## 14.5 Risk Budgeting for Optimized Portfolios

To be completed

• For global minimum variance portfolio, FOCs show that $$MCR_{i}=MCR_{j}$$. Also plugging in solution for $$m$$ shows that vector of MCR is the same.
• For an efficient portfolio with target return equal to the mean return on an asset in the portfolio, the portfolio beta is 1: $$\beta_{i,p}=1$$ when $$\mu_{p}=\mu_{i}$$. Hence, $$CR_{i}=x_{i}\sigma_{p}$$
• For the tangency portfolio, FOCs (from re-written equivalent problem) show that $\frac{\mu_{T}-r_{f}}{\sigma_{T}^{2}}=\frac{\mu_{i}-r_{f}}{\sigma_{i,T}}$

Volatility risk budgets for optimized portfolios have some interesting properties. For example, consider the global minimum variance portfolio allowing for short sales. In chapter 12 we showed that $\mathbf{m}=\frac{\Sigma^{-1}\mathbf{1}}{\mathbf{1}^{\prime}\Sigma^{-1}\mathbf{1}}.$ Plugging this portfolio weight vector into (14.7) gives $\frac{\partial\sigma_{p}(\mathbf{m})}{\partial\mathbf{m}}=\frac{\mathbf{\Sigma m}}{\sigma_{p}(\mathbf{m})}=\frac{\Sigma\Sigma^{-1}\mathbf{1}}{\sigma_{p}(\mathbf{m})\times\mathbf{1}^{\prime}\Sigma^{-1}\mathbf{1}}=\left(\frac{1}{\sigma_{p}(\mathbf{m})\times\mathbf{1}^{\prime}\Sigma^{-1}\mathbf{1}}\right)\times\mathbf{1}=c\times\mathbf{1}$ which is a constant $$c$$ times the one vector where $c=\frac{1}{\sigma_{p}(\mathbf{m})\times\mathbf{1}^{\prime}\Sigma^{-1}\mathbf{1}}.$ As a result, for each asset $$i=1,\ldots,n$$ we have $\mathrm{MCR}_{i}^{\sigma}=c.$ That is, each asset in the global minimum variance portfolio has the same marginal contribution to portfolio volatility. This has to be the case, otherwise we could rebalance the portfolio and lower the volatility. To see this, suppose $$\mathrm{MCR}_{i}^{\sigma}>c$$ for some asset $$i$$. Then we could lower portfolio volatility by reducing the allocation to asset $$i$$ and increasing the allocation to any other asset in the portfolio. Similarly, if $$\mathrm{MCR}_{i}^{\sigma}<c$$ we could lower portfolio volatility by increasing the allocation to asset $$i$$ and decreasing the allocation to any other asset in the portfolio.

Example 2.17 (Volatility risk budgeting for the global minimum variance portfolio)

to be completed

$$\blacksquare$$