Business Finance510
Fall2013
Assignment#1Solutions
Trading and Arbitrage
Part1:Arbitrage and the Law of One Price
Assume for this problem that markets are transactions costs and no short-lling constraints).It’s a week before the Cubs-White Sox game and youfind a market that lls two curities:
•CBS Security which pays$1next week if Cubs win and$0otherwi;currently trading at$0.70
•WS Security which pays$1next week if White Sox win and$0otherwi;currently trading at$0.25
1.What is the risk-free rate if there is no arbitrage?(Do not worry about expressing
this as an annualized percentage.)
Hint:What is the payoffto the risk-free curity if Cubs wins?What is its payoffif White Sox win?Can you u the information given above to construct this curity?
If you buy both the CBS and WS curity,you receive$1next week(regardless of the outcome of the game).This is equivalent to a risk-free ast(becau the payoffis the same regardless of the outcome).The cost of this is$0.95.
(0.95)(1+r f)=1
r f=5.26%
In particular,we do not need to know(and cannot calculate)the probability of each team winning.To infer the probabilities would require some assumptions about the risk premia of the curities.
Some students suggested that both the CBS and WS curity should have an expected return equal to the risk-free rate.Since both of the curities are risky,this is not necessarily true.Some students also inferred a probability of CBS victory from the prices(0.7).It is also not necessarily the ca that the prices are in proportion to the probability of winning.
2.Suppo that the risk-free rate is2%for a week.(i.e.$1invested today pays off$1.02
in a week.)What strategy would you u to take advantage of this situation?
In the previous part,we found a synthetic risk-free ast with a higher interest rate.
We would borrow$0.95at the risk-free rate and buy one unit each of the CBS and WS curities.In one week,we would pay back$0.95(1.02)=$0.969and receive$1 from our CBS/WS portfolio.Overall,there is no cash layout today with a guaranteed payoffof$0.031next week.We would scale this position up as much as possible.
3.What would you expect to happen if CBS=$0.70,WS=$0.25,and r=2%?In
particular,would you expect prices to change?If so,how would they change?If not, why not?
Investors would try to exploit this violation of the law of one price.As investors try to buy the CBS and WS curities and short the risk-free ast,the prices of the OSU and UM curities would ri and the price of the risk-free ast would fall(equivalently, the risk-free rate would ri).
播音员主持人Part2:Buying on Margin and Levering Up
For this part,u hw1data.xls.The sheet labeled“Google”contains end-of-day prices for Google stock
from issuance to the end of2010.For the purpos of this problem,assume that Google pays no dividends.For simplicity,you can also assume that it is possible to buy fractional shares.
1.Suppo that it is the end of the day on December11,2009.The current stock price of
5岁幼儿简短演讲稿Google is$590.51.You have$10,000to invest and decide to buy Google stock.At the end of the day on January11,2010,you decide to ll your position in Google stock.
What is your return?移动硬盘分区
P12/11/09=$590.51
P1/11/10=$601.11
return=601.11−590.51
590.51
=1.80%
2.Now suppo that on December11,2009,you are extremely bullish on Google.Not
only do you want to buy Google stock,but you want to buy on margin.Suppo that your broker requires a margin of50%.Also,your broker charges you interest at a rate of0.03%per day.You decide to invest so that your initial margin is60%.What is the value of stock that you buy?How many shares did you buy?
Margin=10,000
x+10,000=0.6
x=$6,666.67
$16,666.67in stock
幼儿园建议Buy28.2242shares(=$16,666.67/$590.51)
3.Continuing on the above margin question:Suppo that you ll at the end of the day
on January11,2010[31days].What is your return?Compare this to the ca where you did not buy on margin.
(28.2242shares)($601.11per share)=$16,965.85
return=16,965.85−(6666.67)(1.0003)31
10000
−1=2.37%
The return in this ca is larger than when purchasing without margin.
Note that the return is calculated using thefinal owner’s equity(the numerator)and the initial owner’s equity(the denominator).What is relevant in the types of problems is the return that we receive on our investment.(See p.69of your textbook.)
4.Suppo that you instead held Google until February11,2010[62days].What is your
return?Compare this to the ca where you did not buy on margin.What lessons do you learn about buying on margin?
Buy on Dec11at$590.51
Sell on Feb11at$536.40
Equity in account=(28.2242)(536.40)−(6666.67)(1.0003)62=8347.65
return=8347.65−10000
10000
=−16.52%
without buying on margin:return=536.40
590.51
−1=−9.16%
By buying on margin,you amplify both your upside and your downside.It is equivalent to levering up your position.向日葵幼儿园
Part3:Exploiting Mispricing
For this part,u hw1data.xls.The sheet labeled“3Com Palm”contains data on3Com and Palm stock prices.
Suppo that it is March6,2000.You decide to exploit the negative stub of3Com.For each share of3Com stock,a shareholder will eventually receive1.5shares of Palm.Thus, you believe that P3Com≥1.5P P alm.1
Note that this question does not abstract away from the standard short-lling rules:Proceeds from short-lling must be kept on account with your broker and cannot be ud to purcha other shares.
1.Suppo that you believe that3Com is fairly priced and that its stub value is$
its only value comes from Palm).What should Palm’s price be in a market that is both frictionless and free from arbitrage?
March6,2000
P3Com=69.5625
P P alm=63.125
If the stub value of3Com is0,then the law of one price implies that P3Com=1.5P P alm.
If we think that3Com is fairly priced,then P P alm=P3Com
1.5=69.5625
1.5
=46.375.
2.Suppo that you now decide to exploit the negative stub value of3Com.You short
ll150shares of Palm and buy100shares of3Com.You u the3Com shares as collateral in the Palm short sale.What does your balance sheet look like?(i.e.What are your asts,liabilities,and equity?)What is your margin?
Asts Liabilities&Equity
Cash:(150)(63.125)=9468.75Palm Shares:9468.75
营业执照怎么查询3Com Shares:(100)(69.5625)=6956.25Owner’s Equity:6956.25
margin=6956.25
9468.75
=73.47%
1The actual ratio was1.525,but for the purpos of this problem,we will treat it as1.5for simplicity.
3.Jump ahead to March9,2000.What does your balance sheet look like?
March9,2000缅怀英烈
P3Com=68.0625
P P alm=69.375
Asts Liabilities&Equity
Cash:9468.75Palm Shares:(150)(69.375)=10406.25
3Com Shares:(100)(68.0625)=6806.25Owner’s Equity:9468.75+6806.25-10406.25
=5868.75
4.If you are forced to unwind your position on March9,2000,what is your return?2
5868.75
−1=−15.63%
6956.25
5.Why is your return in the previous part negative even though this appears to be a
clear arbitrage trading strategy?What would your return have been if you were able
to hold the position until July28,2000after1.5Palm shares were distributed for each
3Com share?
The forces that drove the mispricing(potentially things like irrational exhuberance)did
not correct and in fact made the mispricing wor.3Note that when the distribution
游戏名称女occurs,you receive150shares of Palm in addition to your100shares of3Com.You
do not trade in your100shares of3Com.
On July28,2000,your balance sheet would have been:
Asts Liabilities&Equity
Cash:9468.75Palm Shares:(150)(37.25)=5587.50
3Com Shares:(100)(12.9375)=1293.75Owner’s Equity:9468.75+1293.75+5587.50 Palm Shares:(150)(37.25)=5587.50-5587.50=10762.50
−1=54.72%
return=10762.50
6956.25
2You might ask why you are force to unwind the position at this point.Think of this as investing money
for a wealthy,butfinancially-unsavvy client.He/she is unable to understand your strategy and is not qu
ite sure if it is an arbitrage.Thus,if the value of your position declines too much,the funds might be withdrawn, forcing you to unwind the position.
3If we take into account the possibility that the distribution would not go through,a decrea in the likelihood of the distribution going through would also cau a negative return.
