Assumptions for Multifactor, Single Index and APT

[Gareth]

The notes don't really say what assumptions the multifactor, single index or APT are.

Am I correct that this is just the MPT assumptions:

- All expected returns, variances and covariances of pairs of assets are known
- Investors make their decisions purely on the basis of expected return and variance
- Investors are risk-averse
- There is a fixed single step time period
- There are no taxes or transcation costs
- Assets may be held in any amount (short selling, infinitely divisible holdings)

Or are there other assumptions that then are required? I can think of one extra for APT: no arbitrage.

Cheers and good luck tomorrow.

 

[ActuaryStudentUK]

APT requires no arbitrage - that's all it says - given that it referred to CAPM just before that in the notes as having 5 additional assumptions over those of MPT you'd think that those applied to APT as well but I don't think they do (but then Im wrong there - see below).

APT is a form of multifactor model so whatever assumptions hold for multifactor models must be the ones to apply.

Thinking about multifactor and single-index models though and looking at the MPT assumptions:
-all expected returns, variances and covariances are known.
This one has got to be true so that you can estimate returns
-all investors base their decisions purely on statistics
Not necessarily true - these models make no effort at determining a portfolio for investors, just giving a value of return on a portfolio or asset. APT also simply provides returns on a portfolio or asset.
You could argue though that if this weren't true, the market wouldn't be efficient and would change our parameters so in that sense it might be required
- investors are non-satiated and risk-averse
Same argument as the last point
-there is a fixed single-step time period
tricky one - I would guess this would be needed since the models are of the same form and if CAPM relies on that then so should the other models
-there are no taxes or transaction costs
This would have to hold or we'd have to exclude them from our returns
-assets may be held in any amount
Again, this one should be needed. Each portfolio is made up of a number of assets, so if we're modelling the returns, I'm guessing it's a continuous parameter indicating the holdings in each asset forming the portfolio.

In that sense, I'd say all of them hold for multifactor and single-index models.

 

[Erik]

This is how I see it.

Assumptions of Single index model:
Returns are correlated only to the return on the market and no other factors, generally.
But if asked what the assumptions underlying SIM is I will write:
E(Ei) =0 for all i
Cov(Ei,Ej) =0 for all i not equal to j
Cov(Ei,Rm) = 0 for all i

(Ei is my error or residual term for the return)

(NB The assumtions of MPT does not hold necessarily)

Assumptions of Multifactor modelReturns are correlated with a set of L indeces.
E(Ei) =0 for all i
Cov(Ei,Ej) =0 for all i not equal to j
Cov(Ei,Ik) =0 for all i and all indeces k

Assumtions of APTThe market is arbitrage free.
We are assuming an appropriate multi index model holds
All investors have the same expectations of returns, variances and covariances.
(Homogenous expectations

Assumptions of CAPM (just for fun)
All assumtions of MPT
Plus 5 extra assumptions that enables us to go from an individual perspective to a market model.

Good luck and hope all goes well
Erik.

 

[ActuaryStudentUK]

I agree with Erik.
Ummm and the MPT assumptions on top of that

 

[Erik]

PS. The whole reasoning behind the existence of the APT is to create a model with less assumptions, since the CAPM is so restrictive because of its assumptions.

The arguments does make sense, but when given a question: I follow these steps:
1. Try to determine which part of the course you are working with (not always as easy as
it sounds)
2. When it seems like a bookwork questions, give bookwork from the core reading if you
can remember any.
3. If you don't remember bookwork, try an argument like the one you did.

The idea is that they can't mark something wrong if its from the core reading and it answers their question. But when they are looking for something from the core reading and you state a general argument, you might not score full marks.

Cheers
Erik

 

[Gareth]

im still counting on the april fools memory stick approach

 

[Erik]

Gareth, are you ever serious.

Just a "thank you" would have been appreciated.
Here I am trying to give you old guys a few tips, while I really should be attempting some Wilkie questions. (I am rubbish in that part)

 

[Gareth]

I was not being disrespectful erik, and apologise if you felt offended.

The comments you made are very helpful indeed.

Also I noticed that that Cov(c_i, I_k) = 0 for i,k seems to be turned into E[c_i(I_k - E(I_k)] = 0 for all i,k under the APT, but I _think_ this is equivalent.

 

[ActuaryStudentUK]

I think that follows from Cov(X,Y) = E[XY] - E[X]E[Y]

 

[Erik]

Correct! They are exactly equivalent!