Synthetic Portfolios: Insights on the CAPM

Synthetic Portfolios: Insights on the CAPM

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I would like to share some interesting things I have discovered about the CAPM. I hope this will be useful to some of you.

The CAPM equation arises from creating a synthetic market portfolio.

Suppose you are considering a project, and you know its expected rate of return. One way to decide if it is a good project is to see if you could get a better return for your risk by investing in the market portfolio.

Now, if you put all your money in the market portfolio you would not earn the same rate of return as the project (you would earn r(m))nor would you experience the same risk (diversied portfolios are less risky). To offset this, we also put some of our money into the bank to offset risk (or borrow money to create more risk).

Our portfolio looks like this:

r(P) = r = (1 - a) * r(f) + a * r(m),

where a is a constant, a mixing weight. Solving for a we find:

a = (r(P) - r(f)) / (r(m) - r(f)),

which should look familiar: it is the definition of β for the portfolio.

What's more, if we find Cov(P) / Variance(M), our other definition of β, we find that it also matches exactly, which suggests the following.

β is the mixing weight for the Market in a synthetic portfolio.

Remember, we created this synthetic portfolio in order to simulate the project we were considering. The portfolio was designed so that r(P) = r, which implies

β(P) = [r - r(f)] / [r(m) - r(f)] = [r - r(f)] / (r(m) - r(f) = β.

The β of the project = β of the simulated portfolio = β (the mixing weight).

Or at least they should be equal. If they are not, you might have an arbitrage opportunity: if the r of your project is greater than the r given in the CAPM, r(f) + β*MRP, then you could short the synthetic portfolio to finance your project for minimal risk.

If it is less, on the other hand, then your money would be better invested in the synthetic portfolio, achieving a greater return for the same risk. You probably can't short your project, so there's no arbitrage in this case but it does tell you that your project sucks and you shouldn't invest in it.

For a project with a given β, its expected rate of return must be at least r(f) + β*MRP to be viable.


A project with a given r must have a β of at most [r - r(f)]/MRP.

This is perhaps the reason why we use the risk-adjusted rate of return. Hope this was helpful!

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