Arbitrage Pricing Theory - 2 Factor Model

Water Cooler Investing
#1

Can someone help me with the following Exercise for understanding APT:

The returns of individual securities are generated by a two-factor model:

R = E[R] + beta1*F1 + beta2*F2

F1 and F2 are market factors with zero expectations.

R is return for the security.

The economy has 4 securities with the following factor betas:

Security Beta 1 Beta 2 E[R]

1 2 3 10%

2 1 4 10%

3 2 1 5%

4 3 1.5 5%

  1. Construct a portfolio using securities 3 and 4 such that by combining them you manage to eliminate all risk, i.e. the portfolio betas you obtain in front of factors F1 and F2 are zero.

  2. Assume now the existence of a riskless asset (T-bill) returning 3%. Devise an arbitrage strategy using your result from part 1.

  3. Can this arbitrage opportunity subsist in the long-run? Explain what would happen to the various rates of return and the prices of the assets.

#2

Do your own homework.

#3

I´m not looking for the solution, I just try to understand the concept. I already know about CAPM and Black´s “extension” (i.e. zero beta portfolio). But I can´t think of equations to solve this particular probem. So if someone could explain what the theory behind the above question is, this would already help me a lot.

#4

Ok, I’ll give you a hint. How much risk would it be to put in $150 in security 3? How much would you expect to make from that?

How about $100 into security 4?

That should get you started…

#5

Thanks a lot!

Return of Security 3: R3 = 5% + 2 (F1) + F2

Return of Security 4: R2 = 5% + 3 (F1) + 1.5 (F2)

Return of a portfolio consisting of Security 3 and 4: Rp = 150 (5% + 2 (F1) + F2)) + 100 (5% + 3 (F1) + 1.5 (F1))

How can I determine the risk and the weights of this portfolio?

#6

Not quite.

The factor risk of 150 of security 3 = [2*(F1 return) * 150] + [1*(F2 return) * 150] = 300 * (F1 return) + 150 * (F2 Return)

Expected return of 150 of security 3 = 150 * 5% = 7.5 + factor effects

The factor risk of 100 of security 4 = [3*(F1 return) * 100] + [1.5*(F2 return) * 100] = 300 * (F1 return) + 150 * (F2 return)

Expected Return of 100 of security 4 = 100 * 5% = 5 + factor effects

So the risk from factors F1 and F2 are identical for 150 of security 3 and 100 of security 4.

But 150 of security 3 has a higher expected dollar return than 100 of security 4 (7.5 vs 5.0).

Note that I’m using the term “risk” in a slightly non-kosher way here. What I really mean by risk here (and couldn’t think of a better way to say it) is “what is the effect of Factor 1 and Factor 2 on the returns of security 3 and security 4.” The key idea here is that if you can create two portfolios with the exact same risk exposures but different expected returns, you can create an arbitrage opportunity by going long the one with greater expected returns and short the one with smaller returns, because you can get the effect of F1 and F2 on one security to offset the the effect of F1 and F2 on the other (and therefore neutralize the risk to the whole portfolio).

Normally, risk in these kinds of problems means the standard deviation or variance, and while it’s possible to solve this problem that way, it’s easier to understand if you just try to see how the effects of factors might neutralize each other. The way that factor variances and standard deviations add is more complicated and if one can solve the problem without going there, so much the better.

How did I get that ratio? Well, if you notice, all of the coefficients for security 3 are 1.5 * the coefficients for security 4. So that means if I have 1.5* secuirity 4 invested in security 3, they will have the exact same exposures to F2 and F4. Sometimes it requries a portfolio of more than one set of securities to neutralize the risk of another, but those problems are typically very difficult to solve unless you have a good handle on linear algebra, and even then it is sometimes tricky.

BTW, this is a tough problem if you haven’t solved it before, which is why I didnt’ just fire back “do your own homework,” however, I also don’t want to just blurt out the answer.

You still need to figure how to construct the arbitrage, which is a little tricky, but interesting. Give it a try

#7

Thanks, now I understand. This is a very good explanation, which helps me a lot. As I understand this, I would short security 4 and buy security 3. The portfolio weights would be as follows:

w(3) = 300%

w(4) = -200%.

But what about the other securities in the market? I thought I have to consider them as well?

If there is a riskless asset available in the market which returns 3%, I would simply short the riskless asset and security 4 and use the proceeds to invest in security 3. As every market participant will act in this way, the expected return of security 3 will decrease (and therefore the betas will change).

What would be a general numerical way in solving for the portfolio weights? I understand that by dividing return3 by sum(return3 & return4), I would get the 40:60 ratio. But how could I obtain the percentages (+300% / -200%)?

#8

As far as I can tell, the other securities are irrelevant. Perhaps there are other problems in your text that use them. Or perhaps the example is to show how arbitrage can work in a larger market of securities.

Actually, the percentages don’t matter so much, as long as they are in a 3/2 or 1.5:1 ratio (techincally, with signs reversed. In practice, there will usually be some additional constraint that fully determines the percent - either that all positions must add up to 100%, or margin constraints, or something. You’ve chosen 300% and -200% so that it all adds up to 100%, and that’s a sensible approach.

So, just to see if you are getting it, how would you do the arbitrage in part 3 of the question? And what expected % return are you getting for all that work?

#9

Maybe securities 1 and 2 are just there for confusion.

As I understand it, the T-bill is mispriced while the portfolio is priced correctly.

For both, the portfolio and the T-bill beta is zero, yet their expected returns are different (5% vs. 3%). In fact, today’s price of the T-bill is too high according to APT.

The arbitrageur could therefore simply short the T-bill (E® = -3%) and buy the portfolio (E® = +5%) with the proceeds. The expected return of this third-security would therefore yield 2%.

The arbitrage portfolio would require no initial investment. But Investments that require no dollar outlay should not earn any return (not even the risk-free rate). To earn the risk-free rate, funds must be committed.

The implication is that this would cause prices to change until the two securities with the same beta sold for the same price. The price of the T-bill will decrease (E® will increase) and the price of security 3 will increase (E® will decrease).

Or is there a more sophisticated approach?

#10

Or you could even built a portfolio of:

50% in T-bill

50% in Security 3.

This would reduce overall beta and therefore you can even achive 7% risk free rate of return.

Prices:

T-bill will increase --> E® will decrease

Security 3 will increase --> E® will decrease

Security 4 will decrease --> E® will increase

#11

lol

#12

Sell $100 worth of Asset 4. When you do that, you’ll have $100 in cash on hand as a result of the sale.

Use that cash to buy $100 worth of asset 3.

You still need $50 more of asset 3 to neutralize the betas you get by shorting asset 4 (remember that you need exposures in the ratio of 3:2), so borrow $50 at 3%.

Effectively, your portfolio is

$150 long Asset 3, expected return, 5% or $7.50

$100 short Asset 4. expected return (cost), -5%, or -$5.00

$50 short Cash, expected return (cost), -3%, or $1.50

After 1 year, the expected returns are:

$7.50 + (-$5.00) + (-$1.50) = $1.00

Note that you haven’t used ANY of your own money to create this portfolio, so it seems like risk-free money. In real life, you will need some money, because most brokers will not let you borrow to short if you have no assets as collateral. Your initial capital is effectively a cushion to keep you solvent in case something goes wrong, so how much you go long or short will be linked to how much capital is available, and how much your broker will let you borrow or leverage, given that capital.

Without those constraints, in theory, you could make that portfolio arbitrarily large.

Also, this is not a true arbitrage, because the $1.00 you get from this portfolio is not truly risk free. Why? Because each asset still has ideosyncratic risks (i.e. risks that don’t come from Factor 1 or Factor 2). So although there is no risk from either F1 or F2, there is asset-specific risk, so that $1.00 is not really risk-free. It may be “market neutral” (assuming that F1 and F2 are the only factors relevant to “the market,” but that is not the same as saying it is risk-free.

This is why this kind of stuff is called “statistical arbitrage.” You’ve statistically eliminated risk from factor exposures, but there is still non-factor risk. And in real life, one of the biggest challenges with this kind of stuff is that the factor exposures tend not to be stable over time, and also the expected returns are not nearly as cut and dry and obvious as they are in a textbook.

That’s different from something like a futures contract, where there actually is an arbitrage with zero-risk possible (assuming everyone performs on their contract as legally required).

It’s also a bit hard to figure out what the percentage return is on something like this, because there really is no natural denominator to use. In practice, the broker constraints link the size of the equity capital available to the size of the long and short positions you can use, so the denominator in a return calculation is usually the amount of margin consumed by the position.