module 1: arbitrage (chapter 3) Flashcards
what is a competitive market for a good?
market in which that good can be bought and sold for the same price (good’s price determines its worth)
what is the valuation principle?
value of an asset to the firm or investors is determined by competitive market price (not always available)
benefits and costs of a decision should be eval using these market prices and when the value of the benefits exceeds value of costs, decision will increase mkt value of the firm
what happens when competitive mkt prices are unavailable?
cannot value decisions w/o accounting for the preferences/views of the decision maker
ex. price of good at retail store is one-sided, can buy good for stated price but can’t sell it to the store for that price
(stated price determines good’s max value since you can buy it for that price, but might value it for less depending on your preferences)
what is the difference in value between money today and money in the future called?
time value of money
how is the rate at which we exchange money today for money in the future determined?
current interest rate, allows us to convert currency at a point in time to another point in time
what is the risk free interest rate (rf)?
interest rate at which money can be borrowed or lent over a period w/o risk
can exchange (1 + rf) dollars in the future per follar today or convert a dollar one period from now into $1/(1+rf) today
what is 1+rf
interest rate factor for risk free cash flows, defines exchange rate over time and has units of “$ in one year per $ today”
ex. an investment opportunity has a cost of 100000 today and a benefit of 105000 in one year- if the interest rate is 7%, we can express our costs as:
cost = (100k today) * (1.07 in one year/1$ today) = 107k in one year
105000 - 107000 = -2000 in one year
we could earn 2000 more in one year by putting our 100k in the bank rather than making this investment. we should reject this investment?
we can determine what the current value of the 105000:
benefit = (105k in one year) / (1.07 $ in one year/1 $ today)
= 105k * 1/1.07
= -98130.84 today
this is also the amnt the bank would lend to us today if we promised to repay 105k in one year, thus it is the competitive market price at which we can buy/sell today an amnt of 105k in one year
net value of the investment:
98130.84 - 100k = -1869.16 today
the negative result indicates we should reject the investment, opting for it would make us 1869.16 poorer today
present vs future value: (1869.16 today) * (1.07/1) = -2000 in one year
discount factors and rates: 1/(1+r) = 1/1.07 = 0.93458
price today of $1 in one year because it provides the discount at which we can purchase money in the future; 1/(1+r) is the one year discount factor
risk free interest rate is also referred to as the discount rate for a risk free investment
building the canada line was projected to be abt 2.05 billion in 2005, projections indicated that costs were rising by about 10% per year. if the interest rate was 3.25%, what would have been the cost of a one year delay in terms of dollars in 2005?
cost in 2006 = 2.05 bill * (1.10) = 2.255 bill
compare to cost of 2.05b in 2005, convert using 3.25%:
2.255 bill in 2006 / ($1.0325 in 2006/1 $ in 2005) = 2.184 bill in 2005
cost of a delay of one year would have been:
2.184 bill - 2.05 bill = 134 mill in 2005
what is net present value?
npv = pv(benefits) - pv(costs)
total of pvs of all project cahs flows
reps value of project in terms of money today
if in exchange for 500 today, you will receive 550 in one year with certainty, if the rf is 8% per year, what is the npv?
pv = 550/1.08 = 509.26 today
509.26 is the amount that we would need to put in the bank today to generate 550 in one year, the pv s the cash cost today of “diy” the amount you need to invest at the current interest rate to recreate cash flow
npv = 509.26 - 500 = 9.26 today
how to detemrine whether investing in a project today will make a firm poorer or richer today using npv?
npv > 0 = RICHER
npv < 0 = POORER
npv = 0 = does not add value and does not make poorer
what is the npv decision rule?
when making an investment decision, take the alternative with the highest npv
choosing this alternative is equiv to receiving its npv in cash today
your firm needs to buy a large amount of office supplies worth 20k today, but the store is offering you the choice of paying one year from today, at the price of 20.5k. if rf is 4%, is this a good deal?
option 1: 20k
option 2: 20.5k/1.04 = 19712
19712 < 20k
take the deal and you’ll pay less
what is the first separation principle?
separation of individual’s consumption preferences from optimal investment decision
- regardless of consumption preferences (present/future), we should always max npv first
- then borrow/lend to shift cash flows thru time to match most preferred consumption spending
- preferences regarding consumption spending pattern thru time are separate from optimal investment decision
supports idea that mgmt should act to max npv since this benefits all shareholders, no matter how they want to spend money at different points in time
also supports idea that shareholders can delegate decision making to managers since mgmt knows that all shareholders always prefer positive npv investments
what is npv and the individual’s consumption preferences?
when comparing projects with diff patterns of prevent and future cash flows, we may have preferences regarding when we want to consume and thus when we want to receive cash
what is arbitrage?
buying and selling equivalent goods to take advantage of a price difference
what is an arbitrage opportunity?
situation where there is a possibility of making a risk free profit without making any investment (free lunch)
due to positive npv, whenever they appear in financial markets, investors race to take advantage of it
those who get to it first and trade quickly will have ability to exploit it, once they place their trades, prices will respond, causing opportunity to disappear
what is a normal market?
competitive market in which there are no arbitrage opportunities
what is the law of one price
if equiv investment opportunities trade simultaneously in diff competitive markets, then they must trade for the same price in both markets
ex. if prices in 2 markets differ, investors will profit by buying in cheaper market and selling in expensive market, equalizing prices
a useful consequence is that when eval costs and benefits to compute npv, use any competitive price to determine a cash value without checking the price in all possible markets
if a bond offers a one time risk free payment to its owner of 10000 in one year’s time and rf = 3%, what can be concluded about its price in a normal market? what if the bond is trading at a diff price?
10000/1.03 = 9708
the bond would trade at a price of 9708 in a normal market
if it is underpriced at 9600:
borrow 9708 at 3% risk free
buy 9600 bond
in one year:
receive 10k from bond
pay bank back 10k
profit: 108 today
if it is overpriced at 9800:
short bond at 9800
invest 9,708 in a bank at 3% risk free
in one year:
9708 grows to 10k
use 10k to buy back bond and close short position
net profit = 10k - 9800 = 200
what is a short sale?
person who intends to sell the security first borrows it from someone who already owns it
that person then must either return the security by buying it back or pay the owner cash flows they would have received
how to calculate bond’s return?
return = (gain at the end of year / initial cost) - 1
return = ((final price - mkt price today / mkt price today) - 1
what is the second separation principle?
when securities trade at no-arbitrage price, the costs and benefits are the same, therefore npv = 0
value is not created by trading securities in a normal mkt but ratehr by th ereal investment projects undertaken by corps (new products, better tech, etc)
can eval decision by focusing on separation of the investment and financing decisions: security transactions in a normal market neither create nor destroy value on their own, therfore we can eval the npv of an invetsment decision separately from the decision the firm makes regarding how to finance the investment or any other security transactions the firm is considering
what is a portfolio
collection of securities
a firm can invest 4 million in a project today that will return 4.25 mill for sure after a year. the firm can either pay for the project entirely with its own cash or it can borrow 2.4 mill fro, a bank at rf = 5% and pay for the remainder of the investment with its own cash.
a) is the project a good investment w/o borrowing
b) is the project a good investment with borrowing
a) yes
t = 0 -> -4
t = 1 -> 4.25
npv = -4 + 4.25/1.05 = 0.0476 > 0
b) yes
t = 0
from bank = 2.4
investment = -4
-2.4
= -4
t = 1
2.4 * 1.05 = -2.52
4.25
2.4 * 1.05 = -2.52 (excess cash)
= 4.25
what is value additivity
consider 2 securities; A & B, and a third security C has the same future cash flows as A and B combined
C = A + B
if value additivity does not hold, there would be an immediate arbitrage opportunity available
implied that since a firm’s cahs flows are the sum of cash flows for all its projects, the value of an entire firm equals the sum of the values of all its projects
-> npv decision rule is therefore consistent with max overall value of firm
how are value additivity and firm value related?
since firm’s cash flows are sum of the cash flows for all projects, value of an entire firm equals the sum of the values of all its projects
price/value of a firm is equal to the sum of the values of all the projects and investments within it
what is the no arbitrage price of a security that pays cash flows of 100 in 1 year and 500 in 2 years?
b1: price today = 94, cf in 1y = 100, cf in 2y = 0
b2: price today = 85, cf in 1y = 0, cf in 2y = 100
b1 provides the 100, so how many b2 do you need to provide the 500
94 * 1 + 85 * 5 = 519
suppose a security with cash flows of 50 in 1 year and 100 in 2 years has a price today of 130. is there an arbitrage opportunity?
b1: price today = 94, cf in 1y = 100, cf in 2y = 0
b2: price today = 85, cf in 1y = 0, cf in 2y = 100
yes!
b1 * 0.5 + b2 * 1 = 47 + 85 = 132
132 - 130 = 2
you can generate a profit by buying the new security and selling the old portfolio
what is a risky cash flow?
could turn out better or worse than what was originally expected
what is risk premium?
expected return - risk free rate
ex. if rf = 4% and a share of the mkt index is selling for 1000 and the expected mkt payoff is 1100, expected return is 10%. so 10% - 4% = 6% therefore risk prem is 6%
consider a risk security A that pays 600 is the econ is strong and 0 if it is weak. what is the no arbitrage price of A?
rf = 4%
in one year one unit of mkt index will be worth 800 if the econ is weak and 1400 if the econ is strong
one unit of mkt index is selling today for 1000
(1000 - (800/1.04))
how do you calculate the pv of a risky cahs flow?
discount expected cash flow at a rate equal to the risk free rate plus suitable risk premium
suppose a risky security A pays 600 if the econ is strong (50% prob) and 0 if it is weak (50%). say that no arbitrage price of A is 230.77. risk free rate is 4%, what is the risk premium of security A
expected return = rf + risk premium
expected payoff = 600 * 0.5 + 0 * 0.5 = 300
expected return = exp. payoff/current price today (no arbitrage price) - 1
= 300/230.77 - 1
risk premium = expected return - rf
= 300/230.77 - 1 - 4%
suppose that security A is trading at a price of either (i) 250 or (ii) 200. show how arbitrage profits can be made in each case
(no arb price = 230.77)
230.77 -> no arbitrage price
250 = short market
200 = buy market
consider another risky security B that pays 600 if the econ is weak and 0 if the econ is strong. what is the no arb price of security B?
rf = 4%
in one year one unit of mkt index will be worth 800 if the econ is weak and 1400 if the econ is strong
one unit of mkt index is selling today for 1000
1400/1.04 - 1000
consider another risky security B that pays 600 if the econ is weak (50% prob) and 0 if the econ is strong (50% prob). what is the no arb price of security B?
300/346.15 - 1 - 4%
how is risk in finance relative to the overall market?
even if a security has highly variable returns, if its returns vary in a way that offsets other risks faced by investors, the security will actually reduce risk
risk premium for a security will be higher the more its returns tned to vary with the overall econ and market. if a security’s returns vary in opposite direction of market, security offers insurance wnad will have a negative risk premium
what are transactions costs
range of no arbitrage prices because arbitrage opportunities are only profitable if the amnt of profit available exceeds transactions costs
suppose that a bond pays 1k after 1 year, the mkt int rate for lending is 2% and the mkt interest rate for borrowing is 2.5%. what is the no arb price range for this bond?
price = 1000/1+r
[1000/1.02, 1000/1.025]
