Management Finance: #3: Arbitrage and Financial Decision Making

[samvara]

Far less brain hurty than last week - readings were quite doable. We did much learning and maths then found our teams and did a practice test to get in the swing of things for next week. My team seems workable.

Learning Objectives!

1. Assess the relative merits of two-period projects using net present value.
2. Define the term “competitive market,” give examples of markets that are competitive and some that aren’t, and discuss the importance of a competitive market in determining the value of a good.
3. Explain why maximizing NPV is always the correct decision rule.
4. Define arbitrage, and discuss its role in asset pricing. How does it relate to the Law of One Price?
5. Calculate the no-arbitrage price of an investment opportunity.
6. Show how value additivity can be used to help managers maximize the value of the firm.
7. Describe the Separation Principle.
8. Calculate the value of a risky asset, using the Law of One Price.

Interest Rates and the Time Value of Money
Consider an investment opportunity with the following certain cash flows.

• Cost: $100,000 today
• Benefit: $105,000 in one year

The difference in value between money today and money in the future is due to the time value of money.

The Interest Rate: An Exchange Rate Across Time
The rate at which we can exchange money today for money in the future is determined by the current interest rate (assuming one risk free rate for the moment). Suppose the current annual interest rate is 7%. By investing or borrowing at this rate, we can exchange $1.07 in one year for each $1 today. Risk–Free Interest Rate (Discount Rate), rf: The interest rate at which money can be borrowed or lent without risk.

• Interest Rate Factor = 1 + r_f
• Discount Factor = 1 / (1 + r_f )

Future Values
To move a cash flow forward in time you must compound it

• FV_n = C x (1 + r)ⁿ

eg: Future Value in 5 years of $1,000 at 6.2% interest = 1,000 x (1 + 6.2)⁵

Present Values
To move a cash flow backward in time you must discount it

• PV_n = C / (1 + r)ⁿ = (for people who hate dividing make n negative) = C x (1 + r)^-n

eg: Present Value of $1,000 to be received in 5 years at 6.2% interest = 1,000 / (1 + 6.2)⁵

Present Value and the NPV Decision Rule
The net present value (NPV) of a project or investment is the difference between the present value of its benefits and the present value of its costs. Only values at the same point in time can be combined or compared.

• Net Present Value (NPV) = PV (Benefits) − PV (Costs) = PV(All project cash flows)

When making an investment decision, take the alternative with the highest NPV. Choosing this alternative is equivalent to receiving its NPV in
cash today.

Choosing Among Projects

Project	Cash Flow (t=0)	Cash Flow (t=1)
A	42	42
B	-20	144
C	-100	225

Rf = 0.20

Project	Cash Flow (t=0)	PV (t=1)	NPV
A	42	42/1.20 = 35	42+35 = 77
B	-20	144/1.20 = 120	-20+120 = 100
C	-100	225/1.20 - 187.5	-100 + 187.5 = 87.5

All three projects have positive NPV, and we would accept all three if possible. If we must choose only one project, Project B has the highest NPV and is the best choice.

What if I don't have $20 (B) to spend, what if I want to have $42 (A) cash right now?

• Choose B anyway, borrow $62, spend $20 on B and keep $42 for whatever you wanted it for.

	Cash Flow (t=0)	Cash Flow (t=1)
Project B	-20	144
Borrow	62	-62 x 1.20 = -74.4
Total	42	69.6

PV = 69.6 /1.20 = 58; NPV = 42 + 58 = 100 - NOTHING CHANGED.

• Regardless of our preferences for cash today versus cash in the future, we should always maximize NPV first. We can then borrow or lend to shift cash flows through time and find our most preferred pattern of cash flows.
• Fisher’s Separation Principle – investment decisions are independent of the consumption preferences of owners/shareholders

Arbitrage and the Law of One Price

• Arbitrage: The practice of buying and selling equivalent goods in different markets to take advantage of a price difference. An arbitrage opportunity occurs when it is possible to make a profit without taking any risk or making any investment.
• eg: scalping tickets
• Normal Market: A competitive market in which there are no arbitrage opportunities.
• Law of One Price: If equivalent investment opportunities trade simultaneously in different competitive markets, then they must trade for the same price in both markets.
• eg: You can buy and resell a few excitingly cheap shares if you have someone who doesn't know they are selling 'cheap' and someone who doesn't know they are buying 'expensive' but they will eventually figure it out and stop doing it. "Eventually" in internet terms is about 5 mins.

No-Arbitrage and Security Prices
Valuing a Security: Assume a security promises a risk-free payment of $1000 in one year. If the risk-free interest rate is 5%, what can we conclude about the price of this bond in a normal market?

• PV ($1000 in one year) = ($1000 in one year) / (1.05 $ in one year / $ today) = $952.38 today
• Price(Bond) = $952.38

What if price of the bond is not $952.38? Say price is less at $940?
First person to notice pounces and buys lots - price rapidly goes up as it gets popular.

What if price of the bond is not $952.38? Say price is more at $960?
First person to notice pounces and sells lots - price rapidly drops as it gets unpopular.

No Arbitrage Price of a Security = Price(Security) = (All cash flows paid = PV by the security)

The NPV of Trading Securities
In a normal market, the NPV of buying or selling a security is zero. Security transactions in a normal market neither create nor destroy value

• Separation Principle: We can evaluate the NPV of an investment decision separately from the decision the firm makes regarding how to finance the investment or any other security transactions the firm is considering.

Valuing a Portfolio
The Law of One Price also has implications for packages of securities. Consider two securities, A and B. Suppose a third security, C, has the same cash flows as A and B combined. In this case, security C is equivalent to a portfolio, or combination, of the securities A and B.

• Price(C) = Price(A + B) = Price(A) + Price(B)

The Price of Risk
Risky Versus Risk-free Cash Flows

• You can invest in a risk-free bond and/or the risky Market Index
• These assets have the below Market Prices (in $) and expected cash flows in one year
• Assume there is an equal probability (50%) of either a weak economy or a strong economy. Rf=4%

Security	Market Price (t=0)	Weak Economy (t=1)	Strong Economy (t=1)
Risk-free bond	1058	1100	1100
Market Index	1000	800	1400

Price(Risk-free Bond) = PV(Cash Flows) = ($1100 in one year) / (1.04 $ in one year / $ today) = $1058 today

Expected Cash Flow (Market Index) = ½ ($800) + ½ ($1400) = $1100

• Although both investments have the same expected value, the market index has a lower value since it has a greater amount of risk.

Risk Aversion and the Risk Premium

Risk Aversion:

• Investors prefer to have a safe income rather than a risky one of the same average amount.

Risk Premium:

• The additional return that investors expect to earn to compensate them for a security’s risk.
• When a cash flow is risky, to compute its present value we must discount the cash flow we expect on average at a rate that equals the risk-free interest rate plus an appropriate risk premium.
• The higher the risk the higher the premium.

Expected return of a risky investment = Expected Gain at end of year / Initial Cost

Market return if the economy is strong

• (1400 – 1000) / 1000 = 40%

Market return if the economy is weak

• (800 – 1000) / 1000 = –20%

Expected market return

• ½ (40%) + ½ (–20%) = 10% so risk premium is 6% (Risk free interest rate is 4%)

PV = E(CF) / (1 + r)ⁿ =1100 / 1.1 = 1000We can use this to do maths (yay!)

Security	Market Price (t=0)	Weak Economy (t=1)	Strong Economy (t=1)
Risk-free bond	?	800	800
Security A	?	0	600
Market Index	1000	800	1400

Given rf of 4%, the market price of the bond is: $800 / 1.04 = $769 today
Therefore the initial market price of security A is $1000 – $769 = $231

Security	Market Price (t=0)	Weak Economy (t=1)	Strong Economy (t=1)
Risk-free bond	769	800	800
Security A	231	0	600
Market Index	1000	800	1400

Expected cash flow of Security A = 0.5(0 + $600) = $300 (that's just an average there, nothing fancy)
Expected return = (300 – 231) / 231 = 30%
Risk premium = 30% - 4% = 26%

<insert another example with a negative risk premium here>

Arbitrage with Transactions Costs
What consequence do transaction costs have for no-arbitrage prices and the Law of One Price?

When there are transactions costs, arbitrage keeps prices of equivalent goods and securities close to each other. Prices can deviate, but not by more than the transactions cost of the arbitrage.

Then we did the test, flailed a tiny bit and ran away

Comments

[ariaflame]

Did you do any sensitivity analysis of the NPVs to determine the possible effect of changes in interest and inflation? Oh, you appear not to have the version that takes inflation into account.

[samvara]

That's probably a bit advanced for this class - or maybe it's not been introduced yet.

[ariaflame]

Might get there later. :) I was actually mostly commenting because for once I actually managed to follow some of what you were talking about. Mostly in the context of using NPV to calculate Renewable Energy systems.

[samvara]

I admit I'm teetering on understanding/not understanding a lot of what I pre-read - the seminar helps and the exercises I do each week are essential.

*cowers*