It is argued that in the commonly used 60 / 40 60 / 40 portfolio ( 60

Question:

It is argued that in the commonly used $60 / 40$ portfolio $(60 %$ stocks, $40 %$ bonds), since stocks are much riskier than bonds, the stocks' relative contribution to the portfolio risk is too high. A remedy is to construct portfolios with equal risk contribution.

For a portfolio $P$ , the marginal contribution of risk due to $i$ th asset with allocation $w_{i}$ is defined as $\partial σ_{P} / \partial w_{i}$ and its risk contribution as $w_{i} \partial σ_{P} / \partial w_{i}$ . A portfolio with equal risk contribution from each asset is called an equal risk parity (ERP) portfolio.

(a) Using $\partial σ_{P}^{2} / \partial w = 2 C w$ and the chain rule, show

$\begin{aligned} \frac{\partial σ_{P}}{\partial w} & = \frac{C w}{σ_{P}} \\ \Rightarrow σ_{P} & = w^{T} \frac{\partial σ_{P}}{\partial w} \end{aligned}$

which shows the decomposition of portfolio risk as the weighted average of the marginal risk contribution of each asset.

(b) For a two-asset portfolio $P, R_{P} = w_{1} R_{1} + w_{2} R_{2}$ , with $R_{i} \sim (μ_{i}, σ_{i})$ and correlation $ρ = ρ (R_{1}, R_{2})$ , solve for the positive weights $w_{1}, w_{2}$ of the ERP portfolio.

(c) Using $R_{Bonds} \sim (3 %, 4 %), R_{Stocks} \sim (8 %, 20 %)$ , and $ρ = 10 %$ , compute the risk and return of the ERP and 60/40 portfolios.