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# P1.T1.401. Arbitrage pricing theory (APT) for well-diversified portfolios

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
P1.T1.401. Arbitrage pricing theory (APT) for well-diversified portfolios

AIMs: Describe the Law of One Price and assess whether an arbitrage situation exists using a multi-factor model. Construct the Security Market Line for a well-diversified portfolio using a single-factor model. Explain how to construct a portfolio to hedge exposure to multiple factors. Describe the Arbitrage Pricing Theory (APT) and the Fama-French three-factor model, and explain the underlying assumptions of each.

Questions:

401.1. Consider the following three well-diversified portfolios that exist in a single-factor economy:

Is there an arbitrage opportunity?

a. No, all three well-diversified portfolios plot on the security market line
b. Yes, an arbitrage includes buying portfolio (A) and selling a combination of (B) and (C)
c. Yes, an arbitrage includes buying portfolio (B) and selling a combination of (A) and (C)
d. Yes, an arbitrage includes buying portfolio (C) and selling a combination of (A) and (B)

401.2. Consider the following multi-factor (APT) model of security returns for a particular stock, along with actual-versus-expected rates of change in the three macro factors:

If we include the "surprises" in the macro factors, what is the expected rate of return for the stock?

a. 5.30%
b. 7.30%
c. 8.40%
d. 9.90%

401.3. Assume two portfolios, (A) and (B), are each well-diversified and the economy has only one factor. The expected return of portfolio (A) is 8.0% and its beta is 1.30. The expected return of portfolio (B) is 6.0% and its beta is 0.90. What is the implied risk-free rate?

a. 1.50%
b. 2.25%
c. 3.00%
d. 5.75%

Answers here:

Last edited by a moderator:

#### khaled binhotan

##### New Member
is there a missing info regarding question 401.1? such as a market price
thanks,

#### secondtimearound

##### New Member
Nicole, I'm unable to find the reading material (study notes) for APT in the FRM Study Planner. I should be looking for this under Part 1 - Foundations of Risk Management, but I cant seem to find it. Can you help point me to it?

R3 > Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 9th Edition
• Chapter 10.................Arbitrage Pricing Theory and Multifactor Models of Risk and Return

Thanks much

#### secondtimearound

##### New Member
Also i checked the calendar and the APT study notes were expected to be published on 1/24..perhaps it is not uploaded yet?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@secondtimearound i'm still working on it. It says target. It's not uploaded yet.

@Nicole M let's talk about removing the calendar, I don't feel like fielding timing status updates all sunday

#### secondtimearound

##### New Member
Thanks for confirming David. I was a bit confused when I saw some questions posted on APT before the study notes were available! This is my first time on BT, so I'm still learning to navigate my way around a plethora of information!

Cheers!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@secondtimearound Okay, great thanks for your patience, sorry I get frustrated, it's just that I have so much support already to provide (actual domain/content support) that the stream of timing updates is really, really distracting for me and stressful. Thanks,

#### nilz

##### New Member
Hello David,

Can you help me solve questions to identify if arbitrage opportunity exists. I got the concept but unable to apply on problems. Example Q 401.1. Consider the following three well-diversified portfolios that exist in a single-factor economy. I have seen many such questions online. Can you please provide me a detailed way to solve such questions mathematically and logically?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nilz

The class of arbitrage opportunities is large, but question 401.1 (which is my variation on Bodie's 10.8) is very narrow due to the two assumptions in the setup "... well-diversified portfolios that exist in a single-factor economy." "Single-factor" suggests we can use the (single-factor) CAPM, and "well-diversified" confirms we can use the security market line (SML) because if an investor is well-diversified, the risk of the non-systematic component is effectively eliminated such that beta (in SML) is the only risk that earns compensation.

Once the assumption tells us that all three of the portfolios should lie on the SML (see http://en.wikipedia.org/wiki/Security_market_line ) the only test we need here is to compare their Treynor ratios: they should all have the same [E(r)-Rf]/beta; if they do not, then one or more of them lies "above" or "below" the SML and the arbitrage is to buy the "cheap" (above SML: higher return for given beta) and sell the "expensive" (below SML: lower return for given beta). I hope that helps,

Thanks!