Sharpe and Jensen Foundations of Risk Management

[Brian_no9]

Hi David, Suzanne,

If you have time, could you please explain to me why Amenc states that if a portfolio is well diversified, the correlation coefficient Ppm is very close to 1? Intuitively, I would have thought that it would approach zero, as the portfolio became more diversified.
Appologies if this is a stupid question, i am just starting my study for May, and don't have a quantative background.

Many thanks for any help you can provide.
Brian

 

[David Harper CFA FRM]

Hi Brian,

Not a stupid question at all, that is an interesting assumption. I would refer to the CAPM/market index model (these ratios are based on the same mean variance framework) that says:
expected[portfolio return] = alpha + beta*[return(market) - riskfree rate] + residual, or
E[R(p)] = Rf + beta*[R(m) - R(f)] + e(p)

I *think* the definition of "well diversified" is that e(p), the idiosyncratic or non-systematic risk [actually the variance of the above, but it's not important here], is reduced to zero due to diversification. This is the free lunch of MPT. By diversifying, you tend the non-systematic risk to zero such that only the systematic risk, beta, remains. Without this assumption (itself the result of assumptions), CAPM could not assert that the expected return is only a function of systematic risk (beta). And the Jensen (not really the sharpe) assumes the same things as CAPM.

So, if the correlation between the portfolio and the market, rho (p,m) were zero, then because beta = cov(p,m)/variance(m) = rho(p,m)*volatility(p)/volatility(m), a correlation of zero would imply beta = 0 and:
E[R(p) | rho = 0] = Rf + 0 = risk-free rate; e.g., like cash would have a correlation of zero, and we'd expect it's return to equal the risk-free rate

But a diversified portfolio, containing all of the risk of the market, should have E[R(p)] ~= E[R(m)] and in order to get E[R(p)] = R(m), we need beta = 1.0 such that:
E[R(p) | beta = 1] = Rf + 1.0*[R(m) - R(f)] = R(m)

please note that correlation, rho(), is different than beta(): it's an input into beta. Further, they are just relationships between pairs, so we can also have correlation[R(p), e(p)], which is assumed to be zero. I hope that helps,

 

[Brian_no9]

Hi David,

Thanks a lot for your help in this matter. I really appreciate it.

Best regards,
Brian.