Download HW Assignment 2

HW: 2
Course: M339D/M389D - Intro to Financial Math
Page: 1 of 2
University of Texas at Austin
HW Assignment 2
Bid-ask spread, transaction costs, simple return, short-sales
Provide your final answer only for the following problems.
Problem 2.1. (1 pt) Source: Problem 1.3.c. from McDonald.
ABC stock has a bid price of $40.95 and an ask price of $41.05. Assume there is a $20 brokerage commission.
Suppose that you buy 100 shares, then immediately sell the 100 shares with the bid and ask prices being as
above. What is your round-trip transaction cost?
(a) About $12.50 gain
(b) About $50.00 gain
(c) About $50.00 loss
(d) About $12.50 loss
(e) None of the above
Solution: (c)
Remember that the terminology bid and ask is formulated from the market makers perspective. Therefore,
the price at which you can buy is called the ask price. Furthermore, you will have to pay the commission to
your broker for the transaction. You pay:
($41.05 × 100) + $20 = $4,125.00
Similarly, you can sell at the market maker’s bid price. You will again have to pay a commission, and
your broker will deduct the commission from the sales price of the shares. You receive:
($40.95 × 100) − $20 = $4,075.00
Your round-trip transaction costs amount to:
$4,125.00 − $4,075.00 = $50
Problem 2.2. Source: FM Exam, November 2000, Problem #24.
Bill and Jane each sell a different stock short for a price of 1000 . For both investors, the margin requirement
is 50%, and interest on the margin is credited at an annual effective rate of 6%. Their respective brokers
keep the short-sale proceeds (without crediting interest).
Bill buys back his stock one year later at a price of P . At the end of the year, just prior to the closing of
the short sale, the stock paid a dividend of X.
Jane also buys back her stock after one year, at a price of (P − 25). At the end of the year, just prior to
the closing of the short sale, her stock paid a dividend of 2X .
Both investors earned an annual effective yield of 21% on their short sales. Calculate P .
(a) 800
(b) 825
(c) 850
(d) 875
(e) None of the above
Solution: (e)
For both the initial margin deposit equals 1000 · 0.5 = 500. So, the both receive 0.06 · 500 = 30 in interest
on the margin account.
Bill’s yield rate satisfies:
30 + (1000 − P ) − X
0.21 =
⇒ 105 = 1030 − P − X ⇒ X = 925 − P.
500
Instructor: Milica Čudina
HW: 2
Course: M339D/M389D - Intro to Financial Math
Jane’s yield rate satisfies:
30 + (1000 − P + 25) − 2X
0.21 =
500
The above two equations imply
⇒
105 = 1055 − P − 2X
P = 950 − 2(925 − P )
⇒
⇒
Page: 2 of 2
P = 950 − 2X.
P = 900.
Please, provide the complete solution to the following problems.
Problem 2.3. An investor buys 10 shares of a non-dividend paying stock for the initial price of $200 per
share. At the end of the investment term (s)he sells the 10 shares for the market price of $205 per share.
What is his/her simple return on this investment?
Solution:
205 − 200
= 0.025.
200
Problem 2.4. An investor buys 10 shares of stock wich pays a continuous dividend with the dividend yield
equal to 0.05. Assume continuous and immediate reinvestment of dividend into the same asset. How many
shares does the investor own 2 years from the original purchase?
Solution:
10e0.05·2 = 11.0517.
Problem 2.5. An investor buys n0 shares of continuous-dividend-paying stock with the aim of owning
exactly 10 shares three later. Assume that the dividend yield is equal to 0.02. Assume continuous and
immediate reinvestment of dividend into the same asset. Calculate n0 .
Solution:
n0 = 10e−0.02·3 = 9.4176
Instructor: Milica Čudina