[R-SIG-Finance] Generating Data for Portfolio Simulation

[Rowe, Brian Lee Yung (Portfolio Analytics)]

Hi Tom,

>From your construction of the problem, you do have a circular
dependency. Your simple factor model is basically CAPM, and if you think
about it from that perspective then you should generate the market
returns independently of your factor model, since you are trying to
explain your asset returns based on the market.

My 2c,
Brian


-----Original Message-----
From: r-sig-finance-bounces at stat.math.ethz.ch
[mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of Tom Smythe
Sent: Tuesday, February 17, 2009 2:24 PM
To: r-sig-finance at stat.math.ethz.ch
Subject: [R-SIG-Finance] Generating Data for Portfolio Simulation


Hi All,

I am looking for a way to generate generate M snapshots of return data
for all securities in an N security universe.  The return for the n-th
security at the m-th snapshot should come from a simple factor model
such that
r[n,m] = alpha[n] + beta[n] * rm[m] + epsilon[n,m],
where rm[m] is the market return, and epsilon[n,m] is noise with zero
mean and given covariance (more on this below).

The issue that I am facing is that I want the market return to be the
weighted sum of the individual security returns.  That is
rm[m] = sum_n b[n] r[n,m]
where b[n] is the ratio of the n-th security's market capitalization
to the total market capitalization, and rm[m] has a given (time
series) mean and variance.

Some other constraints of the problem are that the weighted alpha is
zero
sum_n b[n] alpha[n] = 0
and the weighted beta is one
sum_n b[n] beta[n] = 1.
It would be nice if the noise term epsilon[n,m] had zero column (i.e.,
cross-sectional) mean and zero row (i.e., time series) mean; but both
of these requirements cannot be met simultaneously because they force
the covariance matrix to be singular.  For simplicity, I'd like to
assume that the covariance is non-singular and diagonal, with known
constant diagonal elements.

This appears to be a circular problem.  I seem to be unable to specify
r[n,m] without rm[m], and vice versa.

Can anybody advise me how to generate such factor-model data that is
compatible with the given market return statistics and given noise
covariance matrix?  Unless I'm missing something silly, this seems to
be an under-determined problem (if M >> N), with no unique solution.
However, in my case, any non-trivial but internally consistent
solution will do.

Thanks.

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