FINA4466
Solution to supplementary Questions
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Solution to supplementary Questions
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Question 1
Assume you want to purchase 400 shares of ABC common stock on margin from
your broker. Stock ABC is currently trading at $103 per share. If you only have
$22,000 to invest in this purchase, what is the initial percentage margin?
Ans.:
𝐿. 𝑇. = 𝑄 × 𝑆 − 𝐼. 𝐼𝑛𝑣. = (400 × 103) − 22,000 = 19,200
𝑄 × 𝑆 − 𝐿. 𝑇.
22,000
𝐼. 𝑀 =
=
= 0.5339 = 53.39%
𝑄×𝑆
400 × 103
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Solution to supplementary Questions
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Question 2
Consider a $1,000 par value T-Bill selling at $875 with 135 days to maturity. What
is the equivalent effective annual yield on the T-Bill?
Ans.:
𝐸𝐴𝑌 =
𝐹𝑉
𝑃𝑉
(365/𝑛 )
−1 =
1000
875
365
135
− 1 = 0.4348 = 43.48%
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Solution to supplementary Questions
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Question 3
When a distribution has high kurtosis.
Ans.:
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Standard deviation underestimates risk.
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Solution to supplementary Questions
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Question 4
Assume your tax bracket is equal to 23%. In order for you to be indifferent between
the returns on a taxable corporate bond and a tax-exempt municipal bond paying
18%. How much should the taxable corporate bond offer you?
Ans.:
𝑟𝑁𝑜𝑛−𝑇𝑎𝑥𝑎𝑏𝑙𝑒 = 𝑟𝑇𝑎𝑥𝑎𝑏𝑙𝑒 (1 − 𝑇𝑎𝑥𝑅𝑎𝑡𝑒)
→ 𝑟𝑇𝑎𝑥𝑎𝑏𝑙𝑒 =
𝑟𝑁𝑜𝑛−𝑇𝑎𝑥𝑎𝑏𝑙𝑒
0.18
=
= 0.2338 = 23.38%
(1 − 𝑇𝑎𝑥𝑅𝑎𝑡𝑒) (1 − 0.23)
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Solution to supplementary Questions
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Question 5
You purchased 120 shares of ABC common stock on margin at $100 per share.
Assume the initial margin is 50%, and the maintenance margin is 30%. Below what
stock price level would you get a margin call? Assume the stock pays no dividend;
ignore interest on margin.
Ans.:
𝐼. 𝑀 =
(120 × 100) − 𝐿. 𝑇.
𝑄 × 𝑆 − 𝐿. 𝑇.
→ 0.5 =
→ 𝐿. 𝑇. = 6,000
(120 × 100)
𝑄×𝑆
(120 × 𝑆 ′ ) − 6,000
𝑄 × 𝑆 ′ − 𝐿. 𝑇.
𝑀. 𝑀 =
→ 0.3 =
(120 × 𝑆 ′ )
𝑄 × 𝑆′
→ (120 × 𝑆′) − 6,000 = 0.3 × 120 × 𝑆′
→ 0.7 × 120 × 𝑆 ′ = 6,000
6,000
→ 𝑆′ =
= 71.43
0.7 × 120
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Solution to supplementary Questions
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Question 6
Suppose that you want to sell short 500 shares of stock XYZ. XYZ stock is currently
selling for $120. Your broker has a 48% initial margin requirement. How much cash
or cash equivalent, do you have to deposit if you decide to do the short sale with
your broker?
Ans.:
𝑀. 𝑅. =
(120 × 100) + 𝑀. 𝐴.
𝑄 × 𝑆 + 𝑀. 𝐴.
→ 1.48 =
→ 𝑀. 𝐴. = 28,800
(120 × 100)
𝑄×𝑆
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Solution to supplementary Questions
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Question 7
You purchase a share of XYZ.com stock for $93. One year later, after receiving a
dividend of $8, you sell the stock for $79. What was your holding-period return?
Ans.:
𝐻𝑃𝑅 =
79 + 8 − 93
= −6.45%
93
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Solution to supplementary Questions
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Question 8
The investment manager of a corporate pension fund has purchased a Treasury bill
with 230 days to maturity at a price of $850 per $1,000 face value. Calculate the
bond equivalent yield for the Treasury bill.
Ans.:
𝑟𝐵𝐸𝑌 =
𝐹𝑉 − 𝑃𝑉 365 1000 − 850 365
×
=
×
= 0.28 = 28.00%
𝑃𝑉
𝑛
850
230
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Solution to supplementary Questions
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Question 9
You have been given this probability distribution for the expected rate of return for
ABC stock under different possible states of the economy:
State
of
Economy
Boom
Normal Growth
Recession
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the Probability
Occurrence
0.25
0.35
0.40
of Expected
Return
22%
11%
-15%
ABC
stock
What is the expected rate of return for stock ABC?
Ans.:
𝐸(𝑟𝐴𝐵𝐶 ) = (0.25 × 0.22) + (0.35 × 0.11) + (0.4 × −0.15) = 0.035 = 3.5%
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Solution to supplementary Questions
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Question 10
_______ are real assets.
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Ans.:
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Machines
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Solution to supplementary Questions
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Question 11
New issues of securities are sold in the ________ market(s).
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Ans.:
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Primary
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Solution to supplementary Questions
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Question 12
Consider the rate of return on Portfolio P below:
Year
2018
2019
2020
2021
rP
-13 %
+15 %
- 20 %
11 %
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What will be the value of $100 by the end of 2021 if invested in portfolio P at the
beginning of 2018?
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Ans.:
𝐹𝑉2021 = 100 × (1 − 0.13) × (1 + 0.15) × (1 − 0.2) × (1 + 0.11) = $88.84
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Solution to supplementary Questions
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Question 13
Tania wants to buy a US Treasury Bill that has with 230 days to maturity at a price
of $680 per $1,000 face value. Calculate the bank discount yield for the Treasury
bill.
Ans.:
𝑟𝐵𝐷𝑌 =
𝐹𝑉 − 𝑃𝑉 360 1000 − 680 360
×
=
×
= 0.5009 = 50.09%
𝐹𝑉
𝑛
1000
230
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Solution to supplementary Questions
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Question 14
You purchased 250 shares of common stock on margin for $80 per share. The initial
margin is 60%, and the stock pays no dividend. What would your rate of return be if
you sell the stock at $111 per share after one year? Ignore interest on margin.
Ans.:
(250 × 80) − 𝐿. 𝑇.
𝑄 × 𝑆 − 𝐿. 𝑇.
→ 0.6 =
(250 × 80)
𝑄×𝑆
→ 𝐿. 𝑇. = 0.4 × 250 × 80
→ 𝐿. 𝑇. = 8,000
→ 𝐼. 𝐼𝑛𝑣. = 𝑄 × 𝑆 − 𝐿. 𝑇. = (250 × 80) − 8,000 = 12,000
𝐼. 𝑀 =
(𝑄 × 𝑆1 ) − 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 − 𝐿. 𝐴. −𝐼. 𝐼𝑛𝑣
𝐼. 𝐼𝑛𝑣
(250 × 111) − 0 − 8,000 − 12,000
𝑅𝑒𝑡𝑢𝑟𝑛 =
= 0.6458 = 64.58%
12,000
𝑅𝑒𝑡𝑢𝑟𝑛 =
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Solution to supplementary Questions
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Question 15
You sold short 73 shares of common stock at $66 per share. The initial margin is
47%. Your initial investment was
Ans.:
𝑀. 𝑅. =
(73 × 66) + 𝑀. 𝐴.
𝑄 × 𝑆 + 𝑀. 𝐴.
→ 1.47 =
→ 𝑀. 𝐴. = 2,264.46
(73 × 66)
𝑄×𝑆
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Solution to supplementary Questions
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Question 16
Anna is faced with a risky portfolio P [E(𝑟 ), 𝜎 ] and a risk-free asset. P: E(𝑟 ) =
21% and 𝜎 = 38%. The risk-free rate is equal to 3%. Anna has $250 to invest. She
wants to form a portfolio (C) by investing in portfolio (P) and the risk-free asset.
Anna desires to achieve an expected return equal to 30%. How much money should
Anna invest in portfolio P?
Ans.:
0.3 = (𝑦) (0.21) + (1 − 𝑦)(0.03) = 0.21𝑦 + 0.03 − 0.03𝑦 → 𝑦 =
𝐼𝑛𝑣𝑒𝑠𝑡 ′𝑦′ 𝑖𝑛 𝑃: 1.5 × $250 = $375
0.27
= 1.5
0.18
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Solution to supplementary Questions
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Question 17
Clara is faced with a risky portfolio P [E(𝑟 ), 𝜎 ] and a risk-free asset. P: E(𝑟 ) =
21% and 𝜎 = 38%. The risk-free rate is equal to 3%. Clara has $250 to invest. She
wants to form a portfolio (C) by investing in portfolio (P) and the risk-free asset.
Clara doesn’t want her portfolio standard deviation to exceed 20%. How much
money should Clara invest in the Risk-Free asset?
Ans.:
𝜎𝐶 = 𝑦 𝜎𝑃 → 𝑦 =
𝜎𝐶
0.20
=
= 0.5263
𝜎𝑃
0.38
𝐼𝑛𝑣𝑒𝑠𝑡 (1 − 𝑦) 𝑖𝑛 𝑅𝐹 → 𝐼𝑛𝑣𝑒𝑠𝑡 (1 − 0.5263) 𝑖𝑛 𝑅𝐹 → 𝐼𝑛𝑣𝑒𝑠𝑡 (0.4737) 𝑖𝑛 𝑅𝐹
→ 𝐼𝑛𝑣𝑒𝑠𝑡 ($250 × 0.4737) 𝑖𝑛 𝑅𝐹 → 𝐼𝑛𝑣𝑒𝑠𝑡 ($118.42) 𝑖𝑛 𝑅𝐹
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Solution to supplementary Questions
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Question 18
Brian has the following utility function: U= E(r) - 1/2 A σ2. Brian has a Risk
Aversion Coefficient equal to 3. Assume that Brian is faced with the following
portfolios:
Expected Return
14%
18%
30%
Portfolio A
Portfolio B
Portfolio C
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Risk (SD)
8%
15%
28%
What portfolio should Brian pick to maximize his utility?
Ans.:
1
(3)(0.082 ) = 0.1304
2
1
𝑈𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝐵 = 0.18 − (3)(0.152 ) = 0.14625
2
𝟏
𝑼𝑷𝒐𝒓𝒕𝒇𝒐𝒍𝒊𝒐 𝑪 = 𝟎. 𝟑𝟎 − (𝟑)(𝟎. 𝟐𝟖𝟐 ) = 𝟎. 𝟏𝟖𝟐𝟒
𝟐
𝑃𝑖𝑐𝑘 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝐶
𝑈𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝐴 = 0.14 −
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Solution to supplementary Questions
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Question 19
Brian has the following utility function: U= E(r) - 1/2 A σ2. Brian has a Risk
Aversion Coefficient equal to 3. Assume that Brian is faced with the following
portfolio:
Expected Return
??
30%
Portfolio B
Portfolio C
Risk (SD)
25%
28%
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How much should be the expected return for portfolio B, so that Brian is indifferent
between portfolio B and portfolio C?
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Ans.:
𝑈𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝐵 = 𝑈𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝐶
1
1
(3)(0.252 ) = 0.30 − (3)(0.282 )
2
2
→ 𝐸(𝑟𝐵 ) = 0.1824 + 0.09375 = 0.2762 = 27.62%
→ 𝐸(𝑟𝐵 ) −
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Solution to supplementary Questions
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Question 20
We want to form an index using the two stocks presented in the below table:
Initial Price
(P0)
Stock-A
Stock-B
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Shares
Outstanding
(Q0)
30 Millions
15 Millions
$78
$98
Price at the
end of Period
1 (P1)
$88
$148
Shares
Outstanding
(Q1)
20 Millions
15 Millions
Calculate the rate of return on a price weighted index for the first period (from t = 0
to t = 1)
88 − 78
𝑟𝐴 =
= 0.1282
78
Ans.:
148 − 98
𝑟𝐵 =
98
= 0.5102
78
= 0.4432
78 + 98
98
𝑤𝐵 =
= 0.5568
78 + 98
𝑤𝐴 =
𝑟𝑝𝑟𝑖𝑐𝑒 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑖𝑛𝑑𝑒𝑥 = (𝑤𝐴 × 𝑟𝐴 ) + (𝑤𝐵 × 𝑟𝐵 )
→ 𝑟𝑝𝑟𝑖𝑐𝑒 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑖𝑛𝑑𝑒𝑥 = (0.4432 × 0.1282) + (0.5568 × 0.5102) = 0.3409
= 34.09%
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Solution to supplementary Questions
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Question 21
We want to form an index using the two stocks presented in the below table:
Initial Price
(P0)
Stock-A
Stock-B
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$78
$98
Shares
Outstanding
(Q0)
30 Millions
15 Millions
Price at the
end of Period
1 (P1)
$88
$148
Shares
Outstanding
(Q1)
20 Millions
15 Millions
Calculate the rate of return on an equal weighted index for the first period (from t =
0 to t = 1)
88 − 78
Ans.:
𝑟𝐴 =
= 0.1282
78
148 − 98
= 0.5102
98
𝑟𝑝𝑟𝑖𝑐𝑒 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑖𝑛𝑑𝑒𝑥 = (𝑤𝐴 × 𝑟𝐴 ) + (𝑤𝐵 × 𝑟𝐵 )
→ 𝑟𝑝𝑟𝑖𝑐𝑒 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑖𝑛𝑑𝑒𝑥 = (0.5 × 0.1282) + (0.5 × 0.5102) = 0.3192 = 31.92%
𝑟𝐵 =
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Solution to supplementary Questions
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Question 22
Assume Adam is a risk-neutral investor. Adam wants to invest $200 in either
portfolio A, portfolio B, portfolio C, or portfolio D. Which portfolio should Adam
choose?
Portfolio A
Portfolio B
Portfolio C
Portfolio D
Expected Return
14%
18%
30%
25%
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Ans.:
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Adam should choose portfolio C.
Risk (SD)
8%
15%
28%
35%
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Solution to supplementary Questions
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Question 23
Consider the rate of return on Portfolio P below:
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What is the Geometric Mean achieved on these observed returns?
Ans.:
Year
2018
2019
2020
2021
rP
-10 %
+15 %
- 20 %
32 %
1
𝐺. 𝑀. = [(1 − 0.1)(1 + 0.15)(1 − 0.2)(1 + 0.32)]4 − 1 = 0.0225 = 2.25%
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Solution to supplementary Questions
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Question 24
________ specialize in helping companies raise capital by selling securities.
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Ans.:
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Investment bankers
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25
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