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MATH 134A-量化金融代写
时间：2022-12-05
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
https://canvas.eee.uci.edu/courses/47222/quizzes/234674 1/20
Practice Final Exam
Due Dec 8 at 11pm Points 100 Questions 18 Time Limit None
Instructions
Attempt History
Attempt Time Score
LATEST Attempt 1 201 minutes 10 out of 100
Submitted Dec 4 at 6:09pm
Please note that Canvas will auto-submit your responses to this Final Exam at exactly the time the exam
ends, unless you have manually submitted your Final Exam before that time.
Final Letter Grade Determination
Your final letter grade will be based on a curve that reflects how your performance compares to other
students in the class. The final letter grade is not known until the curve is determined after the final exam
has been graded and all your scores for the quarter are reviewed carefully. In this class we will not drop
the lowest homework or quiz score; all of your work contributes to your final grade as outlined in this
syllabus.
5 / 5 ptsQuestion 1
A cutting-edge pharmaceutical company has developed a new vaccine for
the Coronavirus disease 2019 (COVID-19) . Production of of the vaccine
would require $10 million in initial capital expenditure. It is anticipated that
1 million units would be sold each year for 5 years, and then herd
immunity would be achieved and the mass production of the vaccine
would cease. Each year's production would require 10,000 hours of labor
and 100 tons of raw material.
In the first year, the average wage rate is $30 per hour, the cost of the raw
material is $100 per ton and the vaccine will sell for $3.30 per unit. All
three of these unit prices (wage rate per hour, cost of raw material per ton,
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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and vaccine revenue per unit) will increase each year after the first year
by the inflation rate which is assumed to be 10% per year. All cash flows
(revenues and costs) are assumed to come at the end of each year.
The interest rate is 3% and corporate tax rate is 34% on profit. The initial
capital expenditure can be depreciated in a straight line fashion over 5
years ($2 million per year).
What is the (after-tax) present value of the new vaccine? Please round
your numerical answer to the nearest integer number of dollars.
4,087,435
Correct!
orrect Answers 4,087,435 (with margin: 2,000)
0 / 5 ptsQuestion 2
In lecture we defined quasi-modified duration as the extension of the
concept of duration to the term structure framework.
It is possible to extend the process of immunization to the term structure
framework. A portfolio of bonds designed to fund a stream of obligations
can be immunized against a parallel shift in the spot rate curve by
matching both the present values and the quasi-modified durations of the
bonds and the obligations.
Consider the following stream of future obligations, in dollars,
corresponding to college tuition payments for 2 children, Jay and Rio, as
shown in the table below below:
Year 1 2 3 4 5
JAY 50,000 51,000 52,000 53,000 0
RIO 51,000 52,000 53,000 54,000
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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You are given the spot rate curve is as follows:
YEAR SPOT RATE %
1 2.25
2 2.50
3 2.75
4 3.00
5 3.25
Find a portfolio, consisting of the two bonds below, that has the same
present value as the obligation stream (including tuition payments for both
Jay and Rio) and is immunized against an additive shift in the spot rate
curve:
Bond 1 is a 5-year 6% annual coupon bond with a price of $112.78
Bond 2 is a 4-year 10% annual coupon bond with a price of $126.25
Both bonds have a face value of $100. How many units of Bond 2 should
be held in the immunized portfolio?
Please round your numerical answer to the nearest integer number of
units.
2,522
ou Answered
orrect Answers 4,843 (with margin: 10)
5 / 5 ptsQuestion 3
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
https://canvas.eee.uci.edu/courses/47222/quizzes/234674 4/20
In 1997, the U.S. Treasury issued "Treasury inflation-protected securities"
(TIPS). These are fixed-income securities that are inflation indexed to
protect their value against inflation. Like conventional bonds, they have a
fixed coupon rate and maturity date, but the face value is periodically
adjusted for inflation by multiplying the original face value by the ratio of
the Consumer Price Index (CPI) at the current date to the CPI at the
original issue date. At maturity, the bondholder receives the maximum of
the inflation-adjusted face value or the original face value. Hence, if
deflation occurs, the bondholder is guaranteed not to lose on the face
value. A chart of the historical CPI annual percent changes over the past
20 years, for illustrative purposes only, is provided below.
In October, 2010, you observe the following prices of two 10-year
Treasury inflation-protected securities (TIPS):
TIPS 1: P = 79.66 C = 3% F = 100
TIPS 2: P = 100.00 C = 6% F = 100
where P is the price, C is the coupon rate, and F is the original face value.
Compute the price of a theoretical 10-year inflation-adjusted zero coupon
bond with original face value of 100. Please round your numerical answer
to 2 decimal places. Hint: you do not need to know the precise numerical
values for the CPI annual percent changes in order to solve this problem.
1 1 1
2 2 2
59.32
Correct!
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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orrect Answers 59.32 (with margin: 0.01)
0 / 5 ptsQuestion 4Unanswered
Upon graduation from UCI you decide to buy a home and take out a
variable-rate mortgage. The mortgage value is $1,000,000, the term is 30
years with monthly compounding, and initially the nominal annual interest
rate is 2%. This nominal annual interest rate is guaranteed to be fixed at
2% for 5 years, after which time the rate will be adjusted according to
prevailing rates. The new rate can be applied to the loan either by
changing the monthly payment amount, or by keeping the monthly
payment amount the same and extending the length of the mortgage.
If the interest rate on the mortgage changes to 5% after 5 years, what will
be the new monthly payment that keeps the termination time the same?
Please round your numerical answer to the nearest integer number of
dollars.
ou Answered
orrect Answers 5,098 (with margin: 5)
0 / 5 ptsQuestion 5Unanswered
Upon graduation from UCI you decide to buy a home and take out a
variable-rate mortgage. The mortgage value is $1,000,000, the term is 30
years with monthly compounding, and initially the nominal annual interest
rate is 2%. This nominal annual interest rate is guaranteed to be fixed at
2% for 5 years, after which time the rate will be adjusted according to
prevailing rates. The new rate can be applied to the loan either by
changing the monthly payment amount, or by keeping the monthly
payment amount the same and extending the length of the mortgage.
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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If the interest rate on the mortgage changes to 5% after 5 years, how
many months beyond the original term of 30 years does the mortgage
need to be extended if the monthly payments are to remain the same?
Please round your numerical answer to an integer number of months.
ou Answered
orrect Answers 681 (with margin: 5)
0 / 5 ptsQuestion 6Unanswered
A specialized machine essential for a company's operations costs
$16,000 and has operating costs of $2,000 the first year. The operating
costs increase by $1,000 each year thereafter. We assume that the
operating costs occur at the end of each year. The annual interest rate is
6% and the company plans to stay in operation forever.
You have an option to replace the machine periodically after a period of n
years, where n must be an integer. The replacement cost is $16,000.
Your objective is to select the replacement period n such that the present
value of the total cost is minimized. Assume that due to its specialized
nature, the machine has no salvage value.
What is the optimal replacement period, n? Note n must be an integer.
ou Answered
orrect Answers 6 (with margin: 0)
0 / 2 ptsQuestion 7Unanswered
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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Assume is the variance of the returns of the total stock market and
is the covariance between the returns of risky asset and the returns
of the total stock market.
If the total market includes risky assets and the weight is the weight
of the risky asset corresponding to the market portfolio, then
True orrect Answer
False
0 / 5 ptsQuestion 8Unanswered
In this problem we assume that the annual expected rate of return of the
market portfolio is 12% and the annual risk-free rate is 2%. The standard
deviation of the market portfolio returns is 24%. Assume the market is in
equilibrium such that the Capital Asset Pricing Model (CAPM) holds: the
market portfolio is efficient.
If you have $1,000 to invest, how should you allocate it to achieve an
annual expected return of 18%?
Invest $600 in the risk-free asset and $400 in the market portfolio
Invest $600 in the risk-free asset and sell short $1600 in the market
portfolio
Invest $260 in the risk-free asset and $740 in the market portfolio
Invest $800 in the risk-free asset and $200 in the market portfolio
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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Borrow $600 at the risk-free rate and invest $1,600 in the market portfolio
orrect Answer
Borrow $260 at the risk-free rate and invest $1,260 in the market portfolio
0 / 5 ptsQuestion 9Unanswered
The top 5 stocks in the S&P 500 index, when ranked by market
capitalization, make up 22% of the total market capitalization of the S&P
500 index.
Numerical estimates of the mean (or expected) rates of return values of
these top five stocks, expressed in decimal (not percentage) form, are
listed in the table below.
STOCK TICKER MEAN RETURN
Apple, Inc. AAPL 0.69
Amazon.com Inc. AMZN 0.65
Microsoft Corp. MSFT 0.44
Alphabet, Inc. GOOG 0.36
Facebook, Inc. FB 0.41
The variance of returns (entries on the main diagonal) and covariances
between returns of these top five stocks, expressed in decimal (not
percentage) form, are as follows:
AAPL AMZN MSFT GOOG FB
AAPL 0.21 0.12 0.17 0.13 0.16
AMZN 0.12 0.15 0.12 0.10 0.12
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MSFT 0.17 0.12 0.19 0.14 0.15
GOOG 0.13 0.10 0.14 0.15 0.14
FB 0.16 0.12 0.15 0.14 0.21
You would like to invest $10,000 and seek an expected return of 80%
annually. A friend tells you that this is impossible since the stock with the
highest annual expected return, from the table above, is AAPL with an
expected annual return of 69%. Your friend advises you to invest all the
$10,000 in AAPL.
Assuming there is no risk-free asset available and short selling is
allowed, is it possible to allocate your $10,000 amongst the 5 stocks in
the table to achieve an expected return of 80%?
If so, what is the optimal number of dollars to invest in AAPL such that
the expected portfolio return is 80% while minimizing the portfolio
variance?
If not, then enter an answer of 10,000.
Please round your numerical answer to the nearest integer number of
dollars.
ou Answered
orrect Answers 7,449 (with margin: 50)
0 / 5 ptsQuestion 10Unanswered
The top 5 stocks in the S&P 500 index, when ranked by market
capitalization, make up 22% of the total market capitalization of the S&P
500 index.
Numerical estimates of the mean (or expected) rates of return values of
these top five stocks, expressed in decimal (not percentage) form, are
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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listed in the table below.
STOCK TICKER MEAN RETURN
Apple, Inc. AAPL 0.69
Amazon.com Inc. AMZN 0.65
Microsoft Corp. MSFT 0.44
Alphabet, Inc. GOOG 0.36
Facebook, Inc. FB 0.41
The variance of returns (entries on the main diagonal) and covariances
between returns of these top five stocks, expressed in decimal (not
percentage) form, are as follows:
AAPL AMZN MSFT GOOG FB
AAPL 0.21 0.12 0.17 0.13 0.16
AMZN 0.12 0.15 0.12 0.10 0.12
MSFT 0.17 0.12 0.19 0.14 0.15
GOOG 0.13 0.10 0.14 0.15 0.14
FB 0.16 0.12 0.15 0.14 0.21
Assuming there is no risk-free asset available, suppose you desire to
invest in a portfolio of these 5 stocks by minimizing the variance of your
portfolio of 5 stocks, subject to the constraint that the weightings of each
stock position sum to 1. Note that short positions are permissible.
You are risk-averse and hope that a portfolio with variance less than
or equal to 0.14 is attainable. A friend tells you that this is impossible
since the stocks with the lowest variance from the table above, AMZN or
GOOG, have a variance of 0.15.
Using mean-variance portfolio optimization, determine the absolute
minimum possible portfolio variance.
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Please express your numerical answer in decimal (not percentage) form
and round your numerical answer to two decimal places.
ou Answered
orrect Answers 0.12 (with margin: 0.01)
0 / 5 ptsQuestion 11Unanswered
The top 5 stocks in the S&P 500 index, when ranked by market
capitalization, make up 22% of the total market capitalization of the S&P
500 index.
Numerical estimates of the mean (or expected) rates of return values of
these top five stocks, expressed in decimal (not percentage) form, are
listed in the table below.
STOCK TICKER MEAN RETURN
Apple, Inc. AAPL 0.69
Amazon.com Inc. AMZN 0.65
Microsoft Corp. MSFT 0.44
Alphabet, Inc. GOOG 0.36
Facebook, Inc. FB 0.41
The variance of returns (entries on the main diagonal) and covariances
between returns of these top five stocks, expressed in decimal (not
percentage) form, are as follows:
AAPL AMZN MSFT GOOG FB
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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AAPL 0.21 0.12 0.17 0.13 0.16
AMZN 0.12 0.15 0.12 0.10 0.12
MSFT 0.17 0.12 0.19 0.14 0.15
GOOG 0.13 0.10 0.14 0.15 0.14
FB 0.16 0.12 0.15 0.14 0.21
Assuming there is a risk-free asset available in addition to these 5
risky assets, determine the portfolio weightings for each of these stocks of
the unique fund F as defined in the One-fund theorem of Modern Portfolio
Theory. Note that short positions (negative weights) are permissible
and the sum of all the weights must be equal to one. The annual risk-free
rate is 4%.
What is the optimal portfolio weight for Alphabet Inc. (ticker: GOOG)?
Please express your answer in decimal (not percentage) form and round
your numerical answer to two decimal places.
ou Answered
orrect Answers 0.06 (with margin: 0.02)
0 / 10 ptsQuestion 12Unanswered
In this problem we assume the market portfolio is the S&P 500 index.
The top 5 stocks in the S&P 500 index, when ranked by market
capitalization, make up 22% of the total market capitalization of the S&P
500 index.
Numerical estimates of the beta values of these top five stocks are listed
in the table below.
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STOCK TICKER BETA
Apple, Inc. AAPL 1.16
Amazon.com Inc. AMZN 0.70
Microsoft Corp. MSFT 1.15
Alphabet, Inc. GOOG 0.98
Facebook, Inc. FB 1.01
Furthermore, the variance of returns (entries on the main diagonal) and
covariances between returns of these top five stocks are as follows:
AAPL AMZN MSFT GOOG FB
AAPL 0.21 0.12 0.17 0.13 0.16
AMZN 0.12 0.15 0.12 0.10 0.12
MSFT 0.17 0.12 0.19 0.14 0.15
GOOG 0.13 0.10 0.14 0.15 0.14
FB 0.16 0.12 0.15 0.14 0.21
The variance of the S&P 500 index returns is known to be 0.04.
Suppose we desire to invest in the S&P 500 index portfolio, but find that it
is impractical to invest in all 500 stocks. Instead we choose to invest in the
five stocks in the table in a way that replicates or tracks the S&P 500
index portfolio most closely---in the sense of minimizing the variance of
the difference in returns between our portfolio of 5 stocks and the S&P
500 index portfolio. Note that short positions (negative weights) are
permissible and the sum of all the weights must be equal to one.
What is the optimal portfolio weight for Apple, Inc. (ticker: AAPL)?
Please express your answer in decimal (not percentage) form and round
your numerical answer to two decimal places.
ou Answered
orrect Answers 0.17 (with margin: 0.02)
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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0 / 10 ptsQuestion 13Unanswered
In this problem we assume the market portfolio is the S&P 500 index.
The top 5 stocks in the S&P 500 index, when ranked by market
capitalization, make up 22% of the total market capitalization of the S&P
500 index.
Numerical estimates of the mean (or expected) rates of return values of
these top five stocks, expressed in decimal (not percentage) form, are
listed in the table below.
STOCK TICKER MEAN RETURN
Apple, Inc. AAPL 0.69
Amazon.com Inc. AMZN 0.65
Microsoft Corp. MSFT 0.44
Alphabet, Inc. GOOG 0.36
Facebook, Inc. FB 0.41
Numerical estimates of the beta values of these top five stocks are listed
in the table below.
STOCK TICKER BETA
Apple, Inc. AAPL 1.16
Amazon.com Inc. AMZN 0.70
Microsoft Corp. MSFT 1.15
Alphabet, Inc. GOOG 0.98
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Facebook, Inc. FB 1.01
Furthermore, the variance of returns (entries on the main diagonal) and
covariances between returns of these top five stocks are as follows:
AAPL AMZN MSFT GOOG FB
AAPL 0.21 0.12 0.17 0.13 0.16
AMZN 0.12 0.15 0.12 0.10 0.12
MSFT 0.17 0.12 0.19 0.14 0.15
GOOG 0.13 0.10 0.14 0.15 0.14
FB 0.16 0.12 0.15 0.14 0.21
The variance of the S&P 500 index returns is known to be 0.04.
Suppose we desire to invest in the S&P 500 index portfolio, but find that it
is impractical to invest in all 500 stocks. Instead we choose to invest in the
five stocks in the table in a way that replicates or tracks the S&P 500
index portfolio most closely---in the sense of minimizing the variance of
the difference in returns between our portfolio of 5 stocks and the S&P
500 index portfolio. Simultaneously, we desire to choose the portfolio
weights so that the portfolio mean return is equal to 0.10.
Note that short positions (negative weights) are permissible and the sum
of all the weights must be equal to one.
What is the optimal portfolio weight for Alphabet, Inc. (ticker: GOOG)?
Please express your answer in decimal (not percentage) form and round
your numerical answer to two decimal places.
ou Answered
orrect Answers 1.03 (with margin: 0.1)
0 / 3 ptsQuestion 14Unanswered
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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At what price is the yield of a 10-year, 10% bond equal to the yield of a 5-
year, 4% bond whose price is $102?
Both bonds pay coupons every 6 months and have a face value of $100.
Please round your numerical answer to two decimal places.
ou Answered
orrect Answers 153.79 (with margin: 2)
0 / 5 ptsQuestion 15Unanswered
Upon graduation from UCI you receive a generous gift of $500,000 from a
wealthy relative to buy a house in Irvine. Instead of using the gift to buy a
house, you decide to invest the $500,000 in the S&P 500 index and take
out a mortgage loan for an additional $500,000 to buy the house.
Assume the mortgage has an original principal balance of $500,000 and
has a term of 30 years. The mortgage has an annual interest rate of 3%,
or 0.25% per month. The mortgage compounds monthly. The
downpayment on the house is zero.
Assume your investment in the S&P 500 index grows at 1% per month
and also compounds monthly, for 30 years.
In order to make the fixed monthly mortgage payments that occur at the
end of each month, you sell a portion of your investment in the S&P 500
index at the end of each month to exactly make the required mortgage
payment.
Assume there are zero transactions costs and zero taxes.
After 30 years, your mortgage loan is completely paid off and you own the
house. How much is your investment in the S&P 500 index worth at that
time?
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Between $5 million and $7 million
Between $11 million and $13 million
Between $7 million and $9 million
Less than $5 million
Between $13 million and $15 million
Between $9 million and $11 million orrect Answer
More than $15 million
0 / 5 ptsQuestion 16Unanswered
Consider a market with 100 mutually uncorrelated risky assets that all
have the same identical variance but whose Sharpe ratios are all
different, ranging according to the formula:
Using Markowitz mean-variance optimization, determine the optimal
portfolio weights, assuming a risk-free asset exists. Note the risk-free rate
and variance are unknown.
What is the optimal portfolio weight for the risky asset with the maximum
Sharpe ratio?
Please express your answer in percentage and round your numerical
answer to two decimal places.
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ou Answered
orrect Answers 1.98 (with margin: 0.02)
0 / 10 ptsQuestion 17Unanswered
Consider a popular portfolio management strategy called constant-
proportion rebalancing.
Assume a risky stock price doubles or halves every day, with equal
probability.
You start off with $100 and invest 50% in the risky stock and 50% in cash
which earns zero interest.
At the end of every day, no matter how the stock price changes, you
rebalance your portfolio to be exactly 50% invested in the risky stock and
50% invested in cash.
What is the expected value of this constant-proportion rebalanced
portfolio after 10 days?
Please round your numerical answer to the nearest integer number of
dollars.
ou Answered
orrect Answers 325 (with margin: 10)
0 / 5 ptsQuestion 18Unanswered
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Bond Trading
Consider the United States Treasury Yield Curve today:
The precise yield-to-maturities for 6 tradable zero-coupon U.S. treasuries
are listed here:
30-year Bond 3.1%
25-year Bond 3.0%
20-year Bond 3.0%
15-year Bond 2.5%
10-year Note 2.5%
5-year Note 1.0%
As a bond trader, you are given $100,000 to invest and seek to maximize
the value of your investment in 10 years. You are free to invest in any of
the 6 U.S. zero-coupon treasuries listed in the table above today, and may
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rebalance your portfolio once again in exactly 5 years. After this
rebalancing, you must hold the portfolio for the remaining 5 years.
You may assume the yield curve does not change over the entire 10
year investment period. Note this implies that in 5 years, the 30-year
zero-coupon bond will be a 25-year zero-coupon bond with 3.0% yield-to-
maturity, the 25-year zero-coupon bond will be a 20-year zero-coupon
bond with 3.0% yield-to-maturity, the 20-year zero-coupon bond will be a
15-year bond with 2.5% yield-to-maturity, etc.
What is the maximum possible value of your investment after 10 years?
Please round your numerical answer to the nearest dollar.
ou Answered
orrect Answers 155,515 (with margin: 100)
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Practice Final Exam
Due Dec 8 at 11pm Points 100 Questions 18 Time Limit None
Instructions
Attempt History
Attempt Time Score
LATEST Attempt 1 201 minutes 10 out of 100
Submitted Dec 4 at 6:09pm
Please note that Canvas will auto-submit your responses to this Final Exam at exactly the time the exam
ends, unless you have manually submitted your Final Exam before that time.
Final Letter Grade Determination
Your final letter grade will be based on a curve that reflects how your performance compares to other
students in the class. The final letter grade is not known until the curve is determined after the final exam
has been graded and all your scores for the quarter are reviewed carefully. In this class we will not drop
the lowest homework or quiz score; all of your work contributes to your final grade as outlined in this
syllabus.
5 / 5 ptsQuestion 1
A cutting-edge pharmaceutical company has developed a new vaccine for
the Coronavirus disease 2019 (COVID-19) . Production of of the vaccine
would require $10 million in initial capital expenditure. It is anticipated that
1 million units would be sold each year for 5 years, and then herd
immunity would be achieved and the mass production of the vaccine
would cease. Each year's production would require 10,000 hours of labor
and 100 tons of raw material.
In the first year, the average wage rate is $30 per hour, the cost of the raw
material is $100 per ton and the vaccine will sell for $3.30 per unit. All
three of these unit prices (wage rate per hour, cost of raw material per ton,
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and vaccine revenue per unit) will increase each year after the first year
by the inflation rate which is assumed to be 10% per year. All cash flows
(revenues and costs) are assumed to come at the end of each year.
The interest rate is 3% and corporate tax rate is 34% on profit. The initial
capital expenditure can be depreciated in a straight line fashion over 5
years ($2 million per year).
What is the (after-tax) present value of the new vaccine? Please round
your numerical answer to the nearest integer number of dollars.
4,087,435
Correct!
orrect Answers 4,087,435 (with margin: 2,000)
0 / 5 ptsQuestion 2
In lecture we defined quasi-modified duration as the extension of the
concept of duration to the term structure framework.
It is possible to extend the process of immunization to the term structure
framework. A portfolio of bonds designed to fund a stream of obligations
can be immunized against a parallel shift in the spot rate curve by
matching both the present values and the quasi-modified durations of the
bonds and the obligations.
Consider the following stream of future obligations, in dollars,
corresponding to college tuition payments for 2 children, Jay and Rio, as
shown in the table below below:
Year 1 2 3 4 5
JAY 50,000 51,000 52,000 53,000 0
RIO 51,000 52,000 53,000 54,000
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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You are given the spot rate curve is as follows:
YEAR SPOT RATE %
1 2.25
2 2.50
3 2.75
4 3.00
5 3.25
Find a portfolio, consisting of the two bonds below, that has the same
present value as the obligation stream (including tuition payments for both
Jay and Rio) and is immunized against an additive shift in the spot rate
curve:
Bond 1 is a 5-year 6% annual coupon bond with a price of $112.78
Bond 2 is a 4-year 10% annual coupon bond with a price of $126.25
Both bonds have a face value of $100. How many units of Bond 2 should
be held in the immunized portfolio?
Please round your numerical answer to the nearest integer number of
units.
2,522
ou Answered
orrect Answers 4,843 (with margin: 10)
5 / 5 ptsQuestion 3
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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In 1997, the U.S. Treasury issued "Treasury inflation-protected securities"
(TIPS). These are fixed-income securities that are inflation indexed to
protect their value against inflation. Like conventional bonds, they have a
fixed coupon rate and maturity date, but the face value is periodically
adjusted for inflation by multiplying the original face value by the ratio of
the Consumer Price Index (CPI) at the current date to the CPI at the
original issue date. At maturity, the bondholder receives the maximum of
the inflation-adjusted face value or the original face value. Hence, if
deflation occurs, the bondholder is guaranteed not to lose on the face
value. A chart of the historical CPI annual percent changes over the past
20 years, for illustrative purposes only, is provided below.
In October, 2010, you observe the following prices of two 10-year
Treasury inflation-protected securities (TIPS):
TIPS 1: P = 79.66 C = 3% F = 100
TIPS 2: P = 100.00 C = 6% F = 100
where P is the price, C is the coupon rate, and F is the original face value.
Compute the price of a theoretical 10-year inflation-adjusted zero coupon
bond with original face value of 100. Please round your numerical answer
to 2 decimal places. Hint: you do not need to know the precise numerical
values for the CPI annual percent changes in order to solve this problem.
1 1 1
2 2 2
59.32
Correct!
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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orrect Answers 59.32 (with margin: 0.01)
0 / 5 ptsQuestion 4Unanswered
Upon graduation from UCI you decide to buy a home and take out a
variable-rate mortgage. The mortgage value is $1,000,000, the term is 30
years with monthly compounding, and initially the nominal annual interest
rate is 2%. This nominal annual interest rate is guaranteed to be fixed at
2% for 5 years, after which time the rate will be adjusted according to
prevailing rates. The new rate can be applied to the loan either by
changing the monthly payment amount, or by keeping the monthly
payment amount the same and extending the length of the mortgage.
If the interest rate on the mortgage changes to 5% after 5 years, what will
be the new monthly payment that keeps the termination time the same?
Please round your numerical answer to the nearest integer number of
dollars.
ou Answered
orrect Answers 5,098 (with margin: 5)
0 / 5 ptsQuestion 5Unanswered
Upon graduation from UCI you decide to buy a home and take out a
variable-rate mortgage. The mortgage value is $1,000,000, the term is 30
years with monthly compounding, and initially the nominal annual interest
rate is 2%. This nominal annual interest rate is guaranteed to be fixed at
2% for 5 years, after which time the rate will be adjusted according to
prevailing rates. The new rate can be applied to the loan either by
changing the monthly payment amount, or by keeping the monthly
payment amount the same and extending the length of the mortgage.
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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If the interest rate on the mortgage changes to 5% after 5 years, how
many months beyond the original term of 30 years does the mortgage
need to be extended if the monthly payments are to remain the same?
Please round your numerical answer to an integer number of months.
ou Answered
orrect Answers 681 (with margin: 5)
0 / 5 ptsQuestion 6Unanswered
A specialized machine essential for a company's operations costs
$16,000 and has operating costs of $2,000 the first year. The operating
costs increase by $1,000 each year thereafter. We assume that the
operating costs occur at the end of each year. The annual interest rate is
6% and the company plans to stay in operation forever.
You have an option to replace the machine periodically after a period of n
years, where n must be an integer. The replacement cost is $16,000.
Your objective is to select the replacement period n such that the present
value of the total cost is minimized. Assume that due to its specialized
nature, the machine has no salvage value.
What is the optimal replacement period, n? Note n must be an integer.
ou Answered
orrect Answers 6 (with margin: 0)
0 / 2 ptsQuestion 7Unanswered
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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Assume is the variance of the returns of the total stock market and
is the covariance between the returns of risky asset and the returns
of the total stock market.
If the total market includes risky assets and the weight is the weight
of the risky asset corresponding to the market portfolio, then
True orrect Answer
False
0 / 5 ptsQuestion 8Unanswered
In this problem we assume that the annual expected rate of return of the
market portfolio is 12% and the annual risk-free rate is 2%. The standard
deviation of the market portfolio returns is 24%. Assume the market is in
equilibrium such that the Capital Asset Pricing Model (CAPM) holds: the
market portfolio is efficient.
If you have $1,000 to invest, how should you allocate it to achieve an
annual expected return of 18%?
Invest $600 in the risk-free asset and $400 in the market portfolio
Invest $600 in the risk-free asset and sell short $1600 in the market
portfolio
Invest $260 in the risk-free asset and $740 in the market portfolio
Invest $800 in the risk-free asset and $200 in the market portfolio
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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Borrow $600 at the risk-free rate and invest $1,600 in the market portfolio
orrect Answer
Borrow $260 at the risk-free rate and invest $1,260 in the market portfolio
0 / 5 ptsQuestion 9Unanswered
The top 5 stocks in the S&P 500 index, when ranked by market
capitalization, make up 22% of the total market capitalization of the S&P
500 index.
Numerical estimates of the mean (or expected) rates of return values of
these top five stocks, expressed in decimal (not percentage) form, are
listed in the table below.
STOCK TICKER MEAN RETURN
Apple, Inc. AAPL 0.69
Amazon.com Inc. AMZN 0.65
Microsoft Corp. MSFT 0.44
Alphabet, Inc. GOOG 0.36
Facebook, Inc. FB 0.41
The variance of returns (entries on the main diagonal) and covariances
between returns of these top five stocks, expressed in decimal (not
percentage) form, are as follows:
AAPL AMZN MSFT GOOG FB
AAPL 0.21 0.12 0.17 0.13 0.16
AMZN 0.12 0.15 0.12 0.10 0.12
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MSFT 0.17 0.12 0.19 0.14 0.15
GOOG 0.13 0.10 0.14 0.15 0.14
FB 0.16 0.12 0.15 0.14 0.21
You would like to invest $10,000 and seek an expected return of 80%
annually. A friend tells you that this is impossible since the stock with the
highest annual expected return, from the table above, is AAPL with an
expected annual return of 69%. Your friend advises you to invest all the
$10,000 in AAPL.
Assuming there is no risk-free asset available and short selling is
allowed, is it possible to allocate your $10,000 amongst the 5 stocks in
the table to achieve an expected return of 80%?
If so, what is the optimal number of dollars to invest in AAPL such that
the expected portfolio return is 80% while minimizing the portfolio
variance?
If not, then enter an answer of 10,000.
Please round your numerical answer to the nearest integer number of
dollars.
ou Answered
orrect Answers 7,449 (with margin: 50)
0 / 5 ptsQuestion 10Unanswered
The top 5 stocks in the S&P 500 index, when ranked by market
capitalization, make up 22% of the total market capitalization of the S&P
500 index.
Numerical estimates of the mean (or expected) rates of return values of
these top five stocks, expressed in decimal (not percentage) form, are
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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listed in the table below.
STOCK TICKER MEAN RETURN
Apple, Inc. AAPL 0.69
Amazon.com Inc. AMZN 0.65
Microsoft Corp. MSFT 0.44
Alphabet, Inc. GOOG 0.36
Facebook, Inc. FB 0.41
The variance of returns (entries on the main diagonal) and covariances
between returns of these top five stocks, expressed in decimal (not
percentage) form, are as follows:
AAPL AMZN MSFT GOOG FB
AAPL 0.21 0.12 0.17 0.13 0.16
AMZN 0.12 0.15 0.12 0.10 0.12
MSFT 0.17 0.12 0.19 0.14 0.15
GOOG 0.13 0.10 0.14 0.15 0.14
FB 0.16 0.12 0.15 0.14 0.21
Assuming there is no risk-free asset available, suppose you desire to
invest in a portfolio of these 5 stocks by minimizing the variance of your
portfolio of 5 stocks, subject to the constraint that the weightings of each
stock position sum to 1. Note that short positions are permissible.
You are risk-averse and hope that a portfolio with variance less than
or equal to 0.14 is attainable. A friend tells you that this is impossible
since the stocks with the lowest variance from the table above, AMZN or
GOOG, have a variance of 0.15.
Using mean-variance portfolio optimization, determine the absolute
minimum possible portfolio variance.
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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Please express your numerical answer in decimal (not percentage) form
and round your numerical answer to two decimal places.
ou Answered
orrect Answers 0.12 (with margin: 0.01)
0 / 5 ptsQuestion 11Unanswered
The top 5 stocks in the S&P 500 index, when ranked by market
capitalization, make up 22% of the total market capitalization of the S&P
500 index.
Numerical estimates of the mean (or expected) rates of return values of
these top five stocks, expressed in decimal (not percentage) form, are
listed in the table below.
STOCK TICKER MEAN RETURN
Apple, Inc. AAPL 0.69
Amazon.com Inc. AMZN 0.65
Microsoft Corp. MSFT 0.44
Alphabet, Inc. GOOG 0.36
Facebook, Inc. FB 0.41
The variance of returns (entries on the main diagonal) and covariances
between returns of these top five stocks, expressed in decimal (not
percentage) form, are as follows:
AAPL AMZN MSFT GOOG FB
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AAPL 0.21 0.12 0.17 0.13 0.16
AMZN 0.12 0.15 0.12 0.10 0.12
MSFT 0.17 0.12 0.19 0.14 0.15
GOOG 0.13 0.10 0.14 0.15 0.14
FB 0.16 0.12 0.15 0.14 0.21
Assuming there is a risk-free asset available in addition to these 5
risky assets, determine the portfolio weightings for each of these stocks of
the unique fund F as defined in the One-fund theorem of Modern Portfolio
Theory. Note that short positions (negative weights) are permissible
and the sum of all the weights must be equal to one. The annual risk-free
rate is 4%.
What is the optimal portfolio weight for Alphabet Inc. (ticker: GOOG)?
Please express your answer in decimal (not percentage) form and round
your numerical answer to two decimal places.
ou Answered
orrect Answers 0.06 (with margin: 0.02)
0 / 10 ptsQuestion 12Unanswered
In this problem we assume the market portfolio is the S&P 500 index.
The top 5 stocks in the S&P 500 index, when ranked by market
capitalization, make up 22% of the total market capitalization of the S&P
500 index.
Numerical estimates of the beta values of these top five stocks are listed
in the table below.
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STOCK TICKER BETA
Apple, Inc. AAPL 1.16
Amazon.com Inc. AMZN 0.70
Microsoft Corp. MSFT 1.15
Alphabet, Inc. GOOG 0.98
Facebook, Inc. FB 1.01
Furthermore, the variance of returns (entries on the main diagonal) and
covariances between returns of these top five stocks are as follows:
AAPL AMZN MSFT GOOG FB
AAPL 0.21 0.12 0.17 0.13 0.16
AMZN 0.12 0.15 0.12 0.10 0.12
MSFT 0.17 0.12 0.19 0.14 0.15
GOOG 0.13 0.10 0.14 0.15 0.14
FB 0.16 0.12 0.15 0.14 0.21
The variance of the S&P 500 index returns is known to be 0.04.
Suppose we desire to invest in the S&P 500 index portfolio, but find that it
is impractical to invest in all 500 stocks. Instead we choose to invest in the
five stocks in the table in a way that replicates or tracks the S&P 500
index portfolio most closely---in the sense of minimizing the variance of
the difference in returns between our portfolio of 5 stocks and the S&P
500 index portfolio. Note that short positions (negative weights) are
permissible and the sum of all the weights must be equal to one.
What is the optimal portfolio weight for Apple, Inc. (ticker: AAPL)?
Please express your answer in decimal (not percentage) form and round
your numerical answer to two decimal places.
ou Answered
orrect Answers 0.17 (with margin: 0.02)
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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0 / 10 ptsQuestion 13Unanswered
In this problem we assume the market portfolio is the S&P 500 index.
The top 5 stocks in the S&P 500 index, when ranked by market
capitalization, make up 22% of the total market capitalization of the S&P
500 index.
Numerical estimates of the mean (or expected) rates of return values of
these top five stocks, expressed in decimal (not percentage) form, are
listed in the table below.
STOCK TICKER MEAN RETURN
Apple, Inc. AAPL 0.69
Amazon.com Inc. AMZN 0.65
Microsoft Corp. MSFT 0.44
Alphabet, Inc. GOOG 0.36
Facebook, Inc. FB 0.41
Numerical estimates of the beta values of these top five stocks are listed
in the table below.
STOCK TICKER BETA
Apple, Inc. AAPL 1.16
Amazon.com Inc. AMZN 0.70
Microsoft Corp. MSFT 1.15
Alphabet, Inc. GOOG 0.98
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Facebook, Inc. FB 1.01
Furthermore, the variance of returns (entries on the main diagonal) and
covariances between returns of these top five stocks are as follows:
AAPL AMZN MSFT GOOG FB
AAPL 0.21 0.12 0.17 0.13 0.16
AMZN 0.12 0.15 0.12 0.10 0.12
MSFT 0.17 0.12 0.19 0.14 0.15
GOOG 0.13 0.10 0.14 0.15 0.14
FB 0.16 0.12 0.15 0.14 0.21
The variance of the S&P 500 index returns is known to be 0.04.
Suppose we desire to invest in the S&P 500 index portfolio, but find that it
is impractical to invest in all 500 stocks. Instead we choose to invest in the
five stocks in the table in a way that replicates or tracks the S&P 500
index portfolio most closely---in the sense of minimizing the variance of
the difference in returns between our portfolio of 5 stocks and the S&P
500 index portfolio. Simultaneously, we desire to choose the portfolio
weights so that the portfolio mean return is equal to 0.10.
Note that short positions (negative weights) are permissible and the sum
of all the weights must be equal to one.
What is the optimal portfolio weight for Alphabet, Inc. (ticker: GOOG)?
Please express your answer in decimal (not percentage) form and round
your numerical answer to two decimal places.
ou Answered
orrect Answers 1.03 (with margin: 0.1)
0 / 3 ptsQuestion 14Unanswered
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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At what price is the yield of a 10-year, 10% bond equal to the yield of a 5-
year, 4% bond whose price is $102?
Both bonds pay coupons every 6 months and have a face value of $100.
Please round your numerical answer to two decimal places.
ou Answered
orrect Answers 153.79 (with margin: 2)
0 / 5 ptsQuestion 15Unanswered
Upon graduation from UCI you receive a generous gift of $500,000 from a
wealthy relative to buy a house in Irvine. Instead of using the gift to buy a
house, you decide to invest the $500,000 in the S&P 500 index and take
out a mortgage loan for an additional $500,000 to buy the house.
Assume the mortgage has an original principal balance of $500,000 and
has a term of 30 years. The mortgage has an annual interest rate of 3%,
or 0.25% per month. The mortgage compounds monthly. The
downpayment on the house is zero.
Assume your investment in the S&P 500 index grows at 1% per month
and also compounds monthly, for 30 years.
In order to make the fixed monthly mortgage payments that occur at the
end of each month, you sell a portion of your investment in the S&P 500
index at the end of each month to exactly make the required mortgage
payment.
Assume there are zero transactions costs and zero taxes.
After 30 years, your mortgage loan is completely paid off and you own the
house. How much is your investment in the S&P 500 index worth at that
time?
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Between $5 million and $7 million
Between $11 million and $13 million
Between $7 million and $9 million
Less than $5 million
Between $13 million and $15 million
Between $9 million and $11 million orrect Answer
More than $15 million
0 / 5 ptsQuestion 16Unanswered
Consider a market with 100 mutually uncorrelated risky assets that all
have the same identical variance but whose Sharpe ratios are all
different, ranging according to the formula:
Using Markowitz mean-variance optimization, determine the optimal
portfolio weights, assuming a risk-free asset exists. Note the risk-free rate
and variance are unknown.
What is the optimal portfolio weight for the risky asset with the maximum
Sharpe ratio?
Please express your answer in percentage and round your numerical
answer to two decimal places.
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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ou Answered
orrect Answers 1.98 (with margin: 0.02)
0 / 10 ptsQuestion 17Unanswered
Consider a popular portfolio management strategy called constant-
proportion rebalancing.
Assume a risky stock price doubles or halves every day, with equal
probability.
You start off with $100 and invest 50% in the risky stock and 50% in cash
which earns zero interest.
At the end of every day, no matter how the stock price changes, you
rebalance your portfolio to be exactly 50% invested in the risky stock and
50% invested in cash.
What is the expected value of this constant-proportion rebalanced
portfolio after 10 days?
Please round your numerical answer to the nearest integer number of
dollars.
ou Answered
orrect Answers 325 (with margin: 10)
0 / 5 ptsQuestion 18Unanswered
12/5/22, 1:38 PM Practice Final Exam: MATH 134A LECTURE B (44684)
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Bond Trading
Consider the United States Treasury Yield Curve today:
The precise yield-to-maturities for 6 tradable zero-coupon U.S. treasuries
are listed here:
30-year Bond 3.1%
25-year Bond 3.0%
20-year Bond 3.0%
15-year Bond 2.5%
10-year Note 2.5%
5-year Note 1.0%
As a bond trader, you are given $100,000 to invest and seek to maximize
the value of your investment in 10 years. You are free to invest in any of
the 6 U.S. zero-coupon treasuries listed in the table above today, and may
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rebalance your portfolio once again in exactly 5 years. After this
rebalancing, you must hold the portfolio for the remaining 5 years.
You may assume the yield curve does not change over the entire 10
year investment period. Note this implies that in 5 years, the 30-year
zero-coupon bond will be a 25-year zero-coupon bond with 3.0% yield-to-
maturity, the 25-year zero-coupon bond will be a 20-year zero-coupon
bond with 3.0% yield-to-maturity, the 20-year zero-coupon bond will be a
15-year bond with 2.5% yield-to-maturity, etc.
What is the maximum possible value of your investment after 10 years?
Please round your numerical answer to the nearest dollar.
ou Answered
orrect Answers 155,515 (with margin: 100)
