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 agoRelated 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
ADVERTISEMENT
Latest Questions
- Can you explain to me the differnces in how we paid for the health care (access...
- Discuss proactive and defensive marketing in context to the company position in the hypothetical market structure...
- Write a java code that could be used to show the simulation of a tornado. use...
- A Canadian company issues an 8-year term bond with face value of $1,000 and 6% coupon...
- 1. Consider the following nuclides: &
- In general, teams can be described three ways. Name and define each.
- JAVASCRIPT: Please create an array of student names and another array of student grades. - Create...
ADVERTISEMENT