It is argued that in the commonly used 60/40 portfolio (60% stocks, 40% bonds), since stocks...

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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; is defined as dop/aw; and its risk contribution as wido p/awi. A portfolio with equal risk contribution from each asset is called an Equal Risk Parity (ERP) portfolio. (a) Using do/Ow = 2Cw and Chain Rule, show do p Cw Əw op Op w = дор Əw which shows the decomposition of portfolio risk as the weighted av- erage of the marginal risk contribution of each asset. (b) For a 2 asset portfolio P, Rp = w₁R₁+w₂R₂, with Ri~ (₁,0₁) and correlation p = p(R₁, R₂), solve for the positive weights w₁, W₂ of the ERP portfolio. (c) Using RBonds~ (3%, 4%), RStocks (8%, 20%), and p = 10%, com- pute the risk and return of the ERP and 60/40 portfolios. ~ 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; is defined as dop/aw; and its risk contribution as wido p/awi. A portfolio with equal risk contribution from each asset is called an Equal Risk Parity (ERP) portfolio. (a) Using do/Ow = 2Cw and Chain Rule, show do p Cw Əw op Op w = дор Əw which shows the decomposition of portfolio risk as the weighted av- erage of the marginal risk contribution of each asset. (b) For a 2 asset portfolio P, Rp = w₁R₁+w₂R₂, with Ri~ (₁,0₁) and correlation p = p(R₁, R₂), solve for the positive weights w₁, W₂ of the ERP portfolio. (c) Using RBonds~ (3%, 4%), RStocks (8%, 20%), and p = 10%, com- pute the risk and return of the ERP and 60/40 portfolios. ~