8. Arbitrage-Free Price

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From the course by Indian School of Business

Trading Basics

303 оценки

Indian School of Business

Trading Basics

303 оценки

Course 1 of 5 in the Specialization Trading Strategies in Emerging Markets

The purpose of this course is to equip you with the knowledge required to comprehend the financial statements of a company and understand the various transactions that take place in the stock market so that you can replicate the strategies discovered by the extant academic literature. The first part of the course provides a brief introduction to financial statements and various common filings of firms. You will learn how to obtain information regarding a company's performance from them and use the information to build trading strategies. Next, you are taught basic asset pricing theories so that you will be able to calculate the expected returns of a stock or a portfolio. Finally, you will be introduced to the actual functioning of asset markets, type of players in the market, different types of orders and the efficient ways and opportune time to execute them, trading costs and ways of minimizing them, the concept of liquidity .etc. This knowledge is required to develop efficient algorithm to execute various trading strategies.

From the lesson

Asset pricing theories

Before you start trading, you should first be able to calculate the expected return from a stock. The expected return comes from various asset pricing models such as Capital Asset Pricing Model (CAPM), Arbitrage Pricing Theory (APT) and Fama-French Three Factor Model. The purpose of this module is to equip you with the skills required to calculate the expected returns of a stock (or) a portfolio. After completing this module you will be able to calculate the expected returns of a stock.

1. Expected returns and risk6:27

2. Diversification5:58

4. CAPM Beta6:10

5. Multifactor Models5:55

7. Arbitrage6:32

8. Arbitrage-Free Price6:03

9. Risk Factors4:44

Meet the Instructors

Ramabhadran Thirumalai
Assistant Professor
Indian School of Business

>> Learning outcomes, after watching this video you will be able to solve for

the risk premiums once we have the factor sensitivity.

Determine the arbitrage-free price of an asset

given the prices of other risky assets.

Discuss a key drawback of the APT, the abitrage-free price.

Last time, we saw an example

where Coke was incorrectly priced relatively to Microsoft and Intel.

Let's continue with that example.

In the first step, we have solved for

the factor sensitivities of Microsoft and Intel.

In the second step now, we are going to solve for the risk premium using the APT.

Write the APT for Microsoft and Intel, remember, it says that

the expected return equals the risk period plus beta times the risk premium.

We do not know the is risk free rate and

the risk premium for the business cycle risk factor.

We will have two equations, one each from Microsoft and Intel and

so we can solve for the two unknowns.

For Microsoft we have 0.2 = the risk free rate + 0.8 x the risk factor premium.

For Intel, we have 0.5 = the risk free rate,

plus 3 times the risk factor premium.

Solving, we find that the risk free rate is 9.09%, and

that the business cycle risk factor premium is 13.64%.

Now, for the third step, the APT must hold for

all risky assets, provided the no-arbitrage condition holds.

For Coke, the APT is the expected return of Coke = 9.09%

+ 13.64% times Coke's beta.

This is one equation with two unknowns, how do we go about solving this problem?

Assume that the correct price of Coke today is P.

Let's write the single factor model for Coke in the good and bad states.

In the good state, it is 55 over P- 1 =

Coke's expected return + 0.5 times Coke's beta.

In the bad state it is 15 over p- 1 = Cokes expected

return- 0.5 times Coke's beta.

We can solve for Coke's expected return and beta in terms of P.

Coke's expected return is now 35 over P- 1 and its Beta is 40 over P.

Plugging this back into Coke's APT, we have 35 over

p- 1 = 9.09% + 13.64% times 40 over P.

This is now one equation and one unknown.

Solve for P to get Coke's APT price today to be 27.08.

Let's revisit our arbitrage portfolio that consists of buying one share of Coke and

start selling half share each of Microsoft and Intel.

The future payoffs in the good and bad states are still 5%,

but the cost to set up this portfolio today is now 4.58.

So what is the rate of return on this portfolio?

Without any calculations, we can confidently say that it is 9.09%.

How is that possible, you wonder,

your portfolio is risk-free because it always pays 5 after one period.

It has no risk or uncertainty which means that it is risk-free.

So its return must equal the risk-free rate which is 9.09%.

If it is not, there will be an arbitrage opportunity.

You can verify that 5 over 4.58- 1 is indeed 9.09%.

Remember, we started off by assuming Microsoft and

Intel were correctly priced, and Coke wasn't.

This gave us a risk-free rate of 9.09% and

the factor risk premium of 13.64%.

The problem with arbitrage pricing is that the risk-free rate and

factor risk premium will depend on what we assume are the correctly priced assets and

which one is incorrectly priced.

If we assume that Intel and Coke are correctly priced, and

Microsoft is incorrectly priced, the risk-free rate will be 63.64%.

And the factor risk premium will be negative 4.55%.

Similarly, if we assume that Microsoft and Coke are correctly priced, but

Intel isn't the risk-free rate come's out to negative 9.09% and

the factor risk premium to 36.36%.

You can verify these numbers by following the three steps I have outlined in

this example.

If you look at these numbers, some of them seem absurd.

What does it mean to have a negative risk-free rate or

a negative factor risk premium?

Is it even realistic?

The answer is of course no but

this is the problem with the relative pricing idea that APT uses.

Different sets of assets will give different values for

the risk-free asset and the factor risk premiums, some of which may be absurd.

We have talked about multiple factors affecting expected returns, but

what exactly are these factors?

Next time, we will see a popular set of factors that researchers and

practitioners use.

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