 0.95 0.1975 0.058534178
0.1 0.9 0.195 0.057140179
0.15 0.85 0.1925 0.055823382
0.2 0.8 0.19 0.054589376
0.25 0.75 0.1875 0.053443896
0.3 0.7 0.185 0.052392748
0.35 0.65 0.1825 0.051441715
0.4 0.6 0.18 0.050596443
0.45 0.55 0.1775 0.04986231
0.5 0.5 0.175 0.049244289
0.55 0.45 0.1725 0.048746795
0.6 0.4 0.17 0.048373546
0.65 0.35 0.1675 0.048127435
0.7 0.3 0.165 0.048010416
0.75 0.25 0.1625 0.048023432
0.8 0.2 0.16 0.048166378
0.85 0.15 0.1575 0.048438105
0.9 0.1 0.155 0.048836462
0.95 0.05 0.1525 0.049358383
1 0 0.15 0.05

orchestra answered 1 year ago
Next > < Previous

Related Solutions

In this exercise, we examine the effect of combining investments with positively correlated risks, negatively correlated...
In this exercise, we examine the effect of combining investments with positively correlated risks, negatively correlated risks, and uncorrelated risks. A firm is considering a portfolio of assets. The portfolio is comprised of two assets, which we will call ''A" and "B." Let X denote the annual rate of return from asset A in the following year, and let Y denote the annual rate of return from asset B in the following year. Suppose that E(X) = 0.15 and E(Y)...
CHAPTER 2-4 Exercise 2.24 In this exercise, we examine the effect of combining investments with positively...
CHAPTER 2-4 Exercise 2.24 In this exercise, we examine the effect of combining investments with positively correlated risks, negatively correlated risks, and uncorrelated risks. A firm is considering a portfolio of assets. The portfolio is comprised of two assets, which we will call ''A" and "B." Let X denote the annual rate of return from asset A in the following year, and let Y denote the annual rate of return from asset B in the following year. Suppose that E(X)...
Diversification is “combining negatively correlated assets to reduce, or diversify, risk.” Do you think your investments...
Diversification is “combining negatively correlated assets to reduce, or diversify, risk.” Do you think your investments are adequately diversified? Why or why not? What can you apply from this to improve your personal financial health and lower your risk or increase your return?
How do emotions positively & negatively effect negotiations?
How do emotions positively & negatively effect negotiations?
In this exercise, we examine the effect of the margin of error on determining the sample...
In this exercise, we examine the effect of the margin of error on determining the sample size needed. Find the sample size needed to give, with 99% confidence, a margin of error within ±7. Within ±3. Within ±1. Assume that we use σ˜=20 as our estimate of the standard deviation in each case. Round your answers up to the nearest integer. ME=7 : n= ? ME=3 : n= ? ME=1 : n= ?
1. In this exercise, we examine the effect of the value of the estimated standard deviation...
1. In this exercise, we examine the effect of the value of the estimated standard deviation on determining the sample size needed. Find the sample size needed to give, with 90% confidence, a margin of error within ±4, if the estimated standard deviation is σ˜=60. If the estimated standard deviation is σ˜=20. If the estimated standard deviation is σ˜=10. σ˜=60 : n=? σ˜=20 : n= ? σ˜=10 : n=? 2. In this exercise, we examine the effect of the confidence...
During our first discussion we noticed that P/E and profitability ratios are not necessarily highly correlated—positively...
During our first discussion we noticed that P/E and profitability ratios are not necessarily highly correlated—positively or negatively. Think carefully about this issue and discuss why that is the case. Specifically, think about the role of earning numbers, how they are generated, what their purpose is, how markets view them, other factors that impact the quality of these numbers, and other sources of information that are needed to interpret these numbers correctly.
In this exercise, we examine one of the conditions of the Alternating Series Test. Consider the...
In this exercise, we examine one of the conditions of the Alternating Series Test. Consider the alternating series 1−1+1/2−1/4+1/3−1/9+1/4−1/16+⋯, where the terms are selected alternately from the sequences {1/n} and {−1/n^2}. Explain why the nth term of the given series converges to 0 as n goes to infinity. Rewrite the given series by grouping terms in the following manner: (1−1)+(1/2−1/4)+(1/3−1/9)+(1/4−1/16)+⋯. Use this regrouping to determine if the series converges or diverges. Explain why the condition that the sequence {an}{an} decreases...
In this exercise we examine the effects of overbooking in the airline industry. Ontario Gateway Airlines'...
In this exercise we examine the effects of overbooking in the airline industry. Ontario Gateway Airlines' first class cabins have 10 seats in each plane. Ontario's overbooking policy is to sell up to 11 first class tickets, since cancellations and no-shows are always possible (and indeed are quite likely). For a given flight on Ontario Gateway, there were 11 first class tickets sold. Suppose that each of the 11 persons who purchased tickets has a 20% chance of not showing...
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT