What’s the big deal about Risk Parity?

Abstract

It is often argued in defense of Risk Parity portfolios that they maximize the Sharpe ratio if their securities have identical Sharpe ratios and identical correlations. However, securities have neither identical Sharpe ratios nor this correlation structure. In realistic markets, Risk Parity portfolios do not maximize the Sharpe ratio, do not minimize variance, do not maximize the Information ratio, and do not have any other commonly sought optimal property. So, what’s the big deal about Risk Parity?

Appendix

A portfolio’s standard deviation of return is:

$$\sigma_{\pi } = \sqrt {\underline{\pi }^{\prime } \underline{\underline{\sigma }} \underline{\pi } }$$

(7)

$\sigma_{\pi } \equiv$ Portfolio $\pi$’s standard deviation of return. $\underline{\pi }$ ≡ A column vector containing portfolio $\pi$’s weights. $\underline{\underline{\sigma }} \equiv$ The securities’ covariance matrix.

The vector derivative of the portfolio’s standard deviation with respect to its weights is:

$$\underline{{\frac{{d\sigma_{\pi } }}{{d\underline{\pi } }}}} = \frac{1}{{2\sqrt {\underline{\pi }^{\prime } \underline{\underline{\sigma }} \underline{\pi } } }}2\underline{\underline{\sigma }} \underline{\pi } = \frac{1}{{\sigma_{\pi } }}\underline{\underline{\sigma }} \underline{\pi }$$

(8)

The i’th element of $\frac{{d\sigma_{\pi } }}{{d\underline{\pi } }}$ is the partial derivative of the portfolio’s standard deviation with respect to security i.

$$\underline{{\left( {\frac{{d\sigma_{\pi } }}{{d\underline{\pi } }}} \right)}}_{i} = \frac{{\partial \sigma_{\pi } }}{{\partial \pi_{i} }} = \frac{1}{{\sigma_{\pi } }}\sum\limits_{j} {\sigma_{ij} \pi_{j} } = \frac{{\sigma_{i\pi } }}{{\sigma_{\pi } }}$$

(9)

Equation (9) implies that, for a Risk Parity portfolio, the weighted sum of the partial derivatives of its standard deviation of return is its standard deviation of return. This property stems from the fact that a portfolio’s standard deviation is a homogeneous function of degree 1.

$$\sum\limits_{i} {\pi_{i} \frac{{\partial \sigma_{\pi } }}{{\partial \pi_{i} }}} = \frac{1}{{\sigma_{\pi } }}\sum\limits_{i} {\left( {\pi_{i} \sum\limits_{j} {\sigma_{ij} \pi_{j} } } \right)} = \frac{1}{{\sigma_{\pi } }}\sum\limits_{ij} {\pi_{i} \sigma_{ij} \pi_{j} } = \sigma_{\pi }$$

(10)

A Risk Parity portfolio is defined as a portfolio where the product of a security’s weight,$\pi_{i}$, and its marginal contribution to the portfolio’s standard deviation, $\frac{{\partial \sigma_{\pi } }}{{\partial \pi_{i} }}$, is the same for all of the portfolio’s securities.

$$\pi_{i} \frac{{\partial \sigma_{\pi } }}{{\partial \pi_{i} }} = \pi_{k} \frac{{\partial \sigma_{\pi } }}{{\partial \pi_{k} }}$$

(11)

From Eqs. (9) and (11),

$$\pi_{i} \frac{{\sigma_{i\pi } }}{{\sigma_{\pi } }} = \pi_{k} \frac{{\sigma_{k\pi } }}{{\sigma_{\pi } }}$$

(12)

Equation (12) can be rewritten as:

$$\pi_{i} \sigma_{i\pi } = \pi_{k} \sigma_{k\pi }$$

(13)

Equation (13) can be used as a test to see if a portfolio is a Risk Parity portfolio.

Security i’s beta, $\beta_{i}$, is defined with respect to a reference portfolio. Choose the Risk Parity portfolio to be the reference portfolio.

$$\beta_{i} = \frac{{\sigma_{i\pi } }}{{\sigma_{\pi }^{2} }}$$

(14)

$$\frac{{\sigma_{i\pi } }}{{\sigma_{\pi } }} = \beta_{i} \sigma_{\pi }$$

(15)

Substitute from Eq. (15) into Eq. (12).

$$\pi_{i} \beta_{i} \sigma_{\pi } = \pi_{k} \beta_{k} \sigma_{\pi }$$

(16)

$$\pi_{i} \beta_{i} = \pi_{k} \beta_{k} = k$$

(17)

In view of Eq. (17), it is tempting to say that each security’s contribution to the portfolio’s beta is the same when the Risk Parity portfolio is the reference portfolio for computing the securities’ betas. Since $\pi_{i} \beta_{i}$ depends on all the securities’ weights and characteristics, this characterization is incorrect.

The Risk Parity portfolio’s beta is necessarily 1 if it is the reference portfolio for computing the securities’ betas.

$$\sum\limits_{i} {\pi_{i} \beta_{i} } = nk = 1$$

(18)

$$k = \frac{1}{n}$$

(19)

Therefore,

$$\pi_{i} = \left( {\frac{1}{n}} \right)\left( {\frac{1}{{\beta_{i} }}} \right) = \left( {\frac{{\sigma_{\pi }^{2} }}{n}} \right)\left( {\frac{1}{{\sigma_{i\pi } }}} \right)$$

(20)

A Risk Parity portfolio’s weights are inversely proportional to its securities’ betas (measured with respect to the Risk Parity portfolio) or, equivalently, inversely proportional to its securities’ covariances with the Risk Parity portfolio.

If the securities have identical correlations, then Maillard et al (2010) show that the Risk Parity portfolio’s weights are inversely proportional to the securities’ standard deviations.

$$\pi_{i} = \frac{{\frac{1}{{\sigma_{i} }}}}{{\sum\limits_{j} {\frac{1}{{\sigma_{j} }}} }}$$

(21)

Suppose the securities have the same Sharpe ratios.Equal Sharpe ratios implies that:

$$\frac{{R_{i} }}{{\sigma_{i} }} = \frac{{R_{j} }}{{\sigma_{j} }} = S$$

(22)

$$\underline{R} = \underline{\sigma } S$$

(23)

The maximum Sharpe ratio portfolio’s weights become:

$$\underline{\pi } = \frac{{\underline{\underline{\sigma }}^{ - 1} \underline{R} }}{{\underline{1}^{\prime } \underline{\underline{\sigma }}^{ - 1} \underline{R} }} = \frac{{\underline{\underline{\sigma }}^{ - 1} \underline{\sigma } }}{{\underline{1}^{\prime } \underline{\underline{\sigma }}^{ - 1} \underline{\sigma } }}$$

(24)

This implies that:

$$\underline{\underline{\sigma }} \underline{\pi } = \left( {\frac{1}{{\underline{1}^{\prime } \underline{\underline{\sigma }}^{ - 1} \underline{\sigma } }}} \right)\underline{\sigma } = k\underline{\sigma }$$

(25)

$$\sigma_{i\pi } = k\sigma_{i}$$

(26)

$$\pi_{i} \sigma_{i\pi } = k\pi_{i} \sigma_{i}$$

(27)

With suitable constraints on the securities’ correlations, Eq. (27) meets the criterion of Eq. (13) for a Risk Parity portfolio. For example, Maillard et al (2010) show that this is so for securities with identical correlations. However, the fact that the Risk Parity portfolio is the Sharpe ratio maximizing portfolio in this case, or in other cases with unrealistic correlation assumptions is of no interest.

Suppose the securities’ expected returns are the same.

Equal expected returns implies:

$$R_{i} = R_{j} = R$$

(28)

The maximum Sharpe ratio weights then are the same as the minimum variance portfolio’s weights.

$$\underline{\pi } = \frac{{\underline{\underline{\sigma }}^{ - 1} \underline{R} }}{{\underline{1}^{\prime } \underline{\underline{\sigma }}^{ - 1} \underline{R} }} = \frac{{\underline{\underline{\sigma }}^{ - 1} \underline{1} }}{{\underline{1}^{\prime } \underline{\underline{\sigma }}^{ - 1} \underline{1} }}\frac{R}{R} = \frac{{\underline{\underline{\sigma }}^{ - 1} \underline{1} }}{{\underline{1}^{\prime } \underline{\underline{\sigma }}^{ - 1} \underline{1} }}$$

(29)

These are not generally the Risk Parity portfolio’s weights. For example, zero correlations, a special case of identical correlations, imply that:

$$\pi_{i} = \frac{{\frac{1}{{\sigma_{i}^{2} }}}}{{\sum\limits_{j} {\frac{1}{{\sigma_{j}^{2} }}} }}$$

(30)

The portfolio’s weights are inversely proportional to the securities’ variances of return. These are not the Risk Parity portfolio’s weights.