Solutions to Further Problems
Risk Management and Financial
Institutions
Fourth Edition
John C. Hull
,Chapter 1: Introduction
1.15.
Suppose that one investment has a mean return of 8% and a standard deviation of return of 14%.
Another investment has a mean return of 12% and a standard deviation of return of 20%. The
correlation between the returns is 0.3. Produce a chart similar to Figure 1.2 showing alternative
risk-return combinations from the two investments.
The impact of investing w1 in the first investment and w2 = 1 – w1 in the second investment is
shown in the table below. The range of possible risk-return trade-offs is shown in figure below.
w1 w2 P P
0.0 1.0 12% 20%
0.2 0.8 11.2% 17.05%
0.4 0.6 10.4% 14.69%
0.6 0.4 9.6% 13.22%
0.8 0.2 8.8% 12.97%
1.0 0.0 8.0% 14.00%
1.16.
The expected return on the market is 12% and the risk-free rate is 7%. The standard deviation of
the return on the market is 15%. One investor creates a portfolio on the efficient frontier with an
expected return of 10%. Another creates a portfolio on the efficient frontier with an expected
return of 20%. What is the standard deviation of the return on each of the two portfolios?
In this case the efficient frontier is as shown in the figure below. The standard deviation of
returns corresponding to an expected return of 10% is 9%. The standard deviation of returns
corresponding to an expected return of 20% is 39%.
,1.17.
A bank estimates that its profit next year is normally distributed with a mean of 0.8% of assets
and the standard deviation of 2% of assets. How much equity (as a percentage of assets) does the
company need to be (a) 99% sure that it will have a positive equity at the end of the year and (b)
99.9% sure that it will have positive equity at the end of the year? Ignore taxes.
(a) The bank can be 99% certain that profit will better than 0.8−2.33×2 or –3.85% of assets. It
therefore needs equity equal to 3.85% of assets to be 99% certain that it will have a positive
equity at the year end.
(b) The bank can be 99.9% certain that profit will be greater than 0.8 − 3.09 × 2 or –5.38% of
assets. It therefore needs equity equal to 5.38% of assets to be 99.9% certain that it will have a
positive equity at the year end.
1.18.
A portfolio manager has maintained an actively managed portfolio with a beta of 0.2. During the
last year, the risk-free rate was 5% and major equity indices performed very badly, providing
returns of about −30%. The portfolio manager produced a return of −10% and claims that in the
circumstances it was good. Discuss this claim.
When the expected return on the market is −30% the expected return on a portfolio with a beta of
0.2 is
0.05 + 0.2 × (−0.30 − 0.05) = −0.02
or –2%. The actual return of –10% is worse than the expected return. The portfolio manager has
achieved an alpha of –8%!
, Chapter 2: Banks
2.15.
Regulators calculate that DLC bank (see Section 2.2) will report a profit that is normally
distributed with a mean of $0.6 million and a standard deviation of $2.0 million. How much
equity capital in addition to that in Table 2.2 should regulators require for there to be a 99.9%
chance of the capital not being wiped out by losses?
There is a 99.9% chance that the profit will not be worse than 0.6 − 3.090 × 2.0 = −$5.58
million. Regulators will require $0.58 million of additional capital.
2.16.
Explain the moral hazard problems with deposit insurance. How can they be overcome?
Deposit insurance makes depositors less concerned about the financial health of a bank. As a
result, banks may be able to take more risk without being in danger of losing deposits. This is an
example of moral hazard. (The existence of the insurance changes the behavior of the parties
involved with the result that the expected payout on the insurance contract is higher.) Regulatory
requirements that banks keep sufficient capital for the risks they are taking reduce their incentive
to take risks. One approach (used in the U.S.) to avoiding the moral hazard problem is to make
the premiums that banks have to pay for deposit insurance dependent on an assessment of the
risks they are taking.
2.17.
The bidders in a Dutch auction are as follows:
Bidder Number of shares Price
A 60,000 $50.00
B 20,000 $80.00
C 30,000 $55.00
D 40,000 $38.00
E 40,000 $42.00
F 40,000 $42.00
G 50,000 $35.00
H 50,000 $60.00
The number of shares being auctioned is 210,000. What is the price paid by investors? How
many shares does each investor receive?
When ranked from highest to lowest the bidders are B, H, C, A, E and F, D, and G. Individuals
B, H, C, and A bid for 160, 000 shares in total. Individuals E and F bid for a further 80,000
shares. The price paid by the investors is therefore the price bid by E and F (i.e., $42).
Individuals B, H, C, and A get the whole amount of the shares they bid for. Individuals E and F
gets 25,000 shares each.
Risk Management and Financial
Institutions
Fourth Edition
John C. Hull
,Chapter 1: Introduction
1.15.
Suppose that one investment has a mean return of 8% and a standard deviation of return of 14%.
Another investment has a mean return of 12% and a standard deviation of return of 20%. The
correlation between the returns is 0.3. Produce a chart similar to Figure 1.2 showing alternative
risk-return combinations from the two investments.
The impact of investing w1 in the first investment and w2 = 1 – w1 in the second investment is
shown in the table below. The range of possible risk-return trade-offs is shown in figure below.
w1 w2 P P
0.0 1.0 12% 20%
0.2 0.8 11.2% 17.05%
0.4 0.6 10.4% 14.69%
0.6 0.4 9.6% 13.22%
0.8 0.2 8.8% 12.97%
1.0 0.0 8.0% 14.00%
1.16.
The expected return on the market is 12% and the risk-free rate is 7%. The standard deviation of
the return on the market is 15%. One investor creates a portfolio on the efficient frontier with an
expected return of 10%. Another creates a portfolio on the efficient frontier with an expected
return of 20%. What is the standard deviation of the return on each of the two portfolios?
In this case the efficient frontier is as shown in the figure below. The standard deviation of
returns corresponding to an expected return of 10% is 9%. The standard deviation of returns
corresponding to an expected return of 20% is 39%.
,1.17.
A bank estimates that its profit next year is normally distributed with a mean of 0.8% of assets
and the standard deviation of 2% of assets. How much equity (as a percentage of assets) does the
company need to be (a) 99% sure that it will have a positive equity at the end of the year and (b)
99.9% sure that it will have positive equity at the end of the year? Ignore taxes.
(a) The bank can be 99% certain that profit will better than 0.8−2.33×2 or –3.85% of assets. It
therefore needs equity equal to 3.85% of assets to be 99% certain that it will have a positive
equity at the year end.
(b) The bank can be 99.9% certain that profit will be greater than 0.8 − 3.09 × 2 or –5.38% of
assets. It therefore needs equity equal to 5.38% of assets to be 99.9% certain that it will have a
positive equity at the year end.
1.18.
A portfolio manager has maintained an actively managed portfolio with a beta of 0.2. During the
last year, the risk-free rate was 5% and major equity indices performed very badly, providing
returns of about −30%. The portfolio manager produced a return of −10% and claims that in the
circumstances it was good. Discuss this claim.
When the expected return on the market is −30% the expected return on a portfolio with a beta of
0.2 is
0.05 + 0.2 × (−0.30 − 0.05) = −0.02
or –2%. The actual return of –10% is worse than the expected return. The portfolio manager has
achieved an alpha of –8%!
, Chapter 2: Banks
2.15.
Regulators calculate that DLC bank (see Section 2.2) will report a profit that is normally
distributed with a mean of $0.6 million and a standard deviation of $2.0 million. How much
equity capital in addition to that in Table 2.2 should regulators require for there to be a 99.9%
chance of the capital not being wiped out by losses?
There is a 99.9% chance that the profit will not be worse than 0.6 − 3.090 × 2.0 = −$5.58
million. Regulators will require $0.58 million of additional capital.
2.16.
Explain the moral hazard problems with deposit insurance. How can they be overcome?
Deposit insurance makes depositors less concerned about the financial health of a bank. As a
result, banks may be able to take more risk without being in danger of losing deposits. This is an
example of moral hazard. (The existence of the insurance changes the behavior of the parties
involved with the result that the expected payout on the insurance contract is higher.) Regulatory
requirements that banks keep sufficient capital for the risks they are taking reduce their incentive
to take risks. One approach (used in the U.S.) to avoiding the moral hazard problem is to make
the premiums that banks have to pay for deposit insurance dependent on an assessment of the
risks they are taking.
2.17.
The bidders in a Dutch auction are as follows:
Bidder Number of shares Price
A 60,000 $50.00
B 20,000 $80.00
C 30,000 $55.00
D 40,000 $38.00
E 40,000 $42.00
F 40,000 $42.00
G 50,000 $35.00
H 50,000 $60.00
The number of shares being auctioned is 210,000. What is the price paid by investors? How
many shares does each investor receive?
When ranked from highest to lowest the bidders are B, H, C, A, E and F, D, and G. Individuals
B, H, C, and A bid for 160, 000 shares in total. Individuals E and F bid for a further 80,000
shares. The price paid by the investors is therefore the price bid by E and F (i.e., $42).
Individuals B, H, C, and A get the whole amount of the shares they bid for. Individuals E and F
gets 25,000 shares each.
